Semi-Regular (Archimedean) Polyhedra
.jpg) 20 Triangles make an Icosahedron |
.jpg) Icosahedron again, showing 5 triangles at the vertex |
For the Icosahedron we go back to using triangles: there are 20 in all, with 5 at each vertex.
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What’s next? Well, sadly this is the end of the simple Platonic solids: there are no more like these. But there are lots more to come if we vary the rules a bit: our next step is Semi-Regular (Archimedean) Polyhedra.
As an extra point on the maths…the proof that there are no more of these simple solids drops out simply from trying to build them. Round any vertex the total sum of the angles can’t possibly sum to more than 360 degrees (which is a complete rotation round a single point); and the combinations we’ve seen here cover all the possible options. The table below shows the numbers and the possible outcomes as you try the many different combinations:
| Flat shape | Angle | Number round each vertex | Total degrees | Shape |
| Triangle | 60 | 3 | 180 | Tetrahedron |
| Triangle | 60 | 4 | 240 | Octahedron |
| Triangle | 60 | 5 | 300 | Icosahedron |
| Triangle | 60 | 6 | 360 | no good: it just comes out flat |
| Square | 90 | 3 | 270 | Cube |
| Square | 90 | 4 | 360 | no good: it just comes out flat |
| Pentagon | 108 | 3 | 324 | Dodecahedron |
| Pentagon | 108 | 4 | 432 | no good: it doesn’t work at all |
| Hexagon | 120 | 3 | 360 | no good: it just comes out flat |
And so it falls out – there are only 5 simple polyhedra.
Semi-Regular (Archimedean) Polyhedra
Back to The 5 Regular (Platonic) Solids
Back to Polyhedra
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