## The Math Behind LEGO Building Techniques – Volume 3

Introduction

On August 1, 2023 LEGO released a new set (76262) that creates a reasonably accurate replica of Captain America’s Shield from the MCU (Marvel Cinematic Universe) movies.

The initial reactions to this set were less than positive with some people comparing it unfavorably to the bigger (and more expensive) custom version sold by Bricker Builds.

That particular version is built as a sculpture created by stacking regular plates, which makes it far more sturdy. The official set on the other hand is smaller and seems more flimsy. But there are many reasons to appreciate the new set, not the least of which is the clever way that it uses wedge plates to create the round shape.

This is the second recent set (after the Ideas Globe set 21332 – released on February 1, 2022) that creates a round shape by attaching wedge plates at different angles over an inner frame.

There are indeed some obvious gaps between the wedge plates, but together these plates still do a great job of approximating the curved shape. While this is a relatively new technique as far as official LEGO sets are concerned, AFOLs have been using it for quite a while. The Globe set as you know, is an Ideas set that is based on a design by French AFOL Guillaume Roussel (@disneybrick). Another AFOL Pete Strege (@redcokid) who is a master at this technique, has used it to create an impressive array of MOCs (various buildings with round-shaped domes, a hot air balloon and even a flying pig !).

Clearly, there is some interesting math at play here. I am a bit of a novice with this technique myself, but I couldn’t resist the challenge of trying to reverse engineer the two official sets (the Globe and Captain America’s Shield) to try to uncover some of this math. And so, in this third installment of my series on ‘The Math Behind LEGO Techniques’ (the two previous installments are here and here), we will be delving into the math behind the Globe and Captain America’s Shield. We will try to understand conceptually how the outer shapes of the globe and the shield are put together (not necessarily following the instructions for the actual sets). These sets also have quite a bit of internal reinforcement created using Technic elements, and we will not be getting into all the details of that for this article.

Before we start diving in, we need to get a few prerequisites out of the way – namely polygon geometry, an overview of the Technic pieces used to create the inner frame in the Globe set and of course the wedge plates that are used in conjunction with regular plates to create the outer “skin” in both sets.

Polygon geometry

In this earlier post we have seen how LEGO hinge plates can be used to create angled walls. If we extend this concept further, we should also be able to build a continuous chain of angled wall segments that form a closed shape – namely a polygon. Limiting our discussion to regular polygons (polygons where all the sides and angles are equal), how do we determine what the length of each side should be and ensure that at least one pair of opposite sides of the polygon line up with the LEGO grid (so that we have a way of attaching the polygonal wall to a baseplate) ?

For this, we will have to first review the geometry of polygons and familiarize ourselves with a few basic terms. A regular polygon has n (where n is 3 or more) sides of equal length. The points where any 2 adjacent sides intersect is called a vertex and the angle they make at the vertex interior to the polygon is called the interior angle. In a regular polygon all the interior angles are also equal. Shown below is a hexagon with 6 equal sides that we will be using for illustration purposes.

All regular polygons have a center that is equidistant from each of the vertices and this distance is called the radius. If we draw lines from the center to each of the vertices, we would be dividing the polygon into n identical triangles. The angle made by the sides of the triangle that intersect at the center is called the central angle.

If you draw a circle with the same center and radius as the polygon, it would pass through each of the vertices of the polygon. This circle is called the circumcircle. It is also possible to draw a smaller circle with the same center that passes through the center point of each of the sides. This circle is called the incircle.

The radius of the incircle is equal to the length of lines drawn from the center that meet each of the sides at their center points. These lines are perpendicular to the sides and their length is called the apothem.

If we create a regular polygon using LEGO pieces, we need to ensure that the distance between any pair of opposite sides (which would be twice the apothem) is close enough to a whole number of studs. This will allow us to attach the polygonal wall firmly to a baseplate.

Let us figure out how the apothem can be calculated if we know n (the number of sides) and the length of each side in studs. First, we will start with the central angle. If the polygon has n sides, the central angle would be 360/n. Since the sum of the angles in each of the n triangles would add up to 180Â°, and the two other angles in the triangle are essentially identical halves of the interior angle, we can calculate the interior angle as 180 – 360/n which sometimes written as ((n-2)*180)/n.

If we split one of the n triangles into two mirrored right angled triangles, the tangent of half the central angle would be equal to half the length of each side (the opposite side) divided by the apothem (the adjacent side).

tan(360/2n) = s/2a

which can be simplified to

tan(180/n) = s/2a

So the formula for calculating the apothem ends up being

a = s/(2*tan(180/n))

You would need a scientific calculator for this but you can also use one of the online calculators available to compute the apothem of your polygon. All you have to enter is the side length (in terms of studs) and the number of sides

One thing to note is that when the polygon has a large enough number of sides, the polygon starts to approach a circle and the apothem (a) starts to approach the radius of the circle (r).

The equation for the apothem can be written as

a = s*n/2*(n*tan(180/n))

It is beyond the scope of this article to explain why, but n*tan(180/n) starts to approach Ï€ as n gets larger. We can plug a few different values for n and confirm that the larger n is, the closer n*tan(180/n) gets to 3.14159… (the value of Ï€). For a large enough value of n, we have

a = s*n/2Ï€

s*n is of course the sum of the lengths of the n sides which would approach the circumference of the circle (c) and that gets us to the familiar equation for the circumference of a circle

c = 2Ï€r

And so, if the number of sides of our polygon is large enough (say 30 or more), we can get away with using this simpler equation to calculate the radius or diameter of our shape (as we have done in this article on round shapes).

LEGO Technic

LEGO Technic is a line of building sets that was introduced in 1977. Technic sets are intended for more advanced builders that want to take their LEGO models to the next level by adding moving parts with realistic mechanisms. Technic sets include more than just regular bricks, plates, etc. They also include specialized pieces such as beams, gears, axles and connectors. It is even possible in some cases to add small motors to mechanize the model.

In this article we will only be looking at the Technic elements that are used in the two official sets we will be covering (the Globe and Captain America’s Shield). These barely scratch the surface of the Technic catalog which has hundreds of different types of pieces.

Technic Axles

LEGO cross-axles – commonly referred to as axles are rods with a cross-shaped cross-section. They come in different lengths which are multiples of a stud and are named based on their length in studs (2L, 3L and so on). LEGO typically color codes their axles to make them easier to identify. The even-numbered lengths are in black or red, while the odd-numbered ones are in light bluish grey or yellow. Axles can be handy for connecting together different Technic elements including gears, wheels, etc.

Technic Axle and Pin Connectors

As the name implies, these pieces can be used to make various connections either in a straight line or at an angle. They also have a hole that accepts a Technic pin. For this article, we will only focus on a particular family of angled axle and pin connectors which is made up of 6 types of pieces that are identified with numbers 1 through 6 (the pieces themselves are marked with these numbers). The angles that these pieces create range from 90Â° to 180Â°.

Here is a table showing all the pieces and the angle that each one creates. At first glance, the angles 112.5Â°, 135Â° and 157.5Â° may seem quite random but they really are not. In fact, if you divide the range between 90Â° and 180Â° into 4 equal parts, you would have angles separated by increments of 90/4 = 22.5Â° and the angles 112.5Â°, 135Â° and 157.5Â° start to make a lot more sense.

Now let us look at the dimensions of these connectors. The #2 connector (180Â° angle) is exactly 3 studs from end to end. When we use it to connect two axles together, the axles can go into the connector by as much as one stud on each side.

This leaves exactly one stud in between for the portion that has the Technic hole. The distance from either end of the connector to the center of the Technic hole is 1.5 studs. The same is true for the angled connectors as well, and so if you wanted to have a distance of 6 studs between the holes on two connectors connected to either side of an axle, this axle would have to be 5 studs long.

What practical uses can the different angles that these connectors create, have in LEGO builds ? Consider for instance the interior angle of a regular octagon. It is 180 – 360/8 = 180 – 45 = 135Â°. And so, we can use axles and #4 connectors to create an octagon. Using the apothem calculator we see that the apothem of the octagon is close enough to a whole number of studs (6) when each side is 5 studs long. This means that we have to use 4L axles for the sides of the polygon.

Why does the apothem of the polygon have to be a whole number of studs even when we are using Technic pieces that do not get attached to a baseplate ? The reason is that the frames constructed using Technic elements typically need internal structural reinforcement and it is a lot easier to add this when the distance between the opposite sides of the polygon is a whole number of studs.

Let us next look at a regular polygon with 16 sides which is called a hexadecagon. The interior angle here is 180 – 360/16 = 180 – 22.5 = 157.5Â°. This means that we can use 16 of the #3 connectors to create the regular polygon with 16 sides. Each side can be 4 or 6 studs long for an apothem that is close enough to a whole number (10 or 15 respectively). The latter case is shown in the picture below with 5L Technic axles used for each of the 16 sides.

Connector #5 cannot be used to create a regular polygon. However it can be combined with the other types of connectors to create many different types of polygons. In fact, you can use the following numbering (which differs from the official LEGO numbering) to figure out the different ways in which the angled connectors can be used (the rule is for the numbers to add to a total of 16 to create a closed polygon).

We have already seen how we can use 8 of the #4 connectors (22222222) to create an octagon or 16 of the #3 connectors (1111111111111111) to create a hexadecagon. Using 4 of the #6 connectors (4444) to create a square or a rectangle is also a no-brainer. Some possible combinations that use the #5 connector are 333331, 33334 and 3322222. Shown below is the polygon created using 33334 (#5 and #6 connectors). Note that this is the smallest pentagon with one 90Â° angle and four 112.5Â° angles with side lengths that are each close enough to a whole number of studs (17, 17, 17, 17 and 11).

Technic Bush Pieces

A bush is a piece with a cross-shaped hole that can fit tightly over an axle. It comes in lengths of half a stud and one stud (named 0.5L and 1L respectively). It can be used to space pieces apart precisely in increments of half a stud or to hold pieces on the axle in place by acting as an end stop.

The LEGO catalog also includes variants of some of the axles (3L, 4L, 5L, 8L) that have a stop built in. These are typically available in colors like reddish brown and dark bluish grey that are different from the regular axles.

Technic bricks and plates with axle holes

It is clear that we can connect axles together using the connector pieces to create the internal framework for all kinds of shapes, but there is no way to create the outer skin for these shapes without having studs available to which we can attach LEGO plates. One way to create these studs is by using Technic bricks that have axle holes.

We can pass the axles that make up the frame through these bricks (positioning them correctly using bush pieces if needed) and this gives us a way to then attach LEGO plates to create the outer surface. However, when we are talking about a Technic framework that approximates a curved shape (like a globe), regular rectangular plates alone may not always work. This is where wedge plates can come into play.

Wedge plates

These are variants of regular LEGO plates except they have a portion that is truncated at an angle. There is a large number of wedge plates in the LEGO catalog with sizes ranging from 2×2 to 12×3. Each type of wedge plate (except the ones that create a 45Â° angle) comes in a left and a right variant (which is simply a mirror image of the left variant). Shown below is a sampling of the various wedge plates that are available.

The angle that the slanted edge of each wedge plate creates is easy to calculate using basic trigonometry (the truncated portion is a right triangle and so the tangent of the angle is equal to the opposite side divided by the adjacent side).

Since we know the lengths of the opposite and adjacent sides in terms of studs, we can find the inverse tangent (arctan) and put together an entire table with the angles created by each type of wedge plate.

Now let us see how all these different types of LEGO elements can be put together to create the curved shapes of the Globe and Captain America’s Shield.

The Globe

If you look at the outer shape of the Globe, you will see that the section in the middle (where the equator would be on a real globe) is made up of regular 6×6 plates and there are 16 of them used to approximate the round shape. If this looks familiar, that is because we have already seen a hexadecagon with a side length of 6 studs that can be created using 16 #3 (157.5Â°) connectors.

We need a 5L axle piece for each side of this hexadecagon and if we pass it through bricks with axle holes (see the instructions for the details) such that the studs on these bricks are facing outwards, we can attach the 6×6 plates needed to form the outer surface. Note that Technic 1×3 liftarms are also used in each of the 16 panels and these are connected together using 4L axles with stops for additional reinforcement.

You will notice that in the images that follow, the plates on the outer surface have a sand green color for illustration purposes. The actual globe set uses dark blue, which does not stand out as well in the renders (making it harder to see what is going on).

Now, the globe is spherical and so we need to create this same round shape in the other two dimensions as well. The first thought that comes to mind is – can we also create hexadecagonal rings in the vertical directions using the same #3 connectors ? Yes, definitely. The way to attach these vertical rings to the horizontal one is by passing the horizontal 5L axles through the hole in a #2 (180Â°) connector placed vertically and then attach it to #3 connectors placed on the top and bottom (using 2L axles).

This creates the same 6 stud spacing between the holes in the #3 connectors vertically that we get horizontally by using the 5L axle. So can we just create an intersecting set of rings in the 3 dimensions and attach 6×6 plates to them ?

That sounds reasonable but it doesn’t work very well in practice – especially since there is no good way to close the holes that remain.

Instead of trying of using 6×6 plates throughout, we can instead create 16 wedge-shaped slices attached to the top and bottom of the central ring. These would taper as they reach the poles in each hemisphere. Of course, the best way to create these wedge-shaped slices is by using wedge plates.

But first we have to create an internal frame that we can attach these wedge-shaped slices to. We can use the same method as before to create 8 vertical rings that are attached to the 16 sides of our horizontal ring. Only this time, there is no way these rings can intersect at the poles and so we will start with incomplete rings that stop well short of the poles. We will need to figure out a different way to connect the rings together at the poles.

Let us do some quick back-of-the-envelope calculations to determine what kind of wedge plates we may need. We will again use plates that are 6 studs tall (based on the spacing between the Technic connectors in the vertical direction).

We can use the circle formula for these rough calculations even though the number of sides in our polygon is not very large. The circumference of the circle created by the existing row is 16×6 = 96 studs and its radius works out to 96/2Ï€ which is roughly 15 studs. If we add another row of plates above the existing row, the circumference of the circle at the top edge of this second row would be roughly 80 studs. We can figure this out using the Pythagorean theorem.

The height from the equator to this smaller circle is roughly 3+6 = 9 studs. The line from the center of the sphere to this smaller circle is equal to the radius of the sphere which is the hypotenuse of a right triangle. One of the sides is the height which is 9 studs and that gives us the radius of the smaller circle âˆš(152-92) = 12. The circumference of this smaller circle is 2*Ï€*12 = 75.36. Now, due to the curvature of the sphere the plates will not be perfectly vertical and the actual height will be less than 9 studs and the circumference of the smaller circle will be more than 75.36. And so we should be fine rounding 75.36 to the next higher multiple of 16 which is 80.

80 divided by 16 is 5 and this means that we need each slice to taper from 6 studs to 5 studs. With the wedge plates that are available, it is easier to get a circumference of 80 studs by alternating 6 and 4 studs. We will therefore need to alternate a slice using 6×6 plates with a slice that uses wedge plates to create a taper from 6 to 4 studs (a 4×6 plate with 6×2 wedge plates on either side would do the trick).

For the row above this one the circle at the top edge is even smaller working out to a circumference of roughly 48 studs. Instead of tapering each slice to 3 studs we can get the same circumference by alternating 4 and 2 studs. This means that we alternate a slice that has a 4×6 plate with one that tapers from 6 studs to 2 studs (which is achieved using two 6×3 wedge plates).

Due to the curvature of the sphere, the third row above this would need to taper to an even smaller circle with a diameter of just 6 studs and circumference of about 19 studs which is not a multiple of 16. The designers chose instead to use 10×10 dish pieces at the poles of the globe and this means the third row has to be shorter (just 4 studs tall). For this segment of the internal frame, a 4L axle piece is used instead of the 5L axle and there are clip pieces attached to the #3 connector closest to the pole. This allows us to attach all 16 slices to a round steering wheel piece placed at the poles. The dish piece with a diameter of 10 studs has a rough circumference of 32 studs and this can be approximated using a hexadecagon with 2 studs on one side. Here slices with 2×4 plates are alternated with slices that have two 2×4 wedge plates that together taper from 4 to 2 studs.

The steering wheel pieces have studs in the middle allowing us to top the globe off with the dish pieces at either end, that represent the north and south poles of the globe.

Captain America’s Shield

The design of the LEGO model of Captain America’s Shield is again based on a polygonal shape. This time it is a polygon with 36 sides and each side is 2 studs long. This polygon is built using hinge bricks and plates and forms the main core that the rest of the pieces are attached to, to create the round shape.

The central angle of the polygon is 360/36 = 10Â°. Looking at the table of angles created by wedge plates we see that the 6×2 wedge plates create an angle of arctan(1/6) = 9.46Â° which is close enough to 10Â°. We can confirm this by attaching 6×2 wedge plates to the polygon so they point inwards. As you can see the wedge plates all fit nicely with minimal gaps.

If we create panels that extend outwards from the polygon while preserving the same angle, we can use them to make the circle as big as we need to (as long as we have a way to support each panel). The shield set creates a panel with 4 sections that are blue, red, white and red (going from the center outwards). Each section has to use the 6×2 wedge plate and is therefore 6 studs tall. If we create the round shape using panels that flare in the same direction as shown below, we will end up with an outer edge that is a little jagged.

So the LEGO designers decided to mix things up a bit by creating an alternating pattern of two different types of panels. In each of these panels one of the two middle sections uses regular rectangular plates while the other uses two wedge plates to flare in both directions. This ensures that each panel doesn’t flare too much in any one direction and reduces the jaggedness in the outer edge.

The internal structure that these panels are attached to, is created by using regular 2×14 plates that are attached to the sides of the polygon. These plates are strapped together using 1×4 plates that are attached 11 studs away from the rotation point of the hinges that make up the polygon.

We can confirm that the sides of the right triangle (with an opposite side that is 1 stud long and hypotenuse that is 11 studs long) create the necessary angle (5Â° which is half of the 10Â° angle between the 2×14 plates). sin(5Â°) = 0.087 which is close enough to 1/11 = 0.09.

The apothem of the polygon with 36 sides (and each side that is 2 studs long) turns out to be 11.43 studs (using the online apothem calculator). Given the large number of sides we can simplify our calculations and treat the polygon as a circle with a circumference of 36×2 = 72 studs. The radius of this circle is 72/2*pi = 11.46 studs. The diameter (or the distance between any two opposite sides of the polygon) is close enough to 23 studs. The Captain America Shield set uses Technic axles, bricks with axle holes and Technic 1×2 liftarms that are each half a stud thick to create a core structure that is 19×19 studs when measured between the outermost studs. This structure is attached 2 studs away from the sides of the polygon using 2×8 plates that point inwards for a total spacing of 19 + 2 + 2 = 23 studs.

The curvature of the shield is far more subtle than the one needed for the globe and so instead of using Technic connectors, the designers chose to simply use bricks, plates and interestingly some wedge plates placed sideways to create the internal structure. The panels that form the outer skin just rest on this internal structure.

Each section of the panel uses plates with bar handles and clips allowing it to be hinged independently to achieve the curvature that is needed. The outermost section is also attached to the internal frame again using the same bar handle and clip method.

Conclusion

We have seen how a technique that uses wedge (and normal) plates in conjunction with an assortment of Technic elements to create curved shapes, has been used to great effect in two of the more recent LEGO sets. Are you aware of any other official LEGO sets that use this technique ? Is there a different technique that you would like me to explore in the next installment of this series ? Please provide any feedback you may have in the comments below …

## The Math Behind LEGO Building Techniques – Volume 2

Every so often we come across a technique used in an official LEGO set or a MOC that leaves us scratching our heads. Surely there must be a good explanation for how that technique works, but it isn’t always readily apparent. As it turns out, there is usually some math involved and my goal with this series of posts is to try to delve into the math behind LEGO techniques. The first post in this series can be found here.

In April 2022 LEGO announced two new additions to their Botanical Collection line. I will be talking about one of them (10311 Orchid) in this post. What kind of math could possibly be involved in a set with flowers anyway ? The answer of course lies in the vase. At first glance it almost seems miraculous how the fluted vase is put together with its perfectly round shape and double slopes attached on the outside with nary a gap to be found between them.

The vase was the first thing that caught my eye when I saw the announcement for the Orchid set (although I have to admit that the rest of it is pretty neat too). It is not always easy to create round shapes using LEGO (I have covered a few different ways here). The designer of the Orchid set (Mike Psiaki who is also responsible for the Titanic and Apollo Saturn V sets among others) has come up with a very clever way that relies on the mathematical properties of the 8×8 round plate which we will now examine.

The 8×8 round plate is a relatively new addition to the LEGO catalog joining its smaller siblings (such as the 4×4 and 6×6 round plates). But one thing that is neat about the 8×8 round plate is that all the studs on its periphery are located at about the same distance from the edge of the plate and this includes the studs along the diagonals. This is not true for the 6×6 round plate where the studs along the diagonals are inset a little bit from the edge of the plate.

To understand why, let us consider the diagonal distance between two studs which is âˆš2 times a stud dimension (0.8 cm) which equals 1.13 cm. On the 6×6 round plate, the distance between the outermost studs along the diagonals is 3 x 1.13 = 3.39 cm = 4.24 studs whereas on a 8×8 round plate it is 5 x 1.13 = 5.65 cm = 7.06 studs which is very close to the 7 stud separation between the outermost studs along the horizontal and vertical axes.

What this allows us to do, is place technic bricks along the outer edge of the 8×8 round plate with technic pins sticking out by about the same amount on all sides. Of course, the 1×1 technic bricks along the diagonals need to be rotated by 45 degree angles. The 8×8 round plate with 8 technic pins sticking out is just one of 4 layers that are stacked and rotated using turntables to make up the core of the vase.

The angle of rotation of the even layers (layers 2, 4) is half of 45 degrees or 22.5 degrees. This places the 8 technic pin locations on the even layers exactly halfway between the 8 technic pin locations on the odd layers. When all 4 layers are stacked we have a total of 16 columns of technic pins spaced evenly along the circumference of the core.

16 technic 1×7 liftarm pieces can be attached to these technic pins allowing us to then attach the double slope pieces needed to create the fluted appearance. Notice that with the 4×4 macaroni tiles added on the top, the height of each layer is 5 plates and this ensures the correct vertical spacing between the technic pins when the layers are stacked (5 plates as we know from the basic SNOT equation is equivalent to 2 studs).

Why do we even need the liftarm pieces ? Why not use SNOT bricks instead of the technic bricks and attach the double slope pieces to them directly ? We again need some math to help explain this. The diameter of the 8×8 round plate is 8 studs or 20 plates. With the liftarm pieces (which are 1 stud thick) attached on both sides, the total diameter becomes 20+2.5+2.5=25 plates.

The circumference of the core is now Ï€ x 25 = 78.5 plates. Now divide that by 16 and you get 4.9 plates which is very close to the 5 plate (2 stud) width of each double slope piece. Now we see how the liftarm pieces help make the inner core just large enough to attach the double slope pieces all around with almost no gaps showing between them.

Speaking of turntables, they have also been used in multiple official sets to build angled walls. An example is the Spring Lantern Festival set (80107) where 4Ã—4 turntables are used to attach an arched footbridge at an angle over a koi pond.

A LEGO turntable consists of a base (2Ã—2 or 4Ã—4) which can be attached like a normal plate and a top that can swivel freely by a full 360 degrees. The 2Ã—2 base requires a matching top element while the 4Ã—4 base can accommodate a variety of compatible elements including a 4Ã—4 round plate.

Even when we are using turntables to create an angled wall (or a structure attached at an angle, as in this example), we are essentially creating a right angled triangle that satisfies the Pythagorean Theorem (which I have covered in more detail here). The sides of this triangle intersect at the axes of rotation on the turntables (the center points of top plates). The Pythagorean Triple used in this case is (6,8,10).

As we have seen here, we don’t have a lot of options to choose from if we just limit ourselves to Pythagorean Triples. Most applications of the Pythagorean Theorem use the smallest and most common triple (3,4,5) as in the Boutique Hotel set covered here or a multiple like (6,8,10) used in the Spring Lantern Festival set.

However in certain situations, it is possible to fudge the math a bit and get away with a triple (set of 3 numbers) that is not a Pythagorean triple strictly speaking, but is close enough for practical purposes. I like to call these “near triples”.

When we create angled walls using “near triples”, it is always a good idea to use elements like hinges that naturally have a little bit of wiggle room. This minimizes the strain that you are putting on the LEGO elements when you make the connections for the angled section.

I have used “near triples” like (5,5,7) and (7,7,10) often in my builds and some of these have the added advantage of allowing us to create walls at 45 degree angles (which is not possible with Pythagorean Triples). I wasn’t aware of any official sets that used “near triples” until I looked through the instructions for the Corner Garage 10264 modular set that LEGO released in early 2019.

One notable aspect of this modular is that a large part of the front faÃ§ade of the garage is built at a 45 degree angle. There is also an awning above a gas station island that is attached perpendicular to that faÃ§ade (and ends up being at a 45 degree angle relative to the base).

If we dig into the instructions for this set, we see that the angled section is built on a 2×16 plate that is attached at a 45 degree angle using 1×2 rounded plates. The total length of the angled section is 17 studs measured between the studs at the two connection points.

If we think of this as the hypotenuse of a right triangle, the other two sides would each be 12 studs long (if you picture horizontal and vertical lines drawn along the LEGO grid from the studs at the two connection points, they would intersect at a stud that is 12 studs away in each direction).

Now, (12,12,17) is not a Pythagorean triple strictly speaking. But the length of the hypotenuse in a right triangle where the other two sides are 12 is âˆš(122 + 122) = 16.97 which is close enough to 17. The 1×2 rounded plates act like hinges and provide a firm connection while allowing a little bit of wiggle room.

“Near triples” also come into play in the way the awning is attached at a 45 degree angle (relative to the baseplate). But it is a little less obvious how the math works here. Taking a closer look, we see that the awning is 16Ã—10 studs wide with rounded corners. It is attached to the angled wall of the garage using a hinge assembly consisting of a 1Ã—3 tile with 1 finger on top (attached to the angled wall) and a 1Ã—2 brick with 2 fingers (which is incorporated into the awning itself).

The awning is supported by two vertical posts that are created using technic axles and axle connectors. These posts connect the awning to the gas station island but the island itself is connected to the base in just one spot. There is a good reason for that.

The gas station island has two 4×4 round plates that are 6 studs apart (center to center) but there is no way to connect both these at a 45 degree angle given that there are no “near triples” with 6 as the biggest number. So the designers chose to connect one side of the gas station island to a 2×2 tile with a hole (this attaches to the bottom of the 4×4 round plate to form a turntable) and leave the other side unconnected (it simply rests on the 2×2 black turntable base).

We can look at the side that actually has a connection and try to figure out the math that is involved. The connection point is exactly 13.5 studs from the line connecting the ends of the angled faÃ§ade. This can be broken up into two “near triples”. First, we start with the right triangle representing the original “near triple” (12,12,17).

If we draw a line from the vertex (corner) opposite the longest side (or hypotenuse) so it intersects the hypotenuse at a right angle, we would get two identical right triangles with dimensions (8.5,8.5,12). This is also a “near triple” and so we can think of 8.5 out of 13.5 to be one of the legs (shorter sides) of this smaller right triangle. The remaining 5 is the hypotenuse (longest side) of another “near triple” (3.5,3.5,5). Pretty cool, right ?

One of the best things about LEGO is that there are literally an infinite number of ways in which you can put pieces together to create a model. There’s usually some trial and error involved in designing a model and you may stumble upon certain techniques that work surprisingly well, even if you are not cognizant of the underlying math.

It is quite fascinating (at least for me) to try to understand why something works the way it does. I am hoping that this series of posts ends up piquing your interest as well. If there is any particular technique in an official set or MOC that you would like me to cover in a future installment of this series, please be sure to let me know in the comments section.

Happy Building !

## The Math Behind LEGO Building Techniques – Volume 1

Have you come across a LEGO technique used in an official set or a MOC that left you wondering “how does that even work ?”. It turns out there is a pretty good explanation for every technique out there as long as you don’t mind getting your hands dirty with a little math. In this new series of posts, I will try to use math to explain how some of these techniques work (suggestions are always welcome for the techniques I should cover next).

Let’s start first with the latest modular building set from LEGO – the Boutique Hotel 10297 which was released on January 1, 2022. The first thing that draws your attention to this modular building is its unusual triangular shape. LEGO as we know is based on a regular square grid of stud locations and so how did they pull this shape off and can we figure out the math behind it ?

If you read my post on angled walls, you will see that the trick to placing LEGO pieces at any angle other than 0 or 90 degrees relative to the LEGO grid is by ensuring that the studs at the two ends of the brick or plate line up with studs on the LEGO grid. This only works when the resulting right angled triangle satisfies the Pythagorean theorem (a2+b2=c2). The smallest Pythagorean triple (set of numbers that satisfies the theorem) is (3, 4, 5) and that is the one the Boutique Hotel uses. But it uses the triple in a way that is not immediately apparent. In fact, if you look at the angled walls in this build, you can count not one but 6 separate (3, 4, 5) triangles including two (4 and 6) that even intersect each other.

Given that the hypotenuses (longest sides) of triangles 1-4 are in a straight line connected together using long plates, we don’t even need to connect the outer ends of the triangles 1 and 4. The hypotenuses of triangles 5 and 6 are at right angles to the line connecting the hypotenuses of triangles 1-4.

Even more interesting is the fact that the floor sections separating the 3 levels of the building as well as the roof section are built using regular plates and wedge plates and somehow their angled sides line up perfectly with the angled walls of the building. Coincidence ? I think not. We will need to get into some basic trigonometry to explain this.

In a right angled triangle for either of the smaller angles, the ratio of the length of the side opposite the angle to the side adjacent to it is called the tangent and this ratio is fixed for a given angle regardless of the size of the triangle. If we already know the ratio, we can figure out the angle using the inverse of the tangent function or arctan (available in most scientific calculators). In a (3, 4, 5) triangle, the tangent of the smallest angle is 3/4 = 0.75 and the arctan of that is 36.8 degrees.

If you look at one of the floor sections or the roof section of the Boutique Hotel, you will see that they use two mirrored 6×3 wedge plates to create the angled side. Each wedge plate has a right triangle with a tangent of 2/6 = 0.333 and the arctan of that is 18.4 degrees. So it makes sense that two of these wedge plates would give us a combined angle of 36.8 degrees matching what we have on the angled wall.

We can confirm this using the formula to calculate the tangent of twice a given angle.

Plugging in the numbers, we see that the tangent of twice the angle created by the wedge plate is (2 x 1/3) / (1 – 1/9) = 2/3 x 9/8 = 3/4 which matches the tangent of the angle created by the (3, 4, 5) Pythagorean triple. Pretty neat, eh ?

The way the wedge plates are used here is a common application of the “mirrored hypotenuse” technique also covered in my post on angled walls. This gets around the fact that for an arbitrary right triangle, the length of the hypotenuse is not always a whole number. The hypotenuse for the triangle created by the 6×3 wedge plate is âˆš(6Â² + 2Â²) = 6.32 studs. Even though we cannot place a LEGO element along the hypotenuse, we can create an angled wall by mirroring the right triangle along the hypotenuse and placing LEGO elements along the other two sides of the second triangle. The two triangles have to be held together using hinge plates and this is exactly what is done in the Boutique Hotel set.

The next technique we will be looking at is one that Jason Pyett from Playwell Bricks came up with. He stumbled upon a very interesting way of putting pieces together to create what he calls the Magic Circle and there is definitely something magical about how everything meshes together so nicely with no visible gaps. But how does it work ?

While Jason’s entire build is pretty ingenious, let us focus on how the 8 identical sections of the magic circle fit together so perfectly. If you take a closer look, you will see that the sloped portion of a 45 degree slope piece in section A matches the height of a brick and the little lip on the 45 degree slope sitting above it in section B.

All LEGO slope pieces have a lip at the base of the slope that is about half a plate high. It has been suggested that this lip stems from limitations in the injection molding process used to manufacture LEGO bricks and that may very well be the case. But the half plate lip also makes a lot of sense geometrically speaking, at least for the 45 degree slope (which has been around since the earliest days of LEGO).

A right angled triangle with a 45 degree angle is called the special right triangle. This is because the third angle is also a 45 degree angle and the two sides that make up the right angle are of equal length. In a 45 degree slope piece, if you think of the sloped portion as the hypotenuse (longest side) of a special right triangle, the other two sides have to be of equal length. One of the sides is a stud (or 2.5 plates) wide horizontally and so the other has to be 2.5 plates measured vertically. Given that a brick is 3 plates tall, that leaves us with a lip that is 3 – 2.5 = 0.5 plate tall.

Referring to the Pythagorean theorem (a2+b2 = c2) if two sides of the right triangle are 1 stud each, the length of the hypotenuse (or the angled side) c = âˆš(a2+b2) = âˆš2 studs or 1.414 x 0.8 = 1.13 cm. That is how long the sloped portion of the 45 degree slope piece in section A would be. On the other side in section B, is a brick height which is 3 plates or 0.96 cm plus the half plate lip which is 0.32/2 = 0.16 cm for a total of 1.12 cm. These two numbers are very close to each other which explains why the magic circle works as well as it does.

Stay tuned for the next installment in this series. Until then, happy building !

## Isn’t That Stud Supposed to be on Top ? No, it’s SNOT – Building Sideways using LEGO

Introduction

Creating your own builds using LEGO bricks can sometimes be a lot of hard work. There’s definitely some blood, sweat and tears involved and not to forget, some SNOT. The SNOT I am referring to here is not what you are probably thinking of, but a technique that is used by LEGO builders to build sideways (instead of stacking bricks one on top of the other). SNOT is an abbreviation (though not the most elegant one) for “Studs Not On Top”.

If you look through the bricks that make up any recent LEGO set, there is a good chance that you will see some bricks that have studs (bumps) not just on the top but on their sides as well. These are the bricks that are designed to facilitate sideways building, but they are a relatively recent development in the overall history of LEGO (most of them were introduced in just the last 10-15 years).

The geometry of LEGO bricks

Before we get into the how and why of SNOT, it may be useful to get a little refresher on LEGO bricks and their geometry. The most common LEGO elements can be classified into two categories – bricks and plates. Bricks are the basic building blocks of LEGO while plates are their thinner counterparts. Yes, there are tiles too but tiles are essentially plates with no studs.

While we may think of the 1×1 brick as the most basic LEGO element, one interesting piece of trivia is that the first brick that was actually invented and patented back in 1958 by Godtfred Kirk Christiansen was the 2×4 brick. In the 60+ years since the invention of these plastic bricks, LEGO has grown to become the biggest toy company in the world. But one thing that hasn’t changed in all these years is the size of the brick itself. The LEGO bricks being made today are fully compatible with the bricks made during the earliest days of LEGO.

There are a few different units that have been used to measure the dimensions of LEGO elements such as LU (LEGO Units) and LDU (LDraw Units), not to forget centimeters (metric units work better than US customary units like inches). I have always preferred to use a simpler unit – the thickness of a plate (which is 0.32 cm and is equivalent to 2 LUs or 8 LDUs).

Using this unit, the height of a brick (0.96 cm) is 3 plates. The width of a 1×1 brick (0.8 cm) which we usually refer to as 1 stud is equivalent to 2.5 plates. 1 stud also represents the stud pitch (or the distance center to center between any two adjacent studs on a LEGO brick or plate). Keep in mind that all these are nominal measurements. The actual width of a 1×1 brick is more like 0.78 cm (to allow clearance between bricks when they are placed abutting each other).

So clearly, a 1×1 brick is a little taller than it is wide and the ratio of the height to width is 6/5. This is an important ratio to remember when you are building sideways. What does this 6/5 ratio mean in practical terms anyway ? Let’s start with the width of a 1×6 plate which is 6 studs = 6 x 2.5 plates = 15 plates. This is the same as the height of a wall built by stacking 5 bricks on top of each other (each brick is 3 plates high x 5 = 15 plates).

If our wall used bricks with studs on their sides, could we just turn the 1×6 plate on its side and attach it to the face of the wall ? Not really. If we had a stack of 5 bricks with studs on their sides, we would have 5 studs on the face of the wall and the spacing between these studs would be equal to the height of each brick which is 3 plates. This would not match the spacing between the 6 studs on the 1×6 plate which as we saw before is 1 stud = 2.5 plates.

It is useful to think of the rectangular face of the 1×1 brick as being made up of a square that is 2.5 plates on a side that sits above a sliver that is 0.5 plates high. Why, you may ask. If you look closer at a brick with a stud on its side, you will see that the stud is centered in the square portion and that way if you were to attach a 1×1 plate to the face of the brick, its upper edge would be flush with the top of the brick (not including the stud). The plate would cover the square portion but not the 0.5 plate high sliver below it.

Replace the 1×1 plate with a 1×2 plate and you would have an overhang that is the 1 extra stud (2.5 plates) minus 0.5 plate = 2 plates high. So you would need to add 2 plates below the brick to get the height of the stack (3 plates + 2 plates) to match the width of the 1×2 plate which also happens to be 2 studs. This illustrates a way to get the studs on the face of our wall to line up with the studs (every other stud in this case) on the brick or plate we are attaching sideways. We can simply sandwich two layers of plates between the layers of bricks with studs on their sides and this way, the spacing between the studs is 1 brick height + 2 plates = 5 plates = 2 studs.

Some relatively recent additions to the LEGO catalog even allow us to have a full complement of studs on the face of our wall. For instance, there are special SNOT bricks available now (like modified brick 32952) that are only 5 plates tall and have 2 rows of studs on their sides. Even if we use regular bricks with studs on their sides, we can replace the 2 layers of plates with elements like the modified plate 99206 that are 2 plates thick but have studs on their sides.

Now that we have seen how the geometry of LEGO bricks comes into play for sideways building, let us take a closer look at the most common types of SNOT elements and how they can be used.

LEGO sets going back to the early 1970s included sideways building in some rudimentary form – including some techniques (like wedging a plate vertically in the space between studs) that would be considered illegal now. But this was done in a very limited fashion and purely for decorative purposes. Sideways building didn’t come into its own until the first LEGO brick with a stud on its side was introduced and that was the headlight brick.

The arrival of the headlight brick back in 1980 was pretty unremarkable. LEGO designers had no good way of attaching headlights to the cars and other vehicles that were a part of the Classic Town sets being released around that time. The “headlight” brick (also known as the Erling Brick) was invented by LEGO designer Erling Dideriksen to solve this problem. It is basically a 1×1 brick with a stud on one of its sides (in addition to the stud on the top).

One interesting thing about this brick is that the stud on the side is recessed by half a plate. In fact, the entire top square portion of the face of the brick is recessed (and so, that top portion is only 2 plates deep) while the bottom “sliver” is intact creating a notch that is half a plate high. I am assuming this was done to ensure that the “headlight” plate would not stick out too much.

Even more interesting is the fact that the headlight brick has a square cut-out on the back which is essentially an anti-stud allowing it to be attached like a regular brick even when it is turned by 90 degrees. This is probably an indication that Dideriksen had more applications in mind for this brick than its nickname would suggest.

More than 40 years later, the “headlight” brick remains one of the most interesting and unique LEGO elements with applications that no one could have imagined back when it was invented. It has also paved the way for an array of sideways building techniques that go well beyond the original intent of SNOT bricks. These techniques help create details and shapes in LEGO models that would simply not be possible by stacking bricks the normal way (one on top of the other).

Given the unique geometry of headlight bricks, there are quite a few interesting ways in which they can be joined to each other. In the first case, you can use 4 headlight bricks to create a SNOT square (one that has studs on 4 sides) and 8 headlight bricks to create a SNOT cube (with studs on all 6 sides).

In the second case, headlight bricks can be used to create something that I am sure many people recognize – the logo for the now retired LEGO Creator Expert theme.

Bricks with studs on their sides

After the introduction of the headlight brick in 1980, it took a surprisingly long time for sideways building aka SNOT to go “mainstream” and be widely adopted in official LEGO sets. In fact, the regular 1×1 brick with one stud on its side (which I consider to be one of the most basic SNOT elements) didn’t make an appearance until 2009. However, the somewhat less versatile 1×1 brick with studs on all 4 sides has been around a lot longer (since 1985).

Rounding out the family of 1×1 SNOT bricks are the bricks with studs on two opposite sides (introduced in 2004) and two adjacent sides (introduced in 2017). There are larger (1×2 and 1×4) counterparts of some of these SNOT bricks that are also a part of the LEGO catalog along with some relatively new bricks that have been designed especially for SNOT. These new bricks are 5 plates tall with two rows of studs on their sides.

As you can imagine, each type of SNOT brick is suited to specific applications in sideways building and while I can’t possibly cover each one here, I can provide examples from my own builds showing how I used each of these types of SNOT bricks.

My first example is some simple SNOT I used for the base section of the Empire State Building. Here the windows needed to be slightly recessed compared to the walls and an easy way to achieve that is by attaching tiles to the wall sections between the windows (so instead of pushing the windows in, I pushed the rest of the wall out). With the 1/230 scale I was using, each floor was 5 plates high and this was perfect for SNOT. I had layers with bricks sandwiched between layers with plates. I used the 1×1 bricks with 2 studs on adjacent sides in the corners and 1×1 bricks with a single stud on their side everywhere else and attached 1×6 tiles to the wall sections between the windows to achieve the recessed windows effect.

It was a little trickier to create the same effect on the base of my model of the Hearst Tower. Here the scale is bigger (1/156) and calls for 7 plates per floor. To be able to attach 1×8 tiles to the faces of the wall, I needed to somehow get the studs on the faces of the walls to be 5 plates apart even if each floor was 7 plates high. I ended up having to mix bricks and plates within the same layer (so to speak) to achieve this.

Here’s another example from the base of my model of 70 Pine Street. I used the 1×1 bricks with studs on 2 adjacent sides in the corners and 1×1 bricks with studs on 2 opposite sides to create the columns that separate the different bays of windows.

Bricks with studs on all 4 sides were used extensively in the past when the other types of 1×1 bricks with studs on their sides did not exist. But in most of those cases, we can now get away with using the newer bricks with 1 or 2 studs on their sides. Applications that actually use the studs on all 4 sides are limited at least in architectural builds. The bricks can possibly be used in spires on the top of a building or to create an octagonal column as shown below.

Bricks with studs on 1 side and 2 adjacent sides can also be combined to create SNOT cores that can have plates and curved slopes attached to them to create various shapes like cylinders, spheres, etc.

Plates with studs on their sides

Plates with studs on their sides may be relatively new but they have proved to be very useful for sideways building. To understand where they fit in, let us revisit the basic SNOT wall. In order to get the right spacing between studs on the face of the wall, you need to sandwich two layers of plates between layers of bricks with studs on one side. But there are no studs facing outwards on the plates themselves.

If we could turn the two layers of plates into a single element that also has studs facing out, we would end up with something like the plates with studs on their side. These modified “plates” are actually two plates thick and have studs on their side allowing them to complement bricks with studs on their side and create a 5-plate high stack that has two rows of studs. The advantage this approach has over using the special SNOT bricks that are 5 plates high is that the modified plates give us the ability to use long plates in the back and strap the SNOT section to the bricks on either side (for greater structural stability).

The plates with studs on their side are currently only available in 2×2 (part 99206) and 2×6 (part 87609) versions but I am sure more types will be added in the future. The smaller of the two (99206) has been indispensable in some of my models – I have used it in the roof sections of the Chrysler Building and 40 Wall Street.

Brackets

A LEGO bracket is essentially a 1×1 or 1×2 plate with studs on its side except that these studs are on an extension that is perpendicular to the plate. This extension is sized like a normal plate with anywhere from 1 to 8 studs (depending on the type of bracket) but it is only half as thick.

In a regular bracket the top edge of the extension is flush with the top of the plate (not including the studs) while in an inverted bracket the bottom edge of the extension is flush with the bottom edge of the plate. Here’s a selection of some of the types of brackets that are available in the LEGO catalog

Half plate offsets

Half plate offsets (not to be confused with half stud offsets) can be used to offset or shift a LEGO element by half a plate (which is a smaller amount than half a stud). To understand why they may be needed, consider the fact that the LEGO system is based on a square grid where each square is 1 stud (or 2.5 plates) on a side. But when we build sideways, the smallest increment that is normally available is 1 plate. This can sometimes leave us with a half plate gap or misalignment, especially if our SNOT portion occupies an odd number of studs.

This is usually not something we have to worry about when we are using SNOT to add minor details to the exterior of our model. But when we are trying to seamlessly integrate a SNOT section into a model that is mostly built with studs on top, we need to pay close attention to how the elements that are placed sideways line up with the LEGO grid. As long as we can get the SNOT section to occupy an even number of studs, that works out to a whole number of plates (2 studs as you recall is equivalent to 5 plates). For instance, my model of the Taj Mahal needed an accent stripe going around the main doorway and I created the vertical portion of this stripe by attaching sand green plates sideways. I was able to create a gap in the wall around the doorway that was 2 studs wide and fill it with a brick and a tile placed sideways in addition to the sand green plate used for the accent stripe itself (so the SNOT portion has a total thickness of 5 plates).

However, when we are dealing with an odd number of studs, we may end up with a half plate gap that we need a way to fill. We could eliminate the gap if we had the equivalent of a brick with a stud on its side but with the stud pushed out by half a plate. It is possible to create just that using two other types of SNOT elements – brackets and headlight bricks. Brackets as you recall have extensions that are a half plate thick. When you combine brackets with regular bricks and/or plates you can create the equivalent of bricks with studs on their side but these studs would be offset by half a plate. Headlight bricks on the other hand have studs on their sides that are recessed by half a plate. Attaching a plate to the front of the headlight brick would again create a brick with a half plate offset.

Here is another example that is loosely based on the roof section of my model of the Chrysler Building. We are creating rounded tapers on each side by attaching curved slopes sideways. For aesthetic reasons we need the curve to flow smoothly from the straight portion that is built with studs on top. If we use 2 studs for the SNOT portion on each side, we can attach a brick sideways along with the curved slope itself (which is 2 plates thick when combined with a 1×1 plate) for a total thickness of 5 plates.

But suppose there is a window in the middle that leaves us with only 1 stud on each side for the SNOT portion. 1 stud is equivalent to 2.5 plates but the curved slope (combined with a 1×1 plate) is only 2 plates thick which will cause the curved portion to be inset by half a plate relative to the straight portion. Adding a plate to the SNOT section does not help because it would now make the curved portion stick out by half a plate.

What we need here is a half plate offset. We can use brackets (inverted brackets in this case) to push out the SNOT section by half a plate and get the curved slope to line up correctly with the straight wall below it.

Half stud offsets can also be used to eliminate the jaggedness in slopes built by cascading multiple cheese slope pieces. The so-called “cheese” slopes are basically 1×1 slope pieces that get their name from the fact that they resemble little wedges of cheese. Each cheese slope piece is two plates high with a lip at the base of the slope that is half a plate high.

If we were to simply stagger each successive cheese slope by 2 plates, we would not have a slope that is smooth because of the stair-stepping caused by the half plate lip. What we need is a way to stagger each cheese slope by 2 – 0.5 = 1.5 plates. We can take advantage of the geometry of basic LEGO bricks to get the 1.5 plate offset we need. A regular brick is 2.5 plates deep and 1 plate less than that is 1.5 plates.

Of course, we can also use the half plate offsets that can be created using headlight bricks and brackets to achieve the same effect.

Quarter plate offsets

If we line up a headlight brick, a regular brick with a stud on one side and a bracket (attached to two 1×1 plates), we will see that the studs on the front have successive half plate offsets.

Is it possible to create an even smaller offset – say a quarter plate offset ? This may be purely academic at this point (given that I haven’t found many applications for this technique at least in my models), but it is possible to combine half stud offsets (using jumper plates) with SNOT to create quarter plate offsets. Recall that 1 stud is 2.5 plates and so half a stud is 1.25 plates. Suppose we start with 2 identical SNOT bricks (say headlight bricks) placed next to each other. If we offset one by half a stud using a jumper plate and attach a 1×1 plate to the front of the other, we would end up with studs in the front that are 1.25 – 1 = 0.25 plate apart.

Combining half plate offsets with quarter plate offsets, we can create an entire sequence with each step offset by a quarter plate. Here’s how the math works for this sequence

1. Headlight brick (2 plates) + 1 plate = 3 plates
2. Half-stud offset using jumper plate (1.25 plates) + headlight brick (2 plates) = 3.25 plates
3. Brick with stud on side (2.5 plates) + 1 plate = 3.5 plates
4. Half-stud offset using jumper plate (1.25 plates) + brick with stud on side (2.5 plates) = 3.75 plates
5. Regular plate (2.5 plates) + bracket (0.5 plate) + 1 plate = 4 plates
6. Half-stud offset using jumper plate (1.25 plates) + regular plate (2.5 plates) + bracket (0.5 plate) = 4.25 plates

Stud reversal

In general when we build sideways, we turn the direction of the studs by 90 degrees relative to their normal orientation. But there may be some situations where we need to turn the direction of the studs by 180 degrees or essentially reverse their direction. See for instance my model of 40 Wall Street where the green pyramidal roof is made up of 4 triangular panels that are angled using hinges. Each of these panels is built in two halves with studs facing opposite directions and the two halves are joined together using bricks with studs on 1 side.

There are many different ways (some legal, others not) to reverse studs but here are a few that use the SNOT elements we have covered here.

Conclusion

I have tried to present SNOT from a limited perspective based on the models I have built. But this is just the tip of the iceberg, so to speak. As you have probably noticed, I have only used basic system elements (bricks, plates, brackets) for SNOT but there is an array of other SNOT techniques available that use Technic and other specialized elements like plates with clips, lamp holders, etc. For a far more exhaustive overview of SNOT in its various forms, I would highly recommend the series of articles by Oscar Cederwall that has been posted on Bricknerd.

(Note – This is an expanded version of an earlier post â€“ this time all the pictures have also been redone).

## Don’t Underestimate That Jumper Plate – Half Stud Offsets in LEGO Models

Introduction

What are half stud offsets and why do we need them ? Before we get into that, let’s start with the basics. A stud is not just the bump you see on the top of a LEGO piece that allows you to connect it to other LEGO pieces. It is also a unit of measurement representing the distance between two adjacent bumps (or knobs) on a LEGO piece (measured center to center). The basic building block of the LEGO system is a 1×1 brick which is exactly 1 stud wide and 1 stud deep (though it is a little taller than 1 stud). A 2×4 brick is 2 studs wide and 4 studs deep. All basic LEGO pieces (bricks and plates) have footprints that are multiples of 1 stud.

When you build with LEGO, you can only place each piece such that its studs (bumps) line up with the studs on the layer that is immediately below. If you start with a 32×32 baseplate, you are limited to a 32×32 grid of possible locations where you can place the pieces in your next layer and these locations are separated by increments of 1 stud (the measurement unit). Say you are building a wall where a section needs to be recessed. The smallest amount you can normally set the recessed section back is one stud. If you are looking to create a more subtle effect, would it be possible to set the recessed section back by just half a stud instead of a full stud ? Yes, and that is exactly what jumper plates allow you to do.

A 1×2 jumper plate (part 15573) has a single stud that is located exactly halfway between where the two studs on a regular 1×2 plate would be located. You can use it to set back (or offset) your wall section by half a stud as shown in the picture below.

Half stud offsets in one and two dimensions

While a 1×2 jumper plate allows you to create a half stud offset in one dimension (either front to back or sideways), you can do the same in two dimensions using a 2×2 jumper plate (part 87580). This jumper plate has a single stud exactly in the center of where the 4 studs on a regular 2×2 plate would be located.

LEGO has recently expanded their catalog to also include “double jumper” counterparts for their 1×2 and 2×2 jumper plates. They are the 1×3 jumper plate (part 34103) with 2 studs and the 2×4 jumper plate (part 65509) with 2 studs.

Creating recessed wall sections, windows, etc.

As we have seen, the 1×2 jumper plates are perfect for setting back sections of a wall or a window by half a stud to add more subtle detail. You may need another set of jumper plates above the window (a second half-stud offset to counter the first one) to get back to the normal alignment of the studs as you continue to build the wall above the window.

Here are examples of half stud offsets from two of my skyscraper models. In the case of the Empire State Building, I used jumper plates to create recessed wall sections at the top of the building.

I did something similar in my model of 70 Pine Street. Here, jumper plates were used to create recessed windows.

Centering elements with odd vs. even number of studs

Another great application for jumper plates is centering an element with an odd number of studs relative to something with an even number of studs or vice versa.

The top portion of the Empire State Building has narrow windows (one stud wide based on the scale I was using) that needed to be centered relative to the windows below them (which were each 2 studs wide). I was able to use 1×2 jumper plates to center the smaller windows relative to the bigger windows below them.

In the digital model of the Blue Mosque that I built, I needed to have 3 smaller arched window openings inside a bigger arch. I used jumper plates to center the 3 smaller arches (with a total outside dimension of 7 studs) inside bigger arch with an opening that is 6 studs wide.

Smoother tapers using jumper plates

An obvious application for jumper plates is to help achieve smoother tapers than is possible with regular bricks and plates. Like many of the other skyscrapers built during the early 1930s, the Empire State Building and 70 Pine Street have top sections that taper as they lead up to their spires. I used 2×2 jumper plates for the tapers in my models of these buildings.

My models for other buildings like the Transamerica Pyramid and John Hancock Center required more extensive use of half stud offsets given that the entire buildings have tapered shapes.

The Transamerica Pyramid in San Francisco is a skyscraper with a square base that gradually tapers to a point at the top. Based on the scale I was using, I needed to taper the model from 28 studs to 7 studs over 42 floors. With regular bricks and plates, the smallest amount I would have been able to taper the model is 2 studs (one stud on each side) approximately every 4 floors. Using 2×2 jumper plates I was able to taper the model by just one stud (half stud on each side) every 2 floors. This definitely minimized the jaggedness and resulted in a smoother taper.

However, there is one downside to using half stud offsets and this became apparent on the model of the Transamerica Pyramid. The pyramid of this building is flanked by “wings” on two sides which are structures that hold the elevator shaft and stairwell. In my model, the wings needed to rise vertically without any tapers and this meant I would have normal wall sections intersecting the pyramid walls that are tapered using half stud offsets. In this situation, half stud gaps are unavoidable (because the bricks that make up the tapered pyramid are no longer in a straight line vertically). As you can see in the picture below, I did my best to plug these gaps by attaching tiles to the “wings”.

Tapering by unequal amounts in the two dimensions

My LEGO model of the John Hancock Center in Chicago presented a different kind of challenge. In order to accurately represent the proportions of the real building, I needed to taper the wide and the narrow sides of the model by different amounts in terms of studs. So I couldnâ€™t simply use the 2×2 jumper plates like I did on the Transamerica Pyramid. I had to use a mix of 1×2 jumper plates (oriented two different ways) along with 2×2 jumper plates to taper the model by one stud (half stud on each side) every 6 floors (on the long side) and every 8 floors (on the short side).

To help illustrate this a little better, I have created a simplified example where I taper a building with 12 floors from a base that is 8×6 studs to a top that is 3×3 studs. The longer side goes from 8 studs to 3 studs in 6 steps (and so it needs to be tapered by a stud every 2 floors). The shorter side goes from 6 studs to 3 studs in 4 steps (and needs to be tapered by a stud every 3 floors).

Here’s a breakdown showing the offsets needed (or not) at each floor to achieve this unequal taper in the two dimensions (X and Y).

Other uses of jumper plates

One other interesting thing about jumper plates is that they have “open” studs that allow plates and bricks to be attached on the top without any offsets. So why even bother using jumper plates ? The answer is “clutch power” or the ability of a LEGO piece to hold together tightly to the piece it is attached to. Given that a 1×2 jumper plate has just 1 stud on the top and two anti-studs (or receptacles for studs) on its underside, means that it has higher clutch power on the bottom than it does on the top. This makes it very useful in big models that have multiple sections that need to be put together and taken apart easily. The jumper plate tends to stay attached firmly to the layer below it while allowing the layer above to be separated without much effort. I tend to mix in jumper plates with tiles at the seams between the different sections that make up my skyscraper models (and I have never had to worry about loose pieces coming off while taking the sections apart).

Here’s an example showing the seam between the base in my model of the Empire State Building and the section above it. You can see a sprinkling of jumper plates among the tiles and these allow the section above to be held in place but separated from the base easily.

SNOT with half stud offsets

This SNOT technique doesn’t use jumper plates but I decided to include it here anyway. It makes use of the Technic 1×2 brick with 1 hole (part 3700). You can insert a Technic pin (part 4274) in the hole to create a “stud” and then use SNOT to attach tiles and other elements to the faces of the Technic 1×2 bricks. The tile is essentially offset by half a stud relative to the stud locations on the Technic bricks and helps add subtle detail.

You can see this technique used in the top section of my model of the Empire State Building along with some of the other techniques we covered earlier.

So the next time you build something using LEGO, don’t underestimate those lowly jumper plates. You just don’t know when or where they may come in handy. If you have any other applications for jumper plates that I have not covered here, please feel free to post a comment. Happy Building !

(Note: This is an expanded version of an earlier post â€“ this time all the pictures have also been redone).

## Thinking Outside the Grid – Building Angled Walls using LEGO

Introduction

How many different ways can you attach a 1×6 brick to a baseplate ? There are quite a few as you can imagine, but each time the studs of the brick have to line up with the studs of the baseplate under it. That is just how LEGO works. The studs on the baseplate are in a regular square grid and therefore you can only place your 1×6 brick such that it is parallel to one of the sides of the baseplate. Suppose you are building a castle out of LEGO and one of your walls needs to be at a 45 degree angle, are you out of luck ?

Remember the regular square grid I mentioned earlier ? If you take any stud on the baseplate, it is exactly the same distance away from each of its neighbors on all 4 sides and that distance happens to be 0.8 cm (or a “stud” which is also the basic unit of measurement in LEGO). But the studs in the 4 corners are farther away (the distance is âˆš2 x 0.8 cm = 1.414 x 0.8 cm = 1.13 cm to be exact). That is because in a square, the 4 sides are of equal length but the distance between any two opposite corners is a little longer. Similarly, the distance between any two studs measured at any angle other than 0 or 90 degrees is not guaranteed to be a whole number of studs.

So how can we turn our 1×6 brick to a different angle and still have it attach firmly to the baseplate ?

Angled wall basics

Let’s try a little experiment now – place two 1×1 plates as shown in the picture below. Now, if you place the 1×6 brick diagonally bridging these two 1×1 plates, it works ! The studs at the two ends of the 1×6 brick line up with the studs on the two 1×1 plates. This allows the 1×6 brick to have a good connection to the baseplate (at least at the two ends). The remaining 4 studs on the 1×6 brick still don’t line up with the studs below (now you see why we had to use the 1×1 plates as spacers to raise the 1×6 brick). So what exactly is going on here ?

If you jog your memory back to high school math (if you haven’t gotten there yet, you will just have to take my word for it), the equation a2 + b2 = c2 may seem vaguely familiar. That is the Pythagorean Theorem that defines the relationship between the sides of a right angled triangle.

Let’s take a closer look at where the two 1×1 plates were placed on the baseplate. If we count along the two sides of the baseplate starting from the corner, we see that the 1×1 plates are 3 and 4 studs away from the corner stud and make up two sides of a right angled triangle. Our 1×6 brick is placed along the third (and longest) side, also known as the hypotenuse. The triangle we created satisfies the Pythagorean theorem because 32 + 42 = 9 + 16 = 25 which is equal to 52. Then, does it make sense that we used a 1×6 brick for the longest side ? Yes, because the three sides of the right angled triangle intersect at the studs and the distance that really matters is the distance between the studs at the two ends of the 1×6 brick which is 5 studs.

So the bottom line is that for any brick or plate to be placed at an angle other than 0 or 90 degrees, you need to make sure the resulting triangle satisfies the Pythagorean theorem. Any set of 3 numbers that satisfies this theorem is called a Pythagorean triple and (3, 4, 5) is the smallest such set made up of whole numbers. What are some other Pythagorean Triples ? Listed below are all the Pythagorean Triples with numbers less than or equal to 25. As you can see, there are not many with practical applications in LEGO builds.

Angled walls using hinge elements

If we were to use the method described earlier to build angled walls, there is just no good way to avoid gaps at the corners where the angled wall segments meet the regular wall segments placed along the LEGO grid. An alternative is to use hinge elements.

There are many different types of LEGO hinge elements but the ones we need for angled walls are the ones that swivel – specifically the 1×4 hinge plate that consists of a 1×2 swivel base and a 1×2 swivel top.

The 1×2 plates that make up each of the two halves of the hinge plate are joined at their corners by a hinge that allows the angle between the plates to be changed from 0 to 180 degrees. If you place two 1×4 hinge plates as shown below, you can bridge them with a 1×5 plate (yes, LEGO makes one now !) attached at the top.

We are still creating the same right angled triangle (3, 4, 5) as before, but this time the hypotenuse (angled side) has 5 studs instead of 6. This is because the sides of the right angled triangle now intersect at the corners of the plates rather than the studs.

An example of a LEGO official set that uses the (3,4,5) triple extensively is the Boutique Hotel set (10297). Here you can find not one but 6 separate (3,4,5) triangles used to create the angled walls that give the building its unique triangular shape. More details can be found here.

Angled walls using turntables

A LEGO turntable consists of a base (2×2 or 4×4) which can be attached like a normal plate and a top that can swivel freely by a full 360 degrees. The 2×2 base requires a matching top element while the 4×4 base can accommodate a variety of compatible elements including a 4×4 round plate.

Turntables give us another way to create angled walls and this technique has been used in multiple official LEGO sets. An example is the Spring Lantern Festival set (80107) where 4×4 turntables are used to attach an arched footbridge at an angle over a koi pond.

Even when we are using turntables to create an angled wall (or a structure attached at an angle, as in this example), we are essentially creating a right angled triangle that satisfies the Pythagorean Theorem. The sides of this triangle intersect at the axes of rotation on the turntables (the center points of top plates). The Pythagorean Triple used in this case is (6,8,10).

Near Triples (When close enough is good enough)

The angled wall we built earlier can probably pass for a 45 degree wall but if you take a closer look at it, the smaller angle is more like 37 degrees. A right-angled triangle with a 45 degree angle is called a special right triangle because the third angle also ends up being 45 degrees (the three angles in a triangle have to add up to 180 degrees). This also means that the two sides that make up the right angle have to be of equal length. None of the Pythagorean triples we have seen, has two smaller numbers that are equal. So is it really possible to achieve a 45 degree wall using LEGO ?

Thankfully, the little bit of give that hinge plates have, allows us to use numbers that are close enough to Pythagorean triples. Consider for instance (5,5,7) and (7,7,10) which are “near triples” that allow you to build LEGO walls at 45 degree angles as shown below.

(12, 12, 17) and (17, 17, 24) are a couple of others. There are also “near triples” like (4, 7, 8) and (4, 8, 9) that don’t give you a 45 degree angle but are useful nonetheless.

Official sets have also used “near triples” such as the Corner Garage set (10264) that uses (12,12,17) to create the part of the main faÃ§ade that sits at a 45 degree angle. You can find more details here.

Using jumper plates for even more options

Pythagorean triples (or “near triples”) also work when you multiply all the numbers in a triple by a whole number like 2 or halve them. For instance, take the triple (3, 4, 5) and multiply all the numbers by 2 and you get another triple (6, 8, 10). Similarly, if you take the “near triple” (7, 7, 10) and halve all the numbers, there is reason to believe that (3.5, 3.5, 5) would work. But how do you create a triangle that has sides that are 3.5 studs long ? Using jumper plates, of course !

Combining “near triples” with half stud increments gives you more options for building 45 degree walls. One combination I have found to be particularly useful is (8.5, 8.5, 12) which is half of a “near triple” (17, 17, 24).

Angled Walls – Hearst Tower

The Hearst Tower in New York is a wonderful example of a skyscraper that combines old and new architectural styles. It preserves the faÃ§ade of the original 6-story Art Deco building as its base and adds a modern glass tower on the top. Two of the corners of the base are chamfered and I needed to build 45 degree walls for those two corners. Based on the scale I was using, I needed the angled wall section to be about 7 studs wide. This was a perfect application for the “near triple” (5,5,7).

Angled Walls – Taj Mahal

One of the modern wonders of the world, the Taj Mahal in Agra, India is probably one of the most well known masterpieces of Islamic Architecture. The main structure in the Taj Mahal is a mausoleum that sits on a raised platform. The mausoleum is shaped as a cube with four truncated corners that create the shape of an unequal octagon. To create the chamfered corners in my LEGO model, I used another “near triple” (7, 7, 10).

Angled Walls – Tribune Tower

One of the most beautiful skyscrapers in the world – the Tribune Tower in Chicago was inspired by Neo-Gothic architecture. Its highly ornate crown complete with flying buttresses was designed after the Butter Tower of the Rouen Cathedral in France. My LEGO model of this building required not one but several different types of angled walls. For the chamfered corners of the main tower I used the “near triple” (5,5,7). The two levels of the octagonal crown used two near triples – (8.5, 8.5,12) and (7,7,10) and the flying buttresses were attached to the base of the crown using another near triple (4,7,8).

Angled walls – other techniques

This article would not be complete without at least a mention of a few other ways of building angled walls using LEGO. The techniques we have seen so far create angled walls by placing elements along the hypotenuse (angled side) of a right-angled triangle. But for this to work, the length of the hypotenuse has to be a whole number of studs and this limits our options to Pythagorean triples and near triples.

There are a few other techniques where we don’t actually place any elements along the hypotenuse and can therefore disregard its length. This opens up quite a few other possibilities …

The “mirrored hypotenuse” technique

Let us start with an arbitrary right triangle – say the sides that make up the right angle are 6 studs and 2 studs long. The length of the hypotenuse (angled side) would be âˆš(62 + 22) = 6.32 studs which is not a whole number. It is not possible to place a LEGO element along this side, but if we mirror the right triangle along the hypotenuse, we can create an angled wall by placing LEGO elements along the other two sides of the second triangle. We will need to use hinge plates to join the two mirrored right triangles together.

A more common application of the mirrored hypotenuse technique uses wedge plates to create the two mirrored triangles. The 6×3 wedge plates create triangles that are equivalent to the ones from my earlier example (where the numbers I picked were not so arbitrary, after all). This technique is used to create angled sides in the floor and roof sections of the Boutique Hotel set (10297). You can find more details here.

It is also possible to extend this technique to unequal triangles where the hypotenuse of one triangle is one of the sides that makes up the right angle of the second triangle. The resulting quadrilateral (4-sided shape) would have to satisfy the equation for a Pythagorean quadruple which is a2 + b2 + c2 = d2. The simplest Pythagorean quadruple is (1,2,2,3). Multiply all numbers by 2 and you get (2,4,4,6) which is a Pythagorean quadruple as well.

The neat thing about Pythagorean quadruples is that they can be used to place elements at an angle not just in 2 dimensions but in 3 dimensions as well. Credit goes to hafhead on Flickr for the idea.

The “switched diagonals” technique

The diagonal distance between two studs on a plate may not be a whole number of studs. But if you take a rectangular plate, the distances between the two pairs of studs at opposite corners (1, 3 and 2, 4) are exactly the same. So you can rotate the plate and attach it such that its corners 2 and 4 line up with where the corners 1 and 3 normally would be. Once again you will need to use 1×1 plates as spacers.

If you think of 1-3 and 2-4 as the longest sides of two identical right angled triangles that are mirrored, all you are doing is rotating one of the triangles so that the longest sides are lined up. The angle of rotation would obviously depend on the size of the right angled triangle (the number of studs on the two sides that make up the right angle). It turns out that the closest you can get to a 45 degree rotation is by using a 4×8 plate.

This technique can also be extended to include hinge elements. It looks a little different because this time the diagonals that are switched go all the way to the corners of the plates.

Conclusion

I am sure there are several other ways to create angled walls that I have not been able to cover here. I am hoping the overview I have provided at least points you in the right direction for your own exploration of these techniques. Please let me know if there are other techniques you have used that merit an inclusion here. Happy building !

(Note: This is an expanded version of an earlier post – this time all the pictures have also been redone).

## Squaring the Circle – Building Round Shapes using LEGO

Introduction

LEGO is not a medium that is inherently suited to building round shapes. After all, the basic building block – a 1×1 brick has a square footprint and a LEGO baseplate has studs placed in a regular square grid. And yet, there are many wonderful LEGO creations out there that include round shapes – all kinds of cylinders and even spheres. I did not have much experience building these shapes out of LEGO until I started working on my own version of the Taj Mahal. The focal point of this well-known landmark (which happens to be one of the modern wonders of the world) is its massive dome which sits atop a cylindrical base called the drum (that is an actual architectural term !). The minarets (slender towers) that sit at the 4 corners of the plinth of the Taj Mahal are essentially stacked cylinders too.

Needless to say, I had to take a deep dive into all the techniques out there that can be used to create round shapes using LEGO – I basically scoured the web looking for any information I could find on this topic. Every step of the way, I was amazed and inspired by the endless creativity of the AFOL community (I also tapped into some of the tools created by the Minecraft community). While I can’t claim to have invented any of the techniques listed in this article, I am happy to catalog them here for future reference (giving credit to the inventors wherever possible).

So what exactly is involved in creating round shapes out of LEGO ? The title of this post refers to “squaring the circle” which is an age-old mathematical problem that people tried to solve for centuries until it was proven (in 1882) to be impossible to solve. However it is possible to get close enough by using an approximation of the number Ï€ (pi). In much the same way, all the techniques described here try to use square/rectangular bricks or plates to create the best approximation that is possible of the round shape. The result is never perfect and the limitations of the LEGO medium are always apparent in the jaggedness of the curves and the gaps that you may see, especially when you are looking at your model up close. But the trick is creating a pretty convincing illusion of a round shape at least when you look at your model from a few steps back.

Using SNOT to create small cylinders

For the minarets of the Taj Mahal, I did not want to be limited by the small selection of cylinder pieces that LEGO has. I wanted to try a different way of building cylinders that could be scaled to whatever height I needed. I was already familiar with using curved slopes and SNOT to create round shapes (I had used this technique to build the crown in my model of the Chrysler Building). The same technique can be used to create cylinders with various diameters. All the cylinders in the picture below can be built in height increments of two studs.

The smallest cylinder was just right for the minarets of the Taj Mahal, based on the scale I was using. Ideally the minarets should taper as they rise – the way they do in the real Taj Mahal. Unfortunately there was no easy way to reduce the diameter of the smallest cylinder any further.

Bending LEGO walls to create round shapes

A “brute force” way of building round walls using LEGO is by building straight walls and then bending them to form a circle. The longer your wall, the more flex it will have, making it easier for you to bend it into a complete circle. The number of 1×2 bricks needed in each layer to build a stable round wall tends to be around 72, give or take a few bricks. But I have seen round walls built using far fewer plates in each layer.

Check out the work of Jeff Sanders who specializes in “brick bending”. He has an impressive portfolio of creations, made by bending LEGO brick walls not just into circles but various other shapes as well.

Please note that most brick bending techniques are illegal, strictly speaking, because you are using LEGO elements in ways they are not intended to be used and subjecting them to undue stress and possible damage. An obvious downside to this technique (other than it being illegal) is the fact that it cannot be replicated digitally in stud.io.

A few years ago, I embarked on a LEGO project unlike any other that I had worked on before. I wanted to try doing a studs-up mosaic in a round shape. I came up with my own pattern of intertwined snakes and used a little over 14000 plates to build this round thing that I like to think of as some kind of a vase (because a trash can isn’t as appealing). This sits in a corner somewhere in my basement because I still haven’t figured out a good way to finish this and create something worth displaying. Any suggestions would be welcome, of course !

Mixing regular bricks with round 1×1 bricks to create round walls

This uses the same approach as above – except for the fact that we use 1×1 round bricks (or plates) in our wall to allow it to be bent more readily (and legally !). If we alternate say 1×3 bricks in our wall (1×2 bricks also work) with 1×1 round bricks, the round bricks act like hinges to some extent, allowing the wall to be bent to form a circle. This method allows round walls to be built with smaller diameters than is possible with the previous method. The only downside is that the texture of the wall is uneven due to the 1×1 round bricks. It is possible to use tiles to hide the round bricks and create the effect of a real brick wall (credit goes to Steve DeCraemer on Flickr). The 1×3 bricks here are replaced with 3 headlight bricks with their top studs facing out. These headlight bricks are joined together with a 1×3 plates while 1×4 tiles are attached to the face of the round wall creating the look of a brick wall (which should work great for castle builds).

Round walls using hinge bricks/plates

We have seen that hinge pieces can be very useful for creating angled walls and various polygons (hexagons, octagons, etc). The greater the number of sides in a regular polygon (one where all sides have the same length), the closer it starts to approximate a circle. We can take advantage of this fact to use hinge pieces to create round walls.

One tutorial I came across on this topic is this one by Eggy Pop (flickr user name) – https://www.brickbuilt.org/?p=9015

I have re-created the round wall shown in this tutorial in stud.io using 14 1×4 hinge plates to complete the circle (which is essentially a polygon with 28 two-stud wide sides). One of the neat things about this particular round wall is that its diameter (18 studs on the inside, 20 studs on the outside) is a whole number in terms of studs. As you can see, the hinge plates on 4 sides line up with the LEGO grid allowing the round wall to be attached firmly to a base. So what is special about the number 18 ? Let us see if we can use some math to figure that out. Assuming the inside of the wall is close enough to a circle, its circumference would be the diameter times Ï€ (pi) = 18 x 3.14 = 56.52 studs. Now divide that by 4 (the length of a 1×4 hinge plate) and you get 14.13 which is close enough (for practical purposes) to a whole number 14 – which happens to be the number of hinge plates we used.

Don’t count on the same working for round walls with inner diameters of say 17 or 19 studs. What are some other numbers that work ? The two numbers closest to 18 that work are 14 and 23 (I used 23 for the drum in my Taj Mahal). As you can see from the pictures, these two numbers don’t work as well as 18 and the hinge plates on only 2 sides line up with the LEGO grid (which is usually good enough). Can we explain why ? In the case of 14, the number of hinge plates needed is (14 x 3.14) / 4 which is approximately 11. Since the number of hinge plates is not an even number, you don’t have the hinge plates lining up with the LEGO grid on all 4 sides. In the case of 23, the diameter itself is not even. And so, if you line up the hinge plates with the grid on one axis, the other axis is offset by half a stud (you could use jumper plates to attach the round wall to the base on the other two sides if needed).

One issue with round walls is that when you have just 2 or 4 connection points to the base, the rest of the hinge plates are free to move and so the round shape can easily get distorted. There is no elegant solution I could think of, to get the wall to maintain its shape (I guess you could use long plates as cross members joining each side of the polygon to the one on the opposite side and then have these cross members attached in the middle using turntables). For now, I just built a regular inner wall as close as possible to the round wall, and attached cheese slopes, curved slopes, etc. to the inner wall using SNOT to fill the gaps the best I could. Since the drum has an outer diameter of 25 studs and I needed to center it on a 24×24 base, I ended up using jumper plates. I also needed jumper plates to center the dome (which is built with a core that is 16×16 studs) on the top of the drum.

Building Round Walls Digitally

While it is quite straight-forward to build these round walls using real pieces, how do you do it in stud.io ? The hinge tool allows you to select a piece and rotate it around a hinge point. In this case, the piece being selected would be one half of the 1×4 hinge plate. When you click on the piece with the hinge tool enabled, you will first see a blue arrow. You can click and drag this blue arrow to manually rotate the piece. Or you can click on the blue arrow to get a white text box where you can enter the precise angle to rotate the piece by.

With round walls it is important to make sure all the hinge plates are rotated by just the right amount or the two ends of the wall will not line up correctly to allow you to complete the circle. To figure out the right angle to use, just divide 360 by the number of sides of the polygon you are building which in the case of the first circle (18 studs diameter) would be the number of hinge plates times 2 = 14 x 2 = 28. 360 divided by 28 is 12.85 degrees which is the number you need to enter in the text box. I tend to apply this rotation to one hinge plate and then copy and paste the hinge assembly (both halves together) as many times as I need.

Building spheres by stacking regular bricks or plates

To understand how spheres are built using LEGO bricks, I have found it useful to first look at the world of Minecraft. There are some parallels between Minecraft and LEGO in that both use building blocks with square footprints that are placed on a square grid. One important difference is that the blocks in Minecraft are perfect cubes while a basic 1×1 LEGO brick is not (it is taller than it is wide). More on that later. Anyway, there are a number of resources available in the Minecraft community for building spheres and other shapes using Minecraft blocks and I was wondering if I could leverage some of them for building LEGO spheres.

Before we look at a 3-dimensional sphere, let us look at a basic circle in 2 dimensions. Minecraft enthusiasts often use what is known as a circle chart. It shows the placement of blocks (or pixels) in a square grid that best approximates a circle. The chart shows circles with different diameters and as you can see from the chart, the bigger the diameter, the more convincing the illusion of the round shape is.

In the Minecraft world where the building blocks are perfect cubes, you can also look at one of the circles and think of it as the side view of a sphere. You can think of each row in the chart as one of the layers of blocks in the sphere you are building. So you would essentially start with a layer of blocks placed in a circle having the diameter of the sphere you want to build (that would be the biggest circle you need). You would then stack layers with successively smaller circles (as you go up) until you reach the smallest circle you need. Repeat the same on the bottom (with the circles getting smaller as you go down) and you have a complete sphere. Of course, it could be a lot of work to figure all this out by hand (even with the help of a circle chart). Thankfully, an online tool (Plotz Sphere Generator) can automate this process for you. You just enter the diameter you need and presto, a sphere is generated for you (with 3D and 2D views showing how the sphere is constructed layer by layer).

That sounds simple – right ? Unfortunately, there is a lot more work involved if you want to build this sphere using LEGO bricks. For one, the sphere generator just shows blocks which are equivalent to 1×1 LEGO bricks but you know these bricks just can’t be stacked the way you see in the 3D view. You need to convert the 1x1s into longer bricks that form an interlocked structure that holds together well. You also need to make the walls of the sphere at least 2 studs deep, allowing each successive layer (which is a smaller circle as you move away from the middle layer) to rest on the layer immediately below it. There is one last hitch – if you replace all the blocks in the Minecraft sphere with 1×1 LEGO bricks you will not end up with a perfect sphere. Your sphere will be a little taller than it is wide – just like a 1×1 brick. Thankfully there is also a Plotz Ellipsoid Generator available on the same site and you can use that and compensate for the shape of a 1×1 brick.

For the sphere I built using LEGO bricks, I used Plotz to create an ellipsoid with a width and depth of 36 units and a height of 30 units. Why did I pick a shape that is a little squat compared to a perfect sphere ? Notice that the ratio of height to width of the ellipsoid is 5/6 which is the inverse of the proportions of a 1×1 LEGO brick (which has a height to width ratio of 6/5). And so when we use LEGO bricks which are taller than they are wide, the proportions even out and we end up with a perfect sphere.

The sphere looks good but it is a little blocky. Is there a way to smoothen the curves – perhaps by using LEGO plates which are 1/3rd as tall as bricks ? We go back to the Plotz Ellipsoid Generator and this time the numbers we use have to reflect the shape of a 1×1 LEGO plate with a height to width ratio of 2/5. I used a height of 50 units and a width of 20 units which gives us a height to width ratio of 5/2. This way when we replace the Minecraft blocks with 1×1 LEGO plates (combined into longer plates as needed) we get a perfect sphere. Looking at this sphere from the side, we can see a definite improvement compared to the earlier sphere. The curves are much smoother thanks to the smaller gradations achieved by using plates instead of bricks. But the limitations of this approach are quite apparent when you turn the sphere and look at it from the top. From this view, the curves once again appear blocky. As it turns out, using plates to build the sphere only makes the curves smoother in one of the three dimensions. Another downside to spheres built by stacking bricks or plates is that the undersides of the bricks and plates are visible when you look at the bottom side of the sphere.

Is there a way to achieve smoother curves in all 3 dimensions without any of those pesky undersides of plates being visible ? If we could take just the top part of this sphere that is smoother and somehow use it in all 3 dimensions (all 6 sides), we would have a sphere that looks smooth all around – correct ? That is exactly the idea that AFOL extraordinaire Bruce Lowell had back in 2002. His invention sort of revolutionized the construction of round shapes (not just spheres) using LEGO, earning him the distinction of being the only AFOL (I know of) to have a building technique named after him – the Lowell Sphere !

Lowell Spheres

A Lowell sphere consists of a SNOT cube (with studs in all directions) with 6 identical curved panels (built using LEGO plates) attached on all 6 sides. Each panel is longer than it is wide allowing the 6 panels to interlock perfectly without any visible gaps. Bruce Lowell’s original Lowell Sphere had a diameter of 6.8 studs and it had an inner core that was 4x4x4 studs wide. But in the years that followed, this technique has taken on a life of its own, finding applications in not just spheres but other complex sculptures as well.

While the original 6.8 stud wide sphere is pretty easy to figure out, how do you apply this technique to build bigger spheres ? Another AFOL who has been key to making this technique accessible to everyone is Bram Lambrecht. He developed Bram’s Sphere Generator which allows you to create a Lowell Sphere with any diameter that you need. Just enter the diameter (in increments of 0.2 studs), tweak a few settings and you are ready to save a Ldraw file that can be imported into stud.io. There is an option to use half stud offsets (jumper plates) to get more detail but I have not found that to be practical for larger spheres. One very useful setting is “use alternating layer colors” and with this selected, different colors are used for the layers of plates that make up each of the 6 identical panels. You may wonder why that is useful. The Ldraw file has sub-models for the 6 panels. But when you release (ungroup) one of these sub-models, you will see that each panel is built entirely out of 1×1 plates. So there is some work involved here, going layer by layer and replacing the 1×1 plates with bigger ones. You have to pick the plates in such a way that the entire panel holds together as one unit (by ensuring that the seams between plates don’t line up between successive layers). The good news is that once you have worked on one panel, you can save it back as a sub-model and use that for all the 6 sides. The Ldraw file also doesn’t include the core and so you will have to build that as a SNOT cube with studs in all 6 directions (just a handful of connection points for each of the 6 panels is usually sufficient).

For the rounded dome of the Taj Mahal, I created a Lowell Sphere with a diameter of 27.2 studs. This is made up of a core that measures 16x16x16 studs and 6 curved panels that are each 14 plates thick (so the diameter is 16 studs + 28 plates which is equivalent to 16 + 11.2 = 27.2 studs). Clearly, I didn’t need the bottom part of the sphere and so I had to truncate it. I ended up removing the bottom panel of the sphere entirely and reducing the height of the core by 2 studs to 14 studs. I also had to crop the bottom portions of the curved panels on the 4 sides (since the panels are oriented two different ways, I had to create two variants of the cropped side panels). The bottom portion of the dome which sits on the drum ended up having a diameter of 16 studs + 22 plates = 24.8 studs which was very close to the outer diameter of the drum (25 studs).

The concept of the Lowell Sphere can be extended to other shapes. Lsculpt also developed by Bram Lambrecht, allows you to convert a 3D model into a LEGO sculpture that is structured much like a Lowell Sphere with a SNOT core (that is not necessarily a cube) that has curved panels attached in all 3 dimensions. Here’s a heart sculpture that I built digitally using Lsculpt.

Conclusion

There are several other ingenious ways of creating round shapes using LEGO that I have not been able to cover here. I am hoping that this post at least gives you a starting point for your own exploration into some of these techniques. I would welcome any questions or suggestions that you may have. Happy building !