Clip-based LEGO® Pythagorean triangles: Escaping the grid

In an unexpected bonus addition to his Escaping the Grid series of LEGO® technique articles, brick scholar Arno Knobbe (@legoarno on Instagram) returns to tie up some loose ends. The great potential of Pythagorean triangles was demonstrated in Arno’s last article, but outside the plane, things turn out to work slightly differently. This time, no new math will be introduced (thank goodness!), so no need to pull out your scientific calculator. But an abacus might come in handy.

Lego Baby Astronauts building small triangles from Lego pieces

Image by TobyMac

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In my last article about Pythagorean triangles, I casually dropped a clip-based Pythagorean triangle, as an alternative to the Slope 53 piece. This example raised some questions with a couple of readers, correctly observing that the geometry of such triangles is not obvious. Round about the same time, Tom Loftus reviewed LEGO® Star Wars™ 75409 Jango Fett's Firespray-Class Starship showcasing some innovative clip-based triangles. Given the subtleties involved, it’s worthwhile analysing some of the building tricks involved in detail. Let’s dive in!

Upright triangles

Most of the Escaping discussion so far has been confined to triangles in the plane, where measurements are straightforward: whether you move n modules in one way or in another, it’s always the same distance. However, as you know, a unit LEGO brick is 20% higher than it is wide (factor 1.2), so when you put stuff upright, you’re effectively working with a different measure.

Let’s see how this works in practice with the offending (3, 4, 5) triangle from the last article.

Right angled triangle built vertically from lego pieces. To achieve the diagonal, plates with clips and plates with bars are used

Along the horizontal leg, we measure 3 modules (don’t worry about the clips just yet), so along the vertical leg, we expect 4 modules. However, we’re seeing 3 bricks and a plate here, so what’s this?

Since we need to measure with the same units going across and up, and we’re using ‘modules’ (the width of a unit brick) across, we will need to measure four modules up, and translate that to the appropriate number of bricks and plates. In this case, we divide the 4 modules by a factor 1.2, which gives us 4/1.2 = 3⅓ bricks. As you know, ⅓ brick is simply a plate, so the final vertical distance is 3 bricks and 1 plate, as indeed is the case in the figure.

This triangle doesn’t require an intimate understanding of the geometry of clips yet (we’ll get to that further down). Yes, clips involve an awkward offset in both the horizontal and vertical direction, but since the two white clips are oriented the same way, the offsets cancel out, so we can ignore them.

For the hypotenuse (the diagonal bit), the horizontal offset of the blue modified plates with bars do introduce a relevant offset. Each modified plate involves a half-module offset, so the entire length of the hypotenuse equals 4 + ½ + ½ = 5 modules, as it should.

In some cases, Pythagorean triangle dimensions work out so well that no plates as vertical spacers are even needed, as with these two examples below.

two vertical pythagorean triangles built vertically in Lego: one measures 8, 6, 10 and the other 5, 12, 13

The left example is simply a variation on the (3, 4, 5) pattern: it is twice the (4, 3, 5) triangle. 6 divided by 1.2 happens to be an integer: 5. The same holds for the vertical leg of (5, 12, 13): 12/1.2 = 10. Any triangle with a vertical leg that’s a multiple of 6 has this attractive simplicity.

Double the size of the (8, 6, 10) triangle once more, and you get the below (16, 12, 20) triangle that makes up the roof of 10316 Rivendell. This wonderful set, packed with geometry gems, is designed by LEGO® Senior Designer Mike Psiaki, whose work we’ll see more of further below.

highly decorated Lego balcony with sloped roof from the set the roof of 10316 Rivendell

The two corners forming the hypotenuse can be recognized as red Technic axles with stops, one clearly visible in the bottom-right corner, one tucked away under the roof at the top. Due to the size, all measurements are integer: 16 modules along the balustrade, 10 bricks along the wall (harder to verify visually, but easy to compute: 12/1.2 = 10), and 20 modules along the roof.

In many other cases, we’re not this lucky, though. In some cases, the vertical leg cannot be constructed from mere bricks and plates. For instance, if we flip the (3, 4, 5) triangle on its side, we get a vertical leg of 3 modules: 3/1.2 = 2½. With bricks and plates, we can achieve 2⅓ or 2⅔, but 2½ is right in between, so requires half a plate. And with that, as any seasoned LEGO builder knows, we’re heading into bracket and SNOT territory.

2 vertical lego right angled triangles that both use half plate measurements to achieve the vertical side. One uses a bracket placed sideways, the other uses a headlight brick (part 3070) attached sideways

This image shows two ways of constructing a vertical leg 2½ modules tall. Neither option is very satisfactory.

Plate scale

Given our modest starting point of the (3, 4, 5) triangle, scaling it by a factor 2 or 4 makes sense. But if we would like to use different triangles, for instance to benefit from the different angles they afford, dimensions tend to get prohibitively big.

$A depiction of all the primitive Pythagorean triples (a,b,c) with a and b < 1170 and a odd, where a is plotted on the horizontal axis, b on the vertical. The curvilinear grid is composed of curves of constant m − n and of constant m + n in Euclid's formula, a = m 2 − n 2 , b = 2 m n {\displaystyle a=m^{2}-n^{2},b=2mn}. These curves can be specified parametrically as a = ( 2 i − 1 ) ( 2 j + 1 ) , b = ( ( 2 j + 1 ) 2 − ( 2 i − 1 ) 2 ) / 2 {\displaystyle a=(2i-1)(2j+1),b=((2j+1)^{2}-(2i-1)^{2})/2}, with one of i, j held constant in {1,2,3,...} and the other varied over an interval in R. The oddness of m − n (and of m + n) required for the triple to be primitive is thus built in, with m − n = 2 i − 1 , m + n = 2 j + 1 {\displaystyle m-n=2i-1,m+n=2j+1}. All primitive Pythagorean triples are represented on this grid. The scaled triangles are drawn with their centroids at the corresponding (a,b). Unlabeled grid intersection points correspond to non-primitive Pythagorean triples.$

Adam Cunningham and John Ringland, CC BY-SA 3.0, via Wikimedia Commons

As the diagram demonstrates, Pythagorean triangles increase in size quickly.

But why count in modules? Perhaps we can build triangles from plates?

A very small lego pythagorean triangle of sides 3, 4, 5 long. To build it, start with a 1x1 upward bracket and stack on top two 1x1 plates, a 1x1 brick and a 1x1 plate with clip. Lay this sideways to become the horizontal side. On top of the explosed stud of the bracket now facing upwards, stack a 1x1 plate, a 1x1 brick and a 1x1 plate with clip. The clip should point in the direction of the horizontal side. Then attach a Plate Special 1 x 1 with Handle on 2 Ends (part 78257) between the two clips. A perfect fit!

To start simple, consider this (3, 4, 5) triangle at plate-scale. The fact that this is a legal Pythagorean triangle is not immediately obvious. The hypotenuse is still relatively straightforward: the 2 modules of the modified plate correspond to the desired 5 plates. But why the horizontal leg is 3 plates and the vertical 4 plates remains obscure. In both directions, half-plates seem to be involved (one being the thin part of the grey bracket, one being half a tan plate), so why are these necessary?

A detour into the LEGO® Unit

In order to appreciate the intricacies of this construct, and prepare ourselves for a deep dive into Jango Fett’s craft further down, we need to digress a little. Our first diversion concerns a scarcely used, but very handy LEGO unit of measurement called …drumroll… the LEGO Unit. Most people are familiar with modules, bricks, and plates as units, or sometimes even LDraw Units (LDU), but the LEGO Unit (LU) deserves at least as much renown in my view, if not more.

Let me sell this unit a bit to you, by pointing out some cases where 1 LU is a relevant distance.

4 demonstrations of where a 1 LU measurement can be found within lego bricks

First off, 1 LU is the difference between the height of a 1x1 brick and its width. Thus, in LU, a unit brick is 5 × 5 × 6 LU. The LU also pops up in other crucial areas. To name a few: the thickness of a brick’s wall is 1 LU, as is the upright part of a bracket or panel, the lip at the bottom of a headlight brick, or the height of a stud (not counting the lettering on top). In other words, 1 LU equals half a plate, but you can use it for distances in any direction, not just vertical. Plus, half-plates become awkward when we start discussing ½ LU distances (quarter-plates?).

The LEGO Unit plays a crucial role in the remainder of our story, but we have one more stop on our detour. Here, we encounter the modified plate with clip (and its counterpart, the modified plate with bar). Since clips will be our pivots for building triangles, we need to study a bit the horizontal and vertical offsets involved when using clips.

a lego plate with clip attached to a lego plate with handle. The construction is connected to other plates below, showing that the overall construction is 3 modules wide, with the bar/handle section occupying the central module

The horizontal offset is easy: the gap between two bricks that a clip-and-bar hinge bridges is 1 module, so 5 LU wide, as the image demonstrates. (Before you get carried away: trans-clear plates with bars exist only in our dreams). Hence, the horizontal offset is just half that: 2½ LU.

the same picture but now with measurements overlaid. A dot marks the centre of the handle of the bar. Two parallel lines show where the dot is positioned between the top and the bottom of the plate section of the element: the dot is 1½ LU from the bottom, and ½ from the top.

The vertical offset is less obvious, but for good reasons, as we shall see. Relative to the bottom of the modified plate (for example of 61252, Plate, Modified 1 x 1 with Open O Clip (Horizontal Grip)), the pivot lies 1½ LU up. Measured from the top, this is ½ LU down. For some odd reason, the centre of the clip is ½ LU too high!

The design of the modern bar-and-clip hinge, and its cousins like the locking hinge and Mixel joint, may appear silly – until you tilt them 90 degrees upward.

Examples of two kinds of lego hinge: one is the modern bar/clip, the other is the 1980s "finger" joint hinge. Each example shows two states of the hinge: with each side straight, and with one side at 90 degrees. Both examples of 90 degree bend are surrounded by other lego pieces to show how they fit - or do not fit - into the lego system

As the lefthand constructions demonstrate, the tilted blue plate at the back returns nicely in-grid after being rotated, perfectly lining up with tiles on its left and a bracket on its right.

Compare this to the vintage finger hinge (4276b and 4275b) above right. While the geometry of the finger hinge is much more straightforward (which makes it great for stud-reversal), when rotated, it causes all manner of alignment problems with adjacent parts. The placement of the grey bracket as shown wouldn’t even be physically possible in real life, as its studs would collide with the blue plate.

The beauty of the modern design of hinges and Mixel joints is that the ½ LU offset inherent to bridging the one-module gap is compensated by the ½ LU vertical offset. When you tilt one plate, the two fractions cancel out. In terms of our Pythagorean triangles, this cancellation occurs whenever you combine horizontally and vertically oriented clips, as we’ll see below.

Clip-based triangles

After our scenic detour, we now merge back onto the main highway of our narrative. With our new-found insight, here’s a detailed analysis of our earlier plate-based triangle.

The same image of a small Lego triangle as before, but now with dimensions in LU marked on it. Key measurements are described in the text below.

We now know that the white plate with clip on the left produces a horizontal offset of 1½ LU. Add to that the 6 LU of the sideways brick plus 1 LU for half of the plate, and you get 8½ LU. But the clip at the top reduces the horizontal distance by 2½ LU again, so we get 8½ - 2½ = 6 LU. Which indeed corresponds to the expected 3 plates.

Likewise, we get 1½ + 6 + 2 + 1 – 2½ = 8 LU, which is 4 plates. We already knew that the hypotenuse was 10 LU, so 5 plates, so done and dusted! Note how the ½ LU of the one clip is always resolved by the ½ LU of the other.

Perhaps we can up the ante once more and go truly micro-scale? Let’s build a “baby-pythag” that’s 3 LU wide and 4 LU tall.

Along the bottom, we have ½ LU from the clip, plus 2 LU plus 3 LU (plate and half a brick), minus 2½ LU from the top clip. This indeed makes 3 LU. Vertically: 1½ + 2 + 2 + 1 – 2½ = 4 LU. The Brick Special 1 x 2 Rounded with Center Bars (77808) forms a 5 LU hypotenuse.

Let’s consider some larger triangles in LU-scale. Our triangles so far have had hypotenuses that were multiples of 5 LU, which allows us to use integer modules along the diagonal. If we are going to pick larger Pythagorean triangles (for example from the earlier diagram of primitive Pythagorean triples), we might make our lives easier by picking those that have a multiple of 5 along the hypotenuse. (7, 24, 25) and (33, 56, 65) fit the bill. See if you can do the math…

Lego vertical pythagorean triples of sizes (7, 24, 25) and (33, 56, 65)

Triangles in Jango Fett's Firespray-Class Starship

The "hull" of set 75409 Jango Fett's Firespray-Class Starship under construction. It forms a beautiful oval curve

Multiples of 5 LU along the hypotenuse are convenient, but quite limiting. Too limiting in fact for designer Mike Psiaki, when he was consulted for the advanced geometries of set 75409 Jango Fett's Firespray-Class Starship. The elegant curve of the bottom of this spaceship not only required the introduction of a new 1x8 curved slope, but also the construction of some interesting angles to place the curved slopes just so.

Let’s investigate two of these angles constructed from Pythagorean triangles.

Close up of one section, with the pieces of each segment coloured differently for clarity. A large curve coloured tan is attached using bar/clip joints to the main frame. The triangle this forms has been drawn over the top of the picture, with measurements in LU of each brick

In LU, the triangle that holds the tan subassembly is (30, 16, 34), which is not primitive (meaning there’s a smaller Pythagorean triangle of which this is a multiple). But counting in plates, which apparently is what Mike did, (15, 8, 17) is a primitive Pythagorean triangle.

The arithmetic along the two legs should be straightforward by now (note the various SNOT bricks that connect to the plates on the back). Along the diagonal, we have two clips that jointly are 2 LU thick. Apart from that, we get three varieties of SNOT bricks and a plate to produce a total distance of 34 LU.

Sugar grid and reflected triangles

As a bit of recreational geometry (isn’t that the essence of this series?), let’s see if earlier concepts of the sugar grid and reflected triangles produce anything entertaining. The latter is definitely pertinent here, because at the left of the subassembly, you can recognise a pair of reflected cheese slopes forming an interface with the remaining structure.

The sugar grid doesn’t appear to be present in this design, but as an earlier analysis demonstrated, the (15, 8, 17) triangle corresponds to a 1:4 sugar grid and its 3:5 dual ratio. The latter ratio explains the pair of cheese slopes.

The same picture of the hull of the ship as before but now with two 8x3 lego wedge plates attached to it, forming a kite shape. These wedges are not in the actual set, they show that there is a sugar grid involved in the design but it was not used.

The 1:4 ratio is not used in the actual build, but theoretically, the bottom of the ship could have been covered with a pair of reflected wedges. Lining up these wedge plates is a bit tricky (see how the tapered corner coincides with the centre of the hinge) but could easily have been achieved with some redesigning of the ship’s bottom.

Another curve on the hull, shallower than the last, with its corresponding pythagorean triangle marked.

Angling of the second subassembly is based on the (48, 14, 50) triangle, which again is primitive in plate-scale, but not in LU scale.

Since the hypotenuse is a multiple of 5, the diagonal subassembly doesn’t require any sideways building or SNOT techniques. With a ratio combo of 1:7 and 3:4, I see no obvious opportunities for the sugar grid or reflected triangles.

How to build your own LEGO triangles

Understanding triangles designed by a LEGO Senior Designer is one thing; inventing your own is a different ballgame. Here are some pointers for building clip-based triangles of your own.

Select a Pythagorean triangle from the diagram based on size and angle. If it’s too small, multiply the sides by 2, 3, …
If the hypotenuse is a multiple of 5, you can work with modified plates that lie parallel to the hypotenuse (as in the last example from Fett’s starship). Otherwise, you have to work sideways (first example).
If the horizontal leg is a multiple of 5, you can build the horizontal and vertical leg studs-up. At each corner, place a modified plate with clip in the same orientation.
If the horizontal leg is not a multiple of 5, you have to work sideways for the horizontal leg, and vertical for the vertical leg.
If odd-LU distances are involved (which is often the case), involve brackets or SNOT bricks (regular or headlight).
The building direction of the horizontal leg can be reversed without much impact.
Switching between a ‘hanging’ and ‘standing’ bracket implies a 1 LU increment, so this is a good trick to fill a 1 LU gap.

I find that first lining up the parts in digital software such as BrickLink Studio helps to find the exact geometry. After placing a first clip (on its side), I attach the assembly for the hypotenuse and hinge it to the right angle (as per the length of the two legs). Then I try to line up plates and bricks until the second clips falls into place – without even calculating the required length! This helps guarantee that your candidate triangle is indeed legal, instead of trial and error in real life.

To prove that this technique has application beyond modified plates with clips, I leave you with these (digital) table scraps.

6 further lego triangles, of various sizes, using other pieces for the hypotenuse.

Used for the hypotenuse are: Bar Ladder 7 x 3 with 4 Clips (30095); Panel 4 x 4 x 13 Curved Tapered with Clip at Each End (18969, 43030), Technic Pin Connector Round with 4 Clips (90202); Bar, Parallel Angled with 2 Connecting Bars (Train Hinge) (37494); Arm Mechanical with 2 Clips [Battle Droid] (30377) and Plate Special 1 x 2 with 2 End Towballs (3170).

Happy building!

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27 July 2025