Many of the big thinkers throughout history had a passion for geometry. It preoccupied not just philosophers, mathematicians and astronomers like Plato, Archimedes and Kepler, but also artists such as Leonardo DaVinci and Albrecht Durer. It is useful to have a basic understanding of geometry as an artist, but even more so if you are aspiring to be a great designer. We can learn about the five perfect solids from Plato’s Timaeus, they are often called the Platonic Solids. All of the five shapes look the same viewed from any vertex (corner point), their faces are made of the same regular shape and every edge is identical. Three of the five shapes are made of equilateral triangles, and got their names in function of the number of faces. Tetrahedron has four, octahedron has eight and the icosahedron has twenty equilateral triangles. The other two are the cube, and the dodecahedron, which is constructed of twenty regular pentagonal faces. Plato associated these five perfect solids with the five elements — fire, water, air, earth and “…the fifth construction which God used for embroidering the constellations on the whole heaven.”
The tetrahedron is composed of four equilateral triangles with three meeting at every vertex. Plato associated this solid with the element of Fire because it is the most fundamental of all the regular solids, and also due to the penetrating acuteness of its edges. The Greeks also knew about this solid, they called itpuramis. This is where the pyramids get their name from, and more interestingly the Greek word for fire is pur.
The octahedron is composed of eight equilateral triangles with four meeting at every vertex. Plato associated the octahedron with the element of Air, because he considered it to be the intermediary between the tetrahedron, the element of fire, and the icosahedron, the element of Water. Greek writings attribute the discovery of the octahedron and the icosahedron to Theaetetus of Athens (417 BC – 369 BC). We can often find the tetrahedron, the octahedron and the cube in the mineral kingdom in form of mineral diamonds and fluorite crystals.
The icosahedron is composed of twenty equilateral triangles, five meeting at every corner point. The icosahedron is the largest solid made of equilateral triangles, this led Plato to associate the icosahedron with the element of Water, the densest and the least penetrating of three elements. If you join the two ends of an icosahedron’s edge to the center of the solid, an isosceles triangle is defined. The same as the triangles that make up the faces of the Great Pyramid of Giza.
The cube is the only regular hexahedron, and it is composed of six square faces, with tree meeting at each vertex. Plato assigned it to the element of Earth, due to the stability of its square bases.
The dodecahedron has twelve regular pentagonal faces, three meeting at every vertex. The early Pythagoreans called the dodecahedron “the sphere of twelve pentagons”. Plato associated the previous solids to the elements of Fire, Water, Air, and Earth, about this remaining solid he writes “There remained a fifth construction which God used for embroidering the constellations on the whole heaven.”
These are the five perfect solids, commonly referred to as the Platonic Solids. But the fun doesn’t end here, in fact, this is where it starts really. By experimenting with joining the face-centres of the Platonic Solids something amazing happens. Starting with the tetrahedron, we discover another, inverted tetrahedron. The face-centres of a cube creates an octahedron, and an octahedron creates a cube. The icosahedron and the dodecahedron likewise produce each other. Two polyhedra whose faces and verticles correspond perfectly are known as each other’s duals. The tetrahedron is a self-dual.
In case you are working, or aspiring to work in the field of product design or prop design for entertainment, I would recommend learning more about the compound polyhedra, checking out Kepler’s polyhedra, and the Archimedean solids also. Geometry is a wonderful source of inspiration even if you are not interested in learning it the actual theory behind it, so have fun with it.