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(Φ^{2}) or 2*atan(Φ +1).
The mid radius of the dodecahedron is similarly Φ^{2}/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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