Tag Archives: regular solids

Platonic Solids Part 8: Icosahedron and dodecahedron

Understanding the Duals You may recall from the first article in this series (Platonic Solids Part 1: What are the Platonic solids?) that each platonic solid has a dual. You saw in the post Platonic Solids Part 5: Cube and … Continue reading

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Platonic Solids Part 7: The dodecahedron

If you enjoyed calculating the coordinates of the vertices in an icosahedron, you’re in for a treat! Finding the vertices for a dodecahedron is even harder, largely because a dodecahedron has more vertices. An icosahedron has 20 triangular faces and … Continue reading

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Platonic Solids Part 6: The icosahedron

Finding the Icosahedron’s Vertices Figure 1 shows an icosahedron with its hidden surfaces removed. Figure 2 shows the same icosahedron with hidden surfaces drawn in dashed lines and its vertices labeled a through l. The icosahedron is centered at the … Continue reading

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Platonic Solids Part 5: Cube and octahedron

You may recall from the first article in this series (Platonic Solids Part 1: What are the Platonic solids?) that each platonic solid has a dual. The cube and octahedron are duals of each other. To make a dual from … Continue reading

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Platonic Solids Part 4: The octahedron

Finding the Octahedron’s Vertices A platonic octahedron has six triangular faces. Because the edges of the triangles all have the same length, the triangles are equilateral triangles. If an equilateral triangle has edges of length 1, then its height is … Continue reading

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Platonic Solids Part 3: The cube

A platonic cube has six square faces. The cube’s symmetry makes it easy to find the coordinates of its vertices. For a cube with edge lengths of 1, the vertex coordinates are (±0.5, ±0.5, ±0.5) as shown in Figure 1. … Continue reading

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Platonic Solids Part 2: The tetrahedron

A tetrahedron has four faces that are equilateral triangles. In an equilateral triangle, the sides have the same lengths and they meet at 60 degree angles. Figure 1 shows a tetrahedron. Finding the Tetrahedron’s Vertices It is a trigonometric fact … Continue reading

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Platonic Solids Part 1: What are the Platonic solids?

This is the first in a series of posts about the Platonic solids. These posts will show how to find the corners of six Platonic solids: tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron. Deriving these values requires only algebra and … Continue reading

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