## 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 !

## 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 !