Crochet Geometry

Part 2: The Math

We’re going to keep this as simple as possible, and start by focusing only on the Platonic solids, since they’re likely the most familiar to most people.

These are going to be the easiest to work with because all their faces (the flat parts) are the same, and they’re also all regular polygons. We’re only dealing with equilateral triangles, squares, and regular pentagons for these examples.

We’ll be making each spike starting with a fsc ring, as I mentioned earlier. This fsc ring will outline the perimeter of each face. This means that the length of our ring is mainly determined by the side length of the face, which we’ll measure by number of stitches. I don’t recommend that you go below 3 stitches for this- lower than that and your spikes can be hard to work into and start getting cylindrical rather than spiky.

Now, calculate the perimeter of your faces. Multiply your side length by the number of sides on your face to get the perimeter. For example, if you had a 3 stitch side length and a 5 sided face, your perimeter would be 3 x 5 = 15 stitches around.

Other polyhedra

If you’re making something that has different shaped faces (like an Archimedian solid), the same rules apply. Choose your side length, and then calculate the perimeter of each face based on the side length.

Let’s say you had a polyhedron with triangular, square, and pentagonal faces using a side length of 4. Each triangle has a perimeter of 12 (4×3), squares a perimeter of 16 (4×4), and pentagons a perimeter of 20 (4×5).

Coloring!

If you want your project to be multicolored without two adjacent spikes being colored the same, you won’t need more than four colors to do so. This is due to the four color theorem*, and you should be able to find colorings for many shapes by searching the shape name + “coloring.” Make sure you plan out your colors before you start your project if you’re doing this!

* Technically the four color theorem is only for planar graphs, but the above examples are all planar so it’s fine. Most things you’d make with this are probably going to be planar. If you’re doing a torus shape, for example, you’ll need 7 colors.

Part 2a: The Worse Math

This section will cover in a bit more detail my exact thought process with this technique, and things you’ll need to keep in mind when experimenting with it.

You really don’t have to make the faces of your shapes regular polygons at all. As long as the sides of your faces have integer ratios to each other, you can use it as the base for this technique!

For the above examples, the rhombic triacontahedron on the left can be used for this technique. Each face is a rhombus, so all the sides are the same length. The deltoidal hexecontahedron on the right can’t be used for this since it has side length ratios of 1:(7+sqrt(5))/6.

You also don’t have to use a geometric shape or even something approximating a sphere as your basis for this. Again, the main thing with this is that the side lengths of the faces must be integer ratios of each other. If you’re freehanding spikes you don’t have to worry about this because they’ll always be integer ratios- you can’t have fractional crochet stitches.

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