Golden Ratio and the Platonic Solids

by Nikhat Parveen, UGA

Platonic Solids

The 5 Platonic solids are illustrated above with some of their properties. Euclid, 300 BCE and the Ancient Greeks, in their love for geometry, called these five solids, the atoms of the Universe. In the same way that we today believe that all matter, is made up of combinations of atoms so the Ancient Greeks (also) believed that all physical matter is made up of the atoms of the Platonic Solids and that all matter also has a mystical side represented by their connection with earth, air, fire, water and ether. Similar to our modern atom which shows a nucleus surrounded by electrons in orbits creating spheres of energy, the Greeks felt that these Platonic solids also have a spherical property, where one Platonic Solid fits in a sphere, which alternately fits inside another Platonic Solid, again fitting in another sphere.

The Pythagoreans knew that there were only five regular convex solids, the tetrahedron, cube, octahedron, icosahedron and dodecahedron and each one could be accurately circumscribed by a sphere. The dodecahedron had twelve regular faces, which corresponded to the twelve signs of the Zodiac; it was therefore a symbol of the universe for the Pythagoreans. Moreover, each one of these faces is a pentagon. Euclid described these five regular solids in Book Thirteen of the Elements. They are associated with the name of Plato because of his efforts to relate them to the important entities of which he supposed the world to be made. Plato discusses them in his various dialogues.

The corners of the octahedron fit in the centre of the cube faces. The icosahedron can be inscribed in an octahedron, so that each vertex of the former divides an edge of the latter into the Golden Proportion The icosahedron and the dodecahedron are also uniquely connected with the Golden Proportion by virtue of three intersecting golden rectangles which fit into both. In modern times it has been discovered that the shape of many of the viruses is either icosahedron or cube.

The pentagon (the five-sided figure) is closely related to the icosahedron and its complement the dodecahedron. The diagram shows clearly how the dodecahedron is made up of twelve pentagonal faces. The Icosahedron is made up of triangular faces, but also grouped in 5 as seen at the vertex where 5 triangular faces come together.
Whereas these later 2 mentioned platonic solids are seen in the 5 pointed star the remaining three platonic solids can be found in the 6 pointed star.

Φ occurs many places in the platonic solids.  The dihedral angle on the dodecahedron is 2*atan(Φ), and the dihedral angle on the icosahedron is 2*atan(Φ²) or 2*atan(Φ +1).  The mid radius of the dodecahedron is similarly Φ²/2 or (Φ +1)/2, and the mid radius on the icosahedron is Φ/2. Φ can be found as a factor in the other measurements of these polyhedra too.

If you consider rectangles inscribed in the icosahedron and dodecahedron such that a pair of opposite sides lie along opposite edges of the solid, the ratio of the lengths of these rectangles would be twice the mid radius over the edge length.  This means that three perpendicular golden rectangles can be inscribed in the icosahedron, and three perpendicular rectangles in the ratio Ø ² can be inscribed in the dodecahedron.

The dodecahedron is one of the 5 Platonic solids with 12 pentagonal surfaces.

One of the 5 platonic solids is described by Plato in the "The Phaedo" (110 B.) where he refers to a ball with 12 pentagonal faces and which is the precursor to our modern football, still made with a variable number of pentagons as shown in the adjacent photograph.

The dodecahedron contain 3 Golden Proportion intersecting rectangles, as seen in the adjacent diagram. This later theme has inspired the Laban School of Dancing to base its movements within the rectangles.

The icosahedron fits well as an inscribed solid within the dodecahedron.

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