Note: An updated version of this post has been published here.
If you look at a LEGO baseplate, you will see that the studs on it are arranged in a regular grid with any two neighboring studs exactly 0.8 cm apart (measured center to center). However, this is true only if you are measuring the distance parallel to one of the sides of the baseplate. If you were to measure the distance between two studs that are next to each other diagonally, the distance would be slightly longer (√2 x 0.8 = 1.414 x 0.8 = 1.13 cm to be exact). Herein lies the problem – you can place bricks every which way on the baseplate as long as they are parallel to one of the sides of the baseplate. But as soon as you turn the bricks to an angle different from 0 or 90 degrees, the studs no longer line up.
The trick to placing bricks at a different angle (with good stud connections at both ends) is making sure the triangle you form satisfies the Pythagorean Theorem. Most people haven’t encountered this since high school math, but it basically states that in a right angled triangle, a2 + b2 = c2 where a and b are the adjacent sides that make up the right angle and c is the hypotenuse (or the longest side that is opposite to the right angle). The smallest Pythagorean triple (set of 3 numbers that satisfy the theorem) is 3, 4 and 5 (and that is because 32 + 42 = 9 + 16 = 25 which is equal to 52).
How does this Pythagorean triple (3, 4, 5) translate to LEGO ? Basically, if you place your brick or plate at an angle such that it forms a triangle (3 studs long on one side and 4 studs long on the other), then the studs at the two ends of your brick or plate (but not the ones in the middle) will line up perfectly with the studs directly below. Then, does it make sense that we are using a 1 x 6 plate in the first picture ? Yes, when you consider the fact that the sides of our triangle actually intersect at the centers of the studs, the distance that matters is between the studs at the two ends of the 1 x 6 plate which is 5 studs. The same applies to the other two sides. However, when we use hinge plates to create our triangle, the longest side has 5 studs instead of 6 (because the sides of the triangle now intersect at the corners of our plates).
What are some other Pythagorean triples ? 5, 12, 13 is the next one before we get into much bigger numbers that are not very useful. If we limit ourselves to Pythagorean triples, there are not many to pick from (especially with practical applications in LEGO builds). But if we use LEGO hinge plates to build our angled walls, the little bit of give that these have, allow us to use triples that are not Pythagorean triples (strictly speaking) but are close enough. Most angled wall applications call for the walls to be at 45 degree angles which means the two sides (a and b) of our triangle have to be of equal length. There are no Pythagorean triples where a and b are equal but there are many that are close enough. Consider for instance 5, 5, 7 and 7, 7, 10.
I happened to use 5, 5, 7 to create the angled corners in the base of my model of the Hearst Tower.
Pythagorean triples also work if you multiply all the numbers by 2 or halve them. If you use 7, 7, 10 as an example of something close enough to a Pythagorean triple, then 14, 14, 20 and 3.5, 3.5, 5 should also work. But how do we create a triangle in LEGO with 3.5 studs on its side ? Using jumper plates, of course !
I used angled walls extensively in my model of the Tribune Tower. Here the main tower used 5, 5, 7 for the angled corners. For the octagons in the crown, I used 7, 7, 10 (for the smaller one) and 8.5, 8.5, 12 (for the bigger one) to create the angled sides. The flying buttresses also needed to be attached at an angle (though not 45 degrees) and for that I used another triple that was close enough – 4, 7, 8.