LEGO® Pythagorean triangles: Escaping the grid

Arno Knobbe (@legoarno on Instagram) completes his popular trilogy of guest articles today, gently introducing you to the math behind useful techniques for building at diagonals to the LEGO® grid. If you missed them, you may wish to read the others first – Part 1: building at angles, and Part 2: techniques with reflected wedges.

Building a triangle naturally forces you to escape the LEGO® grid. Sure, you can align 2 sides with the grid – creating a right triangle – but the third side will have to be diagonal. As soon as you try this, you run into trouble: most dimensions will not lead to easy connections at the 2 corners involved, neither with hinge plates nor regular stud-antistud connections. The age-old answer to this challenge, that most builders will be familiar with, is the Pythagorean triangle. Today’s article is my modern take on this millennia-old technique.

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As good-old Pythagoras found out in ancient Greece, you can compute the length of the diagonal in a right triangle (the hypotenuse) using the following formula:

$c^2=a^2+b^2$

It turns out that for certain combinations of a, b, and c, all 3 numbers are integers (whole numbers); these are the so-called Pythagorean triples.

Count the number of white and grey tiles in these two examples, and using the formula above, you’ll find that these triangles are indeed Pythagorean. But try some random numbers for a and b, and you’ll notice that Pythagorean triples are actually not that common: the hypotenuse tends to be irrational (fancy word for a right bother), which in MOC-building terms doesn’t get you very far.

So it pays off to familiarize yourself with a number of Pythagorean triples of modest size, such as these:

• (3,4,5)
• (5,12,13)
• (7,24,25)
• (8,15,17)

These triples are primitive, meaning they are the smallest member of families of triples produced by multiplying with an integer. For example, the Pythagorean triple (6, 8, 10) is (3, 4, 5)’s bigger brother, and hence not primitive.

At this point, you’re probably thinking ‘I know all this, where’s the good stuff?’. So indeed, let’s take it up a notch. Consider the (3, 4, 5) triangle shown below.

The blue side of the right triangle is 3 modules long, the white side 4 modules. The hypotenuse is 5 modules long, such that we can easily connect either end to the other sides using hinge plates. This sounds pretty obvious, but as explained, in most right triangles, the hypotenuse is not a well-behaved integer. So this is not your average right triangle.

I’d like to point out 2 rectangles underneath the diagonal plate; the light grey one being a 1x2 jumper that lines up with 1 of the red plate’s bar ends, and the 1x3 dark grey one consisting of 2 tiles and a 1x1 plate in the middle. Its stud lines up with an antistud on the bottom of the white tile. Two corners of each rectangle line up with grid corners of the red plate, so we have 2 instances of the rectangle trick here, one of ratio 1:2 and one of 1:3.

Since the red plate is now attached in 2 places, the hinge plates are no longer needed. What this leaves us is 2 locations where the grid corners of the diagonal plate coincide nicely with the bottom grid. Actually, there’s a third such location, right where the 2 grey rectangles touch (we’ll need this later on).

The angle α of the lower right corner can be computed from the definition of the tangent function: 

tan α = 4/3, so α = arctan 4/3 = 53.1°

For the opposite angle β, an analogous computation tells us:

tan β = 3/4, so β = arctan 3/4 = 36.9°

I previously pointed out this structure in the 10305 Lion Knights’ Castle set, but didn’t lift the veil entirely. So now you know the whole story: there’s a Pythagorean triangle hidden beneath this elegant tangram of wedge plates.

It won’t surprise you at this stage that there is a prevailing sugar grid of ratio 1:2 (and dual 1:3).

This is how the 1:2 sugar grid plays out, when you align the white offset studs with the corners of the triangle as explained in the last episode. One-wide plates don’t make much sense in the context of sugar grids, but bigger plates rest securely on the grid:

Other ratios

We’ve seen how reflected wedges of ratios 1:2 and 1:3 fit the (3, 4, 5) triangle quite nicely. It turns out that for any ratio of reflected wedges, you can find a Pythagorean triangle with the same angle, such that the reflected wedges can act as a form of cover. 

For example, here’s a Pythagorean triangle covered by 1:4 wedges.

You may have to repeat the basic ratio a few times – in this case four times – but for every ratio, there is a corresponding Pythagorean triangle.

Here’s how you construct the appropriate triangle. Starting from a ratio 1:m, construct the three sides as follows:

• m² − 1 is the number of modules on the first side.
• 2m is the number of modules on the second side.
• m² + 1 is the number of modules on the hypotenuse.

Given this, here’s how we arrive at the dimensions for 1:4 wedges in the image above:

• m² − 1 = 16 − 1 = 15 modules for the first side.
• 2m = 8 modules for the second side.
• m² + 1 = 16 + 1 = 17 modules for the hypotenuse.

The Pythagorean triple thus becomes (15, 8, 17), which is indeed the right triangle used to form the walls in the figure (note that on each of the 3 sides, the wall adds another module thickness).

Don’t have a calculator on hand? Here’s a table showing the triples corresponding to the common wedge ratios. In some cases, the actual triple produced isn’t primitive (it’s too large by a factor of 2). Simply divide those by 2.

Ratio	Angle	Double angle	m2 – 1	2m	m2 + 1	Triple
1:2	26.56°	53.13°	3	4	5	(3, 4, 5)
1:3	18.43°	36.87°	8	6	10	(4, 3, 5)
1:4	14.04°	28.07°	15	8	17	(15, 8, 17)
1:6	9.46°	18.92°	35	12	37	(35, 12, 37)

Scroll back to the top, and you’ll recognize 2 of the triples in this list. The remaining two correspond to the 1:5 and 1:7 ratio, which unfortunately don’t come as wedge plates (TLG, are you reading this?). 

Here’s what these look like in bricks. 

With most of these “Star Destroyers”, there is a bit of a gap in the upper right corner, so you need to get creative to cover this up nicely. Since the dark grey wedges are angled relative to the right-hand side, you’ll need some tiling (shown in blue) to accommodate those.

The triangles have a roof built from reflected wedges of a single ratio, but as we saw with the (3, 4, 5) triangles, you can cover each of the two angles with reflected wedges of the two ratios of its corresponding sugar grid (1:2 and 1:3 in the case of (3, 4, 5)).

This idea applies to any of the other Pythagorean triangles, provided parts for the appropriate ratio and its dual exist. From the previous episode, you’ll remember the 1:4 ratio and its dual 3:5, produced from cheese slopes tipped on their side, like so (I expect a lot of Czech likes here!):

The 1:4 interface lies between the white and red parts, the 3:5 between the red and blue parts. (And yes, the angle between white and blue is 45°, corresponding to the 1:1 ratio of the right corner.)

The dual of the fourth common ratio, 1:6, is 5:7, which unfortunately doesn’t come as any wedge or slope that I’m aware of.

Extended Pythagorean triples

Let’s take it up another notch and revisit our earlier Pythagorean triple with the 2 tricked rectangles.

As pointed out earlier, at the top left corner of the dark grey 1x3 rectangle (where it touches the light grey one), a grid corner of the red plate coincides with a grid corner of the base plate. That means that the lower 3 modules of the red plate and the dark grey rectangle form a rectangle trick. 

Now mentally copy that bit of 1x3 rectangle trick to the top end of the red plate, above the white plate, like so:

We end up with something that’s not a Pythagorean triangle (the distance between the 2 far corners is not an integer), nor is it a rectangle trick (the bottom rectangle has dimensions 4x7, while the red plate is now 1x8).

So what is it? TLG calls it a ‘Pythagorean pair’ (see building instructions of the 10316 Rivendell set, part 3, page 88), which doesn’t entirely convince me. I think Pythagoras was concerned with integer diagonals. Although the diagonals of these 2 rectangles are equal (which is quite remarkable in itself) they’re not rational, let alone integers:

$\sqrt{4^2+7^2}=\sqrt{1^2+8^2}=\sqrt{65}$

 

This extended Pythagorean triangle can be found four times in the LEGO® Star Wars set 75290 Mos Eisley Cantina, as reviewed by Thomas Jenkins.

The Arch 1 x 8 x 2 Raised (16577) forms a 1x8 rectangle that spans the 4x7 void between the two walls. The arch is held in place by 2 hinge plates (colored white above, for contrast) and otherwise just rests on 2 tiles.

In the same way, you can copy the light grey 1x2 rectangle trick to the other side, and you get an even bigger extended Pythagorean triangle. This triangle happens to be Pythagorean, just not a primitive one. With sides (6, 8, 10), it’s double the size of the original.

Clearly, you can continue this extension in either direction as many times as you like, simply alternating between 1x2 and 1x3 rectangles. What you’ll get is a diagonal bar of arbitrary length that is not necessarily Pythagorean, but has ample attachment points in the form of studs beneath it, as well as an abundance of coinciding grid corners on either side of the diagonal plate where you can attach hinges. The example above already has 5 such grid corners, as well as 4 studs.

The same extension trick can be applied to the (15, 8, 17) triangle that’s based on the 1:4 and 3:5 ratios (above). In the red-blue corner, you can recognize a 1x4 rectangle trick that can be copied to the red-white corner to extend the red side by 4 modules.

And at the red-white end, you can identify a 3x5 rectangle trick, although you first need to extend the red side by 2 rows of studs to make the edge 3 modules thick. Again, you can copy the 3x5 rectangle to the opposite (red-blue) end to add another 5 modules on this side:

Slope 53

In the realm of LEGO® Technic, several parts make good use of the attractive properties of Pythagorean triples, but as far as I’m aware (please prove me wrong), there’s only one LEGO® System element that has Pythagorean dimensions: Slope 53 3 x 1 x 3 1/3 with Studs on Slope (6044). It’s an oldie, but a goodie. So good in fact, that I can’t believe it has never featured in a New Elementary article, despite its iconic status amongst hard-core AFOLs. It’s a pleasure and honor to correct this omission today.

Slope 53 quite literally forms a (3, 4, 5) Pythagorean triangle with its base being 3 modules long and its side 4 modules high. Note that due to the unusual proportions of LEGO bricks, 3 modules high is actually equal to 3 bricks and 1 plate high. Along the tilted face of this slope, you find the expected 5 studs.

Slope 53 was in circulation from 1992 to 1999, and appeared in 14 sets. As a result, it’s currently quite rare and comes in only 3 frequent colors: black, yellow, and white (plus a couple very rare ones not found in sets). 

During its 7-year lifespan, Slope 53 proved to be an easy way to prop up elements at an angle of 53.1°, for example to form a simple slide, the back of a deckchair or an awning, but not much more than that. 

With the introduction of several new types of hinges over the years, such an angle could be easily constructed from basic parts, making Slope 53 obsolete for this specific purpose. Still, due to its unusual geometry and row of studs along its slope, it has remained popular among AFOLs. 

©2020 Nick Trotta, reproduced with kind permission

One of its biggest proponents, spaceship builder extraordinaire Nick Trotta, uses excessive quantities of the slope in his Heavenly Strike ship. The left wing alone features a whopping 9 of them! For a breakdown of the interior structure, visit Nick Trotta’s Heavenly Strike page.

Since the early retirement of this remarkable element, after a less-than-remarkable career, the LEGO System landscape has changed drastically, and Slope 53 now affords a number of neat tricks involving SNOT bricks, which only became available well after the turn of the century. Along with the two grids below and above the slope, you can imagine two SNOT grids that service the side of the element, one of the grids being tilted by 53.1°. After this observation, and being familiar with the geometry of (3, 4, 5) triangles, it’s only a small step to the following construct:

The grids below and to the left of the slope are constructed from Brick, Modified 1 x 4 with Studs on Side (30414), introduced one year after Slope 53, and Brick, Modified 1 x 1 x 1 2/3 with Studs on Side (32952), introduced some 16 years later. 

The tilted grid on top requires a little extra attention, since the studs need to have a “low” position, compared to regular SNOT studs that are “high” (just compare to the 4 studs at the bottom). I used five ‘D-SNOT’ bricks (3386, born in 2023), but there are other options, including a good-old headlight brick (4070) on its side.

Just imagine what wonderful creations TLG designers would have produced if Slope 53 and SNOT bricks would have coexisted. Let’s investigate some more options.

Perhaps there’s a sugar grid here? Based on the 1:2 sugar grid’s orientation relative to the (3, 4, 5) triangle, we can plot out the relevant SNOT studs below the slope:

A rotated plate can now be attached to the upright bricks, and it also lines up perfectly with the row of 5 studs on top. If nothing else, this plate (or smaller variations) can be used to lock the slope in place.

In a similar way, there is a SNOT sugar grid located above the slope.

It just baffles me how a 25-year-old part can still buddy up with pieces introduced years down the line, in ways that I’m quite sure weren’t anticipated by the original designers. It’s a testament to the rigor of the LEGO System and its mathematical underpinnings that in LEGO building, you can teach an old dog new tricks!

Given Slope 53’s untapped potential, I think the conclusion is inevitable – TLG should give Slope 53 a second chance!

Conclusion

Today’s discussion on Pythagorean triples concludes this series of forays outside the grid (for now…). We’ve reviewed 4 crucial techniques, 3 well-established and 1 fairly recent innovation (the sugar grid), and I hope I’ve convincingly argued that these 4 techniques are all manifestations of the same phenomenon, with the sugar grid being the glue that binds these techniques together. 

Being privy to the many links that exist between the core LEGO angling techniques should allow you to quickly switch between techniques as you see fit. Imagine building a magnificent palace with a grand ballroom angled according to a ratio of your choice. The floors represent the reference grid, with attractive tiling, while the walls form a Pythagorean triple. To support the reflected wedges of the roof, you place elegant columns in the ballroom according to the sugar grid. Rectangle tricks help to extend the diagonal wall outside the palace for some more architectural showboating.

Despite these sibling techniques forming a wonderful, unified framework, there is a flipside. After toying with some of these techniques for a while, you’ll notice the same angles showing up again and again, most notably 53.1° (related to the 1:2 grid) and 36.9° (1:3). Unless you are willing to build at an excessive scale, the number of unique ratios is quite limited, as are the corresponding angles. If you want to get beyond the set of “sugar angles”, you should apply 2 layers of angling, for example using the nifty Legal LEGO® Angle Finder tool described in the first episode.

With this last adventurous and mildly mathematical excursion outside the LEGO grid, it’s time for me to say goodbye. Thanks for your time – it’s been fun! Happy building.

Our thanks to Arno for this fascinating series! Be sure to follow @legoarno on Instagram for more of his insights and let him know if you build anything with them.

 

READ MORE: Tom Loftus' incredible MOCs using new parts on Pick a Brick

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