Having looked at the flat geometry (two dimensional) of the number Phi, we now find it in the most symmetrical of the three-dimensional solids – the Platonic Solids.

The five regular solids (where “regular” means all sides are equal and all angles are the same and all the faces are identical) are called the five Platonic solids after the Greek philosopher and mathematician, Plato. Euclid also wrote about them.

Coordinates and other statistics of the 5 Platonic Solids

They are the tetrahedron, cube (or hexahedron), octahedron, dodecahedron and icosahedron. Their names come from the number of faces (hedron=face in Greek and its plural is hedra). tetra=4, hexa=6, octa=8, dodeca=12 and icosa=20.

Remember that in these pages Phi is 1 … and phi is 1/Phi = Phi-1 = 0 …

The Solid images can be rotated ( press the button) as can the Stereo views. For the auto-stereographic views, either cross your eyes or keep your eyes focussed in the distance until the two images fuse into one and you see the shape in depth. If you place your mouse on the “rotate” button before you do this then a quick click will make it appear to rotate in 3-dimensions.

The “wire-frame” views are symmetrical plan views of the frame of the object with wire edges and the faces missing, dotted lines being edges that would be hidden by solid faces.

The Dual of a Solid

There are two more important relationships between the dodecahedron and the icosahedron. First, the mid-points of the faces of the dodecahedron define the points on an icosahedron and the mid-points of the faces of an icosahedron define a dodecahedron. The same is true of the cube and the octahedron. If we try it with a tetrahedron, we just get another tetrahedron. Each is called the dual of the other solid where the number of edges in each pair is the same, but the number of faces of one is the number of points of the other, and vice-versa.

Golden sections in the Dodecahedron, Icosahedron and Octahedron

dodecarectangleicosahedron rectangle If we join mid-points of the dodecahedron’s faces, we can get three rectangles all at right angles to each other. What’s more, they are Golden Rectangles since their edges are in the ratio 1 to Phi.

The same happens if we join the vertices of the icosahedron since it is the dual of the dodecahedron.

3 gold rectangles Using these golden rectangles it is easy to see that the coordinates of the icosahedron are as given above since they are:

(0,± 1, ± Phi), (± Phi, 0, ± 1), (± 1, ± Phi, 0).

The Greeks, Kepler and the Five Elements

Kepler The Greeks saw great significance in the existence of just 5 Platonic solids and they related them to the 4 ELEMENTS (fire, earth, air and water) that they thought everything was made from. Together with the UNIVERSE, they associated each with a particular solid.

The astronomer and mathematician, Kepler (1571-1630), shown here as a link to the History of Mathematics web site at St Andrews University, Scotland, justified this as follows:

Of the 5 solids, the tetrahedron has the smallest volume for its surface area and the icosahedron the largest; they therefore show the properties of dryness and wetness respectively and so correspond to FIRE and WATER.

The cube, standing firmly on its base, corresponds to the stable EARTH but the octahedron which rotates freely when held by two opposite vertices, corresponds to the mobile AIR.

The dodecahedron corresponds to the UNIVERSE because the zodiac has 12 signs (the constellations of stars that the sun passes through in the course of one year) corresponding to the 12 faces of the dodecahedron.

Kepler called the golden section “the division of a line into extreme and mean ratio”, as did the Greeks. He wrote the following about it:

“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”

Johannes Kepler, (1571-1630)

Article: Raoul Martens recommends an article in German on Kepler’s interest in the Platonic solids: Die kosmische Funktion des Goldenen Schnitts by Theodor Landscheidt in Sterne, Mond, Kometen, Bremen und die Astronomie zum 75. Jahrestag der Olbers-Gesell-schaft Bremen e.V. Verlag H. M. Hauschild, Bremen 1995.

Source page has a LOT more info.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi3DGeom.html

===

The golden section, also known as the golden ratio or divine proportion, is a mathematical ratio often found in art, architecture, and nature. Here are some solid (three-dimensional) geometrical facts about the golden section:

1. **Golden Rectangles and Solids**: A golden rectangle is a rectangle whose side lengths are in the golden ratio (approximately 1:1.618). When a square is removed from a golden rectangle, the remaining rectangle is also a golden rectangle. Similarly, a golden solid is a three-dimensional solid whose dimensions are related by the golden ratio.

2. **Golden Cuboid**: A golden cuboid is a rectangular prism whose edges are in the golden ratio. Specifically, if the length of the cuboid is a, then the width is approximately 1.618a and the height is approximately 0.618a. The ratio of the length to the width to the height is the golden ratio.

3. **Golden Spiral**: The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every quarter turn it makes. It is often found in natural phenomena such as the shape of certain shells, hurricanes, and galaxies. The spiral can be constructed using nested golden rectangles, where each rectangle is drawn so that its length is the width of the next larger rectangle.

4. **Dodecahedron and Icosahedron**: The regular dodecahedron and icosahedron are two polyhedra that exhibit connections to the golden ratio. In particular, the dodecahedron has 12 faces, 20 vertices, and 30 edges, and the icosahedron has 20 faces, 12 vertices, and 30 edges. The ratios of the number of faces to the number of vertices and the number of vertices to the number of edges in both polyhedra are equal to the golden ratio.

5. **Fibonacci Numbers in 3D**: The Fibonacci sequence, which is closely related to the golden ratio, can also be observed in three-dimensional structures. For example, the number of spirals in certain types of pinecones, pineapples, and sunflower heads often follows Fibonacci numbers, demonstrating the connection between the Fibonacci sequence, the golden ratio, and three-dimensional geometry.

These are just a few examples of how the golden section manifests in three-dimensional geometry, highlighting its prevalence and significance in the natural world and in human-created structures.