Welcome to Apocalypsburg (Set #70840) is one good-looking, super-fun set. Dan observed, and I agree, it somehow manages to walk the line between being for kids and grown-ups at the same time. I love that it takes the tropes of a post-apocalyptic, dystopian society established in science fiction and plays them up for laughs (barbed wire basketball hoop! Crossbow hotdog!). I also love the excellent detailing, the statue’s face and hands, and the clever call backs to the first Lego Movie.

But this isn’t a set review! What I really wanted to talk about is Pythagoras! “Whaaat?” You may ask. “The ancient Greek philosopher and mathematician? What’s he got to do with this?”. Well I’m glad you asked. You may have had trouble making a right-angled triangle out of Lego because the long side (the ‘hypotenuse’) doesn’t line up at one end to click into a hole or onto a stud. That’s because right-angled triangles follow a law famously described by Pythagoras, namely ‘The square on the hypotenuse is equal to the sum of the squares of the other two sides’, or a2+ b2= c2, where a, b and c are lengths of a right angled triangle sides, with c being the hypotenuse. Check this WEBSITE for more info.

Using the 3-4-5 rule you can multiply and upscale the size of the triangle

Because Lego works by the regular spacing of studs, plate thicknesses and pin holes, lengths of elements are always ‘discrete’. That is, their sizes are fixed are not adjustable, not like say a piece of wood is, which can be cut to be any length. So, say you wanted to make a right angled triangle out of three pieces of Lego. You select a 11-long technic liftarm and a fix a five-long piece on the end at a right angle. You go to add another liftarm to complete the triangle but darn, the thing just won’t line up! What’s the problem?Well the actual length of the sides between the corners (the vertices, or turning points) of this triangle you are making are 10 and 4 (count the spaces between the studs or holes, not the holes themselves). If we plug these numbers into Pythagoras’ Law we find that the hypotenuse length (c) is…102+ 42= c2,100 + 16 = c2,c2= 116, c = 10.7703.There’s no piece of Lego that is 10.7703 studs or holes long. So there’s no way to make this triangle out of Lego.

By now you’re wondering if I’m going to get to the point. And here it is. There are some triangles whose hypotenuses are whole numbers, and that you can make out of Lego. The most famous one is the 3-4-5 triangle, and Apocalypsburg is full of them!Just quickly 32+ 42= c2,9 + 16 = c2,c2= 25, c = 5. Boom! Note these lengths can be doubled,tripled, whatever you like, so long as their proportions are maintained.

LEGO triangles using the 3-4-5 rule

The Apocalypsburg set uses the 3-4-5 triangle to fix the angle of the statues’ arm, and the level platforms at the top and halfway down, and for the head, and for a couple of railings too for good measure. Fixing all three corners of the triangles gives the model strength, crucial in a set like this–the buying public would be very upset if the model fell to pieces whenever you tried to pick it up.There are many ways of making Lego models look good, interesting and exciting, and one of these is to include unexpected angles. Apocalypsburg is a masterclass on how to do this. Now you’ve learned the secret of doing it, you can use it too. Here endeth the lesson.

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