Equilateral triangle, square, regular pentagon... a regular polygon on a flat plane can be made with any number of sides, without limit. So can a solid shape — one "where every face is a congruent regular polygon and the same number of faces meets at every vertex" (this is called a Platonic solid) — also be made in infinitely many varieties? Surprisingly, the answer is no. There are exactly 5 Platonic solids in the entire universe — the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, and that's all of them.
To understand why there are only 5, we first need to think about how a solid's vertex forms in the first place. Imagine gathering several regular polygons around a single point on a flat sheet of paper, then lifting that point slightly to fold it into a peak. For the paper to have enough slack left to fold into a point, the sum of the angles gathered at that one point has to be less than 360°. Exactly 360° unfolds completely flat, and past 360° the paper overlaps and can't fold at all. So every single vertex of a Platonic solid has to be in a state where "the angle sum is less than 360°."
Now let's work through each regular polygon's interior angle one by one. An equilateral triangle's interior angle is 60°. Gather three and you get 180°, four gives 240°, five gives 300° — none of these reach 360° yet, so they can all fold. But gather six and you get exactly 360°, which goes flat and can no longer become a solid. So with equilateral triangles, only three cases are possible — 3, 4, or 5 gathered together (making the tetrahedron, octahedron, and icosahedron respectively). A square's interior angle is 90°, so only 3 (270°) works, while 4 (360°) becomes flat (the cube). A regular pentagon's interior angle is 108°, so only 3 (324°) works (the dodecahedron). From the regular hexagon onward, the interior angle is 120°, so even the minimum of 3 already reaches 360°, meaning a solid can't form at all from the start. Working through every case this way reveals that exactly 5 combinations are possible.
These 5 Platonic solids have been known since ancient times. The ancient Greek mathematician Euclid devoted the entire final book of his work "Elements" to proving that there are exactly 5 Platonic solids. The philosopher Plato believed these five shapes symbolized the five elements — fire, earth, air, water, and the universe (aether) — which is why Platonic solids are still sometimes called "Plato's solids" today.
Platonic solids hold another fascinating property. For any Platonic solid, if you take the number of vertices (V), subtract the number of edges (E), and add the number of faces (F), you always get 2 (V − E + F = 2). This is called Euler's formula, and it's actually a much more general property that holds for every convex polyhedron that isn't Platonic solids alone, so long as it isn't caved in. Take the cube as an example — 8 vertices, 12 edges, 6 faces — and 8 − 12 + 6 = 2 checks out exactly.
You can easily find Platonic solids all around you, too. Board-game dice are most commonly cubes, but dice shaped like a tetrahedron, octahedron, dodecahedron, and icosahedron are all sold in real life too. The icosahedron resembles the pattern on a soccer ball (technically, a soccer ball is a semi-regular polyhedron made by trimming the corners off an icosahedron), and in chemistry, some viruses' outer shells and certain molecular structures take on an icosahedral shape as well. The mathematical fact that there are exactly 5 keeps turning up, again and again, throughout nature and in things people make.
On our activity page, you can use sliders to change the number of sides of the regular polygon (n) and the number of faces gathering at a vertex (p) yourself, and see exactly why the shape stops forming the moment the angle sum exceeds 360°. Then rotate all five Platonic solids in 3D and check their number of vertices, edges, and faces for yourself.