Pythagoras and Legos

I think it's safe to say that when we've all started out building with Lego bricks, we've all stuck to the rectilinear grid that's sooo~ conveniently provided by the baseplate (or you had a really dull childhood and didn't build with legos).  Even some of us adult fans of lego are guilty of "sticking to the grid," but today is the day to "Think outside of the block!" (Lego pun #1)

When you sit down for some (as Sheldon puts it) Lego Fun Time, think of each block as it's own right-angled coordinate system of integer-spaced connection points.  Thinking this way, you remember from your really cool high school geometry class that there was this old "stud" (pun #2) named Pythagoras who had quite a bit to say about integers and right angles. It's all conveniently summed up as Pythagorean Triples; that is, sets of integers (a,b,c) that satisfy the equation below.  For your reference, a and b (the two smaller sides that are at a right angle to each other) are called "legs" and c is the "hypotenuse"

Most people are probably thinking "oh man, it's getting mathy, that's about as boring as sorting pieces!" Well, don't be a blockhead (pun #3).  It's easy math, and I'm not asking you to solve equations; you can just look up lists of available Pythagorean triples and use whatever one is convenient for what you're building.  So, what does an old Greek dude have to do with little plastic bricks? Well, Legos only come in integer-sizes, unless, of course, you've cut your pieces >:( , Pythagorean triples are a perfect way to build non-rectilinearly!

There are two ways to use these magic triples: creating "Interior" or "Perimeter" right triangles.  I'll start with Interior, since you can roll with an unmodified list of Pythagorean triples.  For these, you'll need at least two "hinge" pieces to form the angles (this or this; there's probably a couple other pieces that'll work, but these are the most obvious to work with).  The whole idea about using Interior triangles is that the Pythagorean triangle is actually the negative space formed by enclosing a region:

In this example, we've used the (3,4,5) triple.  Your enclosing lengths are equal to the leg and hypotenuse values of your triple, counting in "stud units".  Know what'll make you put on a yellow smiley face? (#4ish) Knowing these triangles can scale! You can double all the lengths of any "primitive triple" (Pythagorean triples where each value is coprime to the others), and it'll still work; (6,8,10), for instance.  

As for Perimeter triangles, you don't need any special pieces, just something to bump your hypotenuse above the studs of your baseplate (I used these for the pic below, but really almost an piece will do).  For these, the bricks themselves form the perimeter of the right triangle.  Now, something REALLY important is our length counting system! Normally, when discussing lego piece sizes, we describe sizes something like "2x4", meaning "two studs by four studs", but the values in the Pythagorean triangles are LENGTH measurements, not UNIT measurements, so for the lego versions of the perimeter triangle to work, we need to measure the distance between studs, not the stud count! This example sums it up:

This example uses the (3,4,5) triangle again, but the lengths, in terms of stud count, is (4, 5, 6).  So, the Pythagorean triples you use when constructing must modified by adding 1 to each value to get to the "standard" Lego stud sizes:

Again, this works for any set (a,b,c) of Pythagorean triples, and can be scaled as well.  

So, those are the basic two uses of Pythagorean triples in Lego building.  

But let's not stop there, our brick bucket isn't empty, yet! (I don't think that counts as a pun...) Strictly speaking, Legos aren't restricted to integer values; with "jumper" plates (this and this, for example), we can make use of the half-integer values, too! These are a little harder to work with, especially since all of the commonly used¹ primitive Pythagorean triples have an odd hypotenuse length.  But, it opens up quite a few possibilities. You can take half of each value of a Pythagorean triple (for interior triangles) or a modified triple (for perimeter triangles) for new stud unit sizes.  Below are half-sized examples of the previous two (3,4,5) triangles: 

Half-sized Interior

Half-sized Perimeter

(sorry, camera position parallax distorts them a bit, but they do line up). For clarity, there are two Jumpers in each photo, one in the hypotenuse and another in the vertically-oriented arm.  

So, theory is great, but how do they "stack up" in practice? You tell me! (that totally counts, btw)

This is the apse of my cathedral (Yes, these photos are shameless self-promotion :P ).  It uses two side-by-side (3,4,5) triangles, that might be better seen in the interior shot, where you have the floor tiling to use as guides for where the hinge connections are made.  Also note that you don't have to form a "triangle" using the interior triangle method, you can use the triangle concept to guide where your hinges need to be anchored to the baseplate.

I put together this is a quick 3-minute MOC just for you all, showing a farmer's beat-up, old cow pasture. It shows the use of two perimeter-method triangles; the left-most fence is the (5,12,13) triple, so of course it has a length of 14 studs. That's the easy (and less interesting) one.  The middle patch of fence shows the halving method on a (5,12,13) triple, so it's length is 7.5 studs, and if you count out the stud placement on the ground it comes out to 3.5x7.  Or the arithmetic, if you prefer :

(we're subtracting 1 rather than adding because we're converting from stud units to length units to use in the Pythagorean equation (the first equation in this post), whereas before, we were doing the reciprocal conversion)

So, much better than always working with square, hard edges! In my opinion, at least...

For your convenience, there are eight triples in particular (and their multiples!) that seem to work very well:  

• (3,4,5)
• (5,12,13)
• (8,15,17)
• (7,24,25)
• (20,21,29)
• (12,35,37)
• (9,40,41)
• (28,45,53)

I cut the list off at these eight simply because anything with larger arms begins to exceed the standard 48x48 stud baseplate size.  But don't let me stop you if you want to attach several plates together!

I'll take a moment to mention a deviation from this "pure" geometry: like all manufactured plastic parts, there are some mold tolerances engineered into the bricks, which means you can "fudge" the numbers a bit.  For instance, (19,11,22) and (6,15,16) are both approximate Pythagorean triples, which work within the "give" of the lego parts.  These little number fudges don't make a whole lot of difference on small scales, but they can be a source of serious tension when they add together, and can really affect the structural integrity of your model.

So, go sit down the best you can with single-jointed, plastic legs, and have a geometric time!  Happy Building!

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1 That'd be an interesting proof: do all primitive Pythagorean triples have an odd-valued hypotenuse? I'll give it some thought after I finish this post.^↩

EDIT: Turns out, yes, the hypotenuse MUST be odd for any primitive Pythagorean triple:
According to this paper, theorem 1.2 shows the hypotenuse can be written:

WLOG, let m be odd (m = 2k + 1), and n be even (n = 2l), where k and l are both integers and selected such that m and n are coprime (though, I state this for completeness, the fact they are coprime is inconsequential to our result).  After some algebra, we get:

Which, of course must be odd, for any pair k and l.  QED.