LEGO® building at angles: Escaping the grid

Today we are excited to publish a guest article from Arno Knobbe (@legoarno on Instagram), an Associate Professor at Leiden University in the Netherlands who attended our building workshop at Skærbæk Fan Weekend 2024. As a fan of both numbers and LEGO® bricks, he is perfectly qualified to tell you everything you ever wanted to know about building “off-grid” with LEGO parts, but were too afraid to ask because it would probably involve math... Fear not! With Arno’s beginners’ introduction to LEGO building at angles and diagonals, it's easy.

When building with LEGO, we usually adhere to the familiar structure of the standard square grid, where everything aligns neatly and predictably. This grid system provides stability and order. However, there are moments when creativity takes over, and we decide to break free from the constraints of this rigid structure. Either because some angled part needs to be matched, or simply because the diagonal bit breaks the monotony of building in “the Matrix”. The LEGO universe offers several techniques for escaping the LEGO grid by building at an angle, which I will be taking you through today.

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Rectangle technique

The most familiar off-grid technique is perhaps the following rectangle trick:

If you take any rectangle, such as the red plate here, and rotate it around its center (marked by the white round plate), you will find that at a particular angle, two of the opposite corners end up where the other two corners used to be. As a result, you can easily attach the plate to parts in the original grid. 

In this example, the red plate used to be right between the two dark tan bricks, and is now attached using hinge plates, locking the plate at a particular angle to the original grid (53.1° to be precise). 

Although fixed, the structure is still quite wobbly, but this is easily resolved by a couple of tiles as support underneath the plate.

The two hinge plates look a bit unusual, and most people will be more familiar with the following:

This is the same rectangle trick using studs at the two corners (I added blue round plates at the top of the rectangles to show where studs at the bottom are being fixed). As these three examples demonstrate, the angle achieved depends on the dimensions of the rectangle, that is, on the number of horizontal and vertical modules that separate the two opposite corners. 

For example, the left-hand rectangle builds on two studs that are separated horizontally by one module and vertically by two. The ratio that determines the angle is thus 1:2. Note that these numbers are less than the outer dimensions of the plate, which are 2x3. That’s because we are using studs, instead of the corners of the plate. So to understand the trick, you need to imagine a slightly smaller rectangle of 1x2 inside the 2x3 plate. In the same way, the other 2x4 and 2x6 plates have ratios 1:3 and 1:5.

The rectangle trick is a versatile tool that has many applications, even in less obvious contexts such as these:

The left-hand example (the smallest instance of the rectangle trick I have come across) works because the diagonal half of the hinge plate is rotated around the center of the white jumper plate. Note that this jumper plate has a hollow stud that lines up nicely with the bit of bar end underneath the hinge plate. The ratio is again 1:2, since the hinge and jumper plate are 1x2. 

The righthand example shouldn’t work, since squares are not strictly rectangles. But as the blue round plates indicate, we’re using only the 2x6 middle strip for the trick (1x5 modules, to be precise), and the remainder of the square just tags along.

An interesting question is what different angles can be produced using this trick. The New E editor forbade me to use math, but heck, here’s a formula anyway. Maybe he won’t notice. 

Here’s how you can calculate the angle achieved for a rectangle with a ratio of a:b:

α = 2 x arctan(a/b)

a/b is simply the ratio between the number of studs a and b, and arctan is “simply” a function that translates ratios into angles. You can find that function on any scientific calculator, or you can just google it (e.g. “arctan(1/2) in degrees”). 

Applying the formula to the 2x3 plate example earlier, we get 2 x arctan(1/2) = 53.1°. The same number applies to the jumper example, but also to the very first example with the 4x8 plate. 4/8 is the same as 1/2, so the angle is the same, 53.1°. This demonstrates that different size plates may produce the same ratio, and thus the same angle. 

So can we achieve arbitrary angles? Well, not really, since the range of ratios/angles is quite limited, if you’re not going to use ridiculously large plates. For example, for plates up to 8x8, the number of available options is only 11. This means that the resulting angle using the rectangle trick can be quite different from the desired angle; up to 8° for shallow angles, which results in a noticeable gap in your builds.

Two-layer technique

Can we do better? Yes, if we are willing to use two instead of one layer of the rectangle trick. 

The first layer, as we know, produces a certain rotation in one direction. A second layer on top of that then produces an additional rotation, either in the same or opposite direction. Adding or subtracting the individual angles produces a much larger range of possible angles, up to about half a degree of error from the desired angle. 

A handy online tool, the Legal LEGO® Angle Finder, produced by math-savvy LEGO enthusiast joeythegreat711, lets you pick a target angle and then suggests the two layers of rectangles to achieve that angle (approximately). 

Legal LEGO® Angle Finder, by joeythegreat711 on desmos.com

The example shown here demonstrates how to achieve an angle very close to 45° (a challenging angle in LEGO building). 

You start off with the green imaginary 6x9 base plate. The blue plate on top of that then produces an angle of 2 x arctan(5/8) = 64.0°. On top of that, place a red 2x7 plate to rotate back by 18.9°, achieving a resulting angle of 45.1°. Not bad. Of course, 6x9 and 2x7 plates don’t exist (yet?), so you should use slightly larger plates and ignore the surplus.

Examples of using the Legal LEGO® Angle Finder

To test the tool a bit, I gave myself a few assignments. First off, could I match the angle of an arbitrary wedge brick, the Wedge 4 x 2 Left (41768)?

The business end of this wedge has a 1:4 ratio, which corresponds to an angle of 14°. So, type in 14° in the Legal LEGO Angle Finder tool, specify a maximum size of 4 studs, and read off the suggested two layers. 

The first layer starts with a 3x4 rectangle, so I take a white 4x4 plate and apply the trick (a bit tricky indeed, but I manage).

The next layer requires a 2x3 plate, but I want it to be a bit larger to reach the target wedge. The top layer can start anywhere on the middle layer, but in this case, I reuse one of the existing pivots. I add the top layer (a red 4x6 plate) and find that Bob is indeed my uncle.

Second assignment: match the 60° angle of a "Nexogon" (Plate Special 6 x 6 Hexagonal with Pin Hole, part 27255) – or 30°, depending on your point of reference. 

The tool starts with the same suggestion for the first layer, a 3x4 plate. The second layer also involves a 2x3 plate to achieve the desired angle (this time rotating in the opposite direction from the previous example). Nailed it.

So how practical is this tool for everyday MOC-building? Well, it’s quite nifty indeed, and the solutions it comes up with are very accurate and quite original sometimes. But I do feel the whole exercise is a bit academic (not necessarily a bad critique, coming from a university professor). What these two somewhat contrived examples demonstrate, is that the tool beautifully solves the rotation problem (getting the angle right), but fails to address the translation problem (getting the offset right). 

In these two assignments, I found I had to resort to a lot of trial and error in order to get the final plate lined up right against the target. So if your to-be-angled plate is out in the open, and doesn’t need to align closely with the rest of your build, this tool can be a nice addition to your LEGO toolbox. But otherwise, be prepared for some frustration.

Sugar Grid technique

So is that all there is to it? Not quite. Let’s take another look at the first example we discussed at the top. 

The two corners of the top plate align perfectly with two corners of the bottom grid, as do the centers of the plates. Perhaps some of the studs on the bottom plate line up with anti-studs of the top plate? As it turns out, they do—a remarkable eleven studs fit snugly underneath the red plate, providing support and locking it securely at the precise angle. 

Here’s where the first six studs go:

The blue plates form a specific pattern known as a sugar grid, a term coined by Chris Enockson from Brick Sculpt that derives from the notion of a sugar cane grid in Minecraft (long story…). This specific sugar grid has a ratio of 1:2, which means that from each stud, you move one module across and two modules up to get to the next stud. (It doesn’t matter whether you do this counting on the original or the rotated grid.)

But I promised eleven studs, so here are the other five:

To find these, you have to find the center of each square of 4 blue studs, and place an offset plate (or 2x2 jumper) there, which then nicely lines up with the tubes at the bottom of the top plate. The white studs again form an (offset) sugar grid. 

To appreciate the grid a bit more, here’s all the studs without the red plate being in the way:

You can in fact place the red plate anywhere on the grid at this angle. You just need to remove the odd white stud that bumps into an edge of the top plate (the blue ones are always fine). 

The interesting property of the sugar grid is that it will accept a plate in two orientations: one aligned with bottom plate, and one rotated at a specific angle.

It’s tempting to think that the angle of the sugar grid is the same as the angle of the top plate, but it’s not. Where the angle of the top plate is 2 x arctan(1/2), the angle of the sugar grid is just half that: arctan(1/2). In this sense, the sugar grid sort of mediates between the top and bottom grid.

The angle of this example, 53.1°, is lucky (well, “deliberate” is more accurate), since it corresponds to a 1:2 sugar grid, which is nice and dense. Different ratios produce different angles, but also sparser grids. As a result, the number of studs that you can place under an average plate drops rapidly with the numbers in the ratio. But modest ratios like 2:3 and 1:4 still work fine.

Could LEGO designers be privy to the sugar grid? I get the impression they are. At least some. Here’s a little evidence from the majestic set 10316 The Lord of the Rings: Rivendell™:

Building instructions ©2023 The LEGO Group

The yellow round plate indicated ends up supporting a plate that extends the tiled grey area to the sides of it. Since the yellow plate is on an angled subassembly (1:6,  18.9°), it never made sense to me that it would form a legal connection. But now it does!

I leave you with this SNOT version of a 1:2 sugar grid, to let you ponder the many possibilities the sugar grid has to offer. There’s plenty more to say about sugar grids and their relation to other standard techniques such as reflected wedges and Pythagorean triples, but that’s a story for another day.

Thanks to Arno for this valuable insight! 

Would you like more articles like this? Please comment if you enjoyed this read, or indeed if you found it too complex – or simple. 

READ MORE: What are the new LEGO® parts in March 2025 sets?

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