Tag: LEGO math

The Math Behind LEGO Building Techniques – Volume 4

This installment in my series “The Math Behind LEGO Building Techniques” (see Volume 1, 2, 3 here) will include an assortment of items related to topics that have already been covered in this blog or my book. They are mostly updates arising from new terms or new LEGO pieces that have been introduced since the last installment was published.

About the “sugar grid” …

You’ve probably been hearing or reading about the “sugar grid” technique for placing LEGO pieces at angles. This term was coined by Chris Enockson who has an excellent and fast-growing Youtube channel Brick Sculpt that talks about building techniques, among other LEGO-related topics. There’s also a highly informative and well-received article by Arno Knobbe on this topic that was posted on New Elementary. So how does the “sugar grid” fit into the framework of angled wall techniques that have been discussed on this blog (and were covered in my book)? Let’s find out!

Mathematically speaking, all the techniques for attaching LEGO pieces at angles other than 0° and 90° involve right triangles. The most common of these techniques involves placing a LEGO brick or plate along the hypotenuse (the longest side which is opposite the right angle) of a single right triangle. For this to work, the length of the hypotenuse has to be a whole number of studs (see Pythagorean triples) or close enough to one (see near triples). However, since the number of useful Pythagorean triples and near triples is quite small, this greatly limits our options and the angles we can create.

There’s another group of techniques that involves two right triangles that share a hypotenuse. We do not place any LEGO pieces along this hypotenuse, and so its length does not have to be a whole number. Instead, we create angled walls by placing LEGO pieces along the sides that make up the right angles in the two triangles. This opens up many more possibilities because it works for right triangles with any arbitrary (whole number) side lengths.

Two of the techniques covered in this blog (the “mirrored hypotenuse” technique and the “switched diagonals” technique) are essentially based on this concept which Chris and Arno have generalized into what they call the “sugar grid”. To better understand how this works, let’s start with a simple 2×3 plate A and consider the right triangle formed by the studs numbered 2, 3 and 4. The two sides that make up the right angle are 2 and 1 studs long (measuring the distance between the studs 2-3 and 3-4). The length of the hypotenuse (2-4) is √22+12 = √5 which, of course, is not a whole number. Next, take a second 2×3 plate B, and consider the right triangle formed by studs 1, 2 and 3. The two identical (but mirrored) right triangles have hypotenuses (2-4 and 1-3) that are identical in length.

So you should be able to rotate plate B such that its hypotenuse 1-3 lines up with the hypotenuse 2-4 on plate A and attach the two 2×3 plates together using 1×1 round plates as spacers. There are in fact, 4 identical right triangles that share the same hypotenuse; two on the original 2×3 plate and two on the rotated 2×3 plate. The angle of rotation is 53.13° which is twice the smaller angle in each right triangle.

This gives you another way to create angled walls without limiting you to Pythagorean triples. You’re still limited to the angles that can be created for various whole number side lengths of the right triangles, but there are many more options available. Arno’s article mentions Legal LEGO Angle Finder which is a tool that helps you pick the right plate sizes to use for the angle you want to create. It’s also possible to stack plates in multiple layers, each with their own rotation, for even more options.

Now, imagine the triangle from the first 2×3 plate repeating across the entire LEGO grid. We can again call this grid A. Similarly, grid B would have the triangle from the second 2×3 plate repeated.

It should be possible to rotate grid B and overlay it on grid A such that the studs at the ends of the hypotenuses of the right triangles line up. The rest of the studs on grid B will not line up with the studs on grid A, but the studs at the ends of the hypotenuses can serve as connection points between the two grids. This sparser grid of connection points allows LEGO elements to be attached in one of two orientations (aligned with either grid A or grid B).

This is the concept of the “sugar grid”. As expected, as the sides of the right triangles increase in length, the “sugar grid” gets sparser (the number valid connection points decreases as a proportion of the total number of available studs in each grid).

So where does the term “sugar grid” come from anyway? Apparently its origins are from Minecraft, which isn’t something I am very familiar with. But I was curious to find the connection anyway. Here’s what I was able to glean from a little bit of research (more experienced Minecrafters can feel free to correct me). In Minecraft, sugarcane farming is important because sugarcane is a valuable resource that can be used to create a variety of other things. Sugarcane obviously needs water to grow.

A common challenge in sugarcane farming is to place sugarcane blocks (represented by the green squares) in the most efficient way such that each one is adjacent to a water block (represented by the blue squares). This can be achieved by tiling any given area with a plus shaped pattern with a water block in the center surrounded by sugarcane blocks on all 4 sides. This way, you end up with the water blocks placed in a grid that resembles the grid of connection points that we saw earlier (the image below shows the sugarcane grid recreated using LEGO 1×1 tiles). Hence the name “sugar grid”. The connection may be tenuous, but I can’t disagree with the fact that “sugar grid” is a catchy and intriguing name to describe the concept.

As Arno shows us in his article, the concept of the “sugar grid” doesn’t limit you to just regular studs-up building. In a clever example, he shows how it can be extended to sideways building aka SNOT as long as you can get the sideways studs to be spaced correctly to conform to the “sugar grid”. The studs here again form the hypotenuses of right triangles with side lengths of 2 and 1 studs.

Arno’s article also mentions an official LEGO set (The Lord of the Rings: Rivendell 10316) that utilizes a “sugar grid” connection. Are you aware of any others?

Technic connector #2.5?

In the most recent volume of The Math Behind LEGO Building Techniques, I had covered the family of Technic angled connectors #1 through #6 that allow you to connect axle pieces at various angles ranging from 90° (#6) to 180° (#2). These connector pieces have been in the LEGO catalog since the late 1990s. So it was quite surprising to see a new member joining this family in 2024 – Technic Axle and Pin Connector Angled #7 – 168.75° (4450). This part has only been used in a handful of sets so far. I have not studied the instructions for these sets to see what the compelling reason was to create a new connector, but it’s definitely an interesting piece.

The one thing that doesn’t make sense with this new connector is the number (#7) assigned to it, and I would even argue that the number should have been #2.5. Confused? If you take a closer look at connectors #2 through #6 and the angles they create, you will see that these angles are separated by increments of 22.5° (which is a quarter of 90°). In fact, for all the connectors except #1 you can get the angle that they create using the formula 225° – n x 22.5° where n is the connector number. If you use this formula for the angle created by the new connector (168.75°), the value of n comes out to be 2.5. But I guess it was easier for LEGO to just name it #7. Here’s an updated table that shows the Technic connectors including connector #7.

NumberPart NumberAngle
132013–
232034180°
332016157.5°
432192135°
532015112.5°
63201490°
74450168.75°

While the angle 168.75° may seem quite random, it really isn’t. It happens to be the interior angle of a regular polygon with 32 sides. You can determine this by solving for n (the number of sides) the equation for the internal angle (n-2) * 180° / n = 168.75° (see more about polygon geometry here). If you use 32 of these connectors and axles you can create a much smoother approximation of a circle than is possible with the other connectors (especially the #3 connector used to create a 16-sided polygon in the Globe set I covered in the article). The resulting diameter is of course, going to be bigger. So is it possible to use the new connector to create a bigger version of the Globe set? The first step would be to create a frame which as it turns out, has a diameter of 64 studs (see the image). It will be a lot more tricky to create the outer skin using wedge plates for this bigger version, but it is something I will be looking into. Stay tuned!

A few new slopes

LEGO continues to expand their catalog of slope pieces and a few recent additions have been particularly noteworthy. They are 3 slopes that are 2, 4 and 6 studs long with no studs on top. The three new additions to the slope family are the 2×1 slope (#5404) that is 2 plates tall, 4×1 slope (#5654) and 6×1 slope (#4569), the last two of which are 3 plates (or 1 brick) tall. If the new 2×1 slope looks familiar, that is because a version with a grille (#61409) also known as the “cheese grater” has been available since 2008. LEGO finally decided to give us a version of this slope piece without the grille.

So what angles do these new slope pieces create? Like most other slope pieces, they have a lip at the bottom of the slope that is half a plate high. The 2×1 slope rises approximately 2-0.5 = 1.5 plates over 2 studs which makes its angle arctan(1.5/5) = 16.7°, but the official name of the slope from Bricklink rounds this off to 18°. This slope also has a groove at the bottom (much like the 1×1 cheese slope) which makes it easier to pry the piece off without ruining your fingernails. Looking at the new 4×1 and 6×1 slopes, they rise 3-0.5 = 2.5 plates over 4 and 6 studs respectively making their angles arctan(2.5/10) = 14° and arctan(2.5/15) = 9.46° respectively (the latter of which is rounded off to 10° in the official Bricklink name).

The half plate lip means that you have to resort to some SNOT to get a smooth slope using two or more of these elements.

Here are all 3 types of slopes side by side showing the subtly different angles they create.

But one interesting side effect of the half plate lip is that the 4×1 and 6×1 slopes have a sloping portion that is 2.5 plates high which make the slopes line up almost perfectly with the 4×2 (#41769, #41770) and 6×2 (#78443, #78444) wedge plates attached sideways.

LEGO has already used these new slopes to great effect in several official sets including Marvel Logo & Minifigures 76313 where all three slopes are used to create the letters in the logo. I am sure MOC builders will also find these new slopes to be a useful addition to their arsenal.

Addendum: Luca Hermann (a very talented builder from Germany) pointed out that in the new 4×1 and 6×1 slopes, the “half plate lip” is a little taller than half a plate. It’s somewhere between 0.25 and 0.3 studs (0.625 and 0.75 plates). This would make the angle of the 6×1 slope more like 9° and may explain the slight stair-stepping that is seen when we use SNOT to try to create a smooth slope using two or more of the 6×1 slopes. Luca has verified this by using 40 of these slopes to create a perfect circle (40 x 9° = 360°) in a recent MOC (shown below is the round base of the Washington Monument that he designed).

Until the next installment, happy building!

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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 √(6² + 2²) = 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 = √(a2+b2) = √2 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 !

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