The Universes Building Blocks. Tetrahedron-Hexahedron-Octahedron-Dodecahedron-Icosahedron

Image Credit:- wikimedia

The Platonic Solids

The Platonic solids are a special type of Polyhedrons (3-D shape), have been depicted throughout history dating back to the late neolithic period about 5200 BC, where representations of these shapes were carved into stone balls.

Image credit:- Wikimedia

They were studied by the ancient Greeks mostly famously, but the name given to them (Platonic) is misleading as they were believed to have been discovered by a friend of Plato called Theaetetus, who derived the mathematical representation.

Image credit:- Extracted from Wikipedia

The name however comes from the fact that these shapes were described in the philosophy of Plato. He wrote about them in his dialogue Timaeus in 360 B.C, each of the solid were assigned an element of nature. The Hexahedron (cube) represents Earth, the Tetrahedron is fire, air is associated with the Octahedron, and water the Icosahedron. The Dodecahedron was described as "being used by God to arrange the constellations on the Heavens", and was also given the assigned the element of Aether (scientists dismiss this term, some may call it space-time) by Aristotle.

The famous Mathematician Euclid extensively described these solids mathematically. He stated that:-

For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length.

This basically means that if you put a sphere around a platonic solid each corner of the solid will touch the surface of the sphere. Also you can nest all five of the platonic solids inside a sphere, all corners of each shape will touch the spheres surface and also touch the surface of every other solid.

Mathematical Properties

The Rules for being a platonic solid are:-

• All faces are convex regular polygons (symmetric 2-d shapes that are flat).
• No faces intersect.
• The same number of faces meet at each of its corners.
• Each vertex has the same interior angle.

Image credit:- Extracted from Wikipedia

The angle associated with the vertices are called dihedral angles, it's the interior angle between two faces that's the same at every vertex. The equation used to determine the interior angle is

where p is number of edges of each face, and q the number of edges meeting at each vertex. This this equation and the information provided in the the table you will be able to confirm the dihedral angles.

Radii of Spheres contained in the Solids

Another interesting aspect of the Platonic solids is that they all have 3 characteristic concentric spheres.

• A circumscribed sphere which intersects all the vertices.
• A midsphere that's tangent to every edge, at the edge's midpoint.
• An inscribed sphere which is tangent to every face at the face center.

The are special radii associated with the spheres. The distance from the center of the polyhedron to the vertice is the circumradius. The distance from the center of the polyhedron to the edge midpoint is called midradius. The distance from the center of the polyhedron to the face center is referred to as the inradius.

The circumradius (R) and the inradius (r) of the solid (p,q), with an edge length (a) are expressed as

where theta is the dihedral angle I have mentioned. The midradius is expressed as

Values of these radii can be seen in the table below. Now I want you notice the constant phi, this is actually the Golden Ratio, derived from the the Fibonacci Sequence (0,1,1,2,3,5,8,13,21,....) that is seen all through nature. To obtain the golden ratio you must take divide a number in the Fibonacci sequence by it's preceding number, for example 21/13 = 1.61538461 . The value of Phi is 1.61803398, so you can notice a difference, the higher you go in the Fibonacci sequence the closer you will get to the true value of Phi when you divide the two numbers. I make this point because Phi is fundamental is nature, I can make a whole post just on that, but these solids contain aspects of the Golden ratio, very interesting indeed.

Below are nice GIF images of each Platonic Solid so you can have a good look at their geometry.

Tetrahedron

Hexahedron

Octahedron

Dodecahedron

Icosahedron

These GIFs were obtained from wikimedia, sourced in the platonic solid reference below.

Platonic Solids in Nature

Nature

The Dihedral angles of the tetrahedron, hexahedron and octahedron can be observed in naturally occurring crystal structure, their lattices have geometry that is platonic like.
On the larger scale of things I will point out that the Great Red Spot on Jupiter is located a position where a tetrahedron would touch the sphere, in the case that one vertex of the tetrahedron is located at the pole, then this is true. There are other example of this in the solar system, but it's not something that has been studied scientifically in depth, I just wanted to make that interesting point. For the open minded keep this fact in mind.

Image credit:- wikimedia

Biology

There is a type of protozoa called Radiolaria, it has been observed that skeletons of these exhibit platonic geometry, for example type was called Circogonia icosahedra, which you can see in the image below.

Image credit:- wikimedia

In fact many varieties of virus exhibit Iscoahedric geometry, herpes is an example, yeah it's gross i know. The structure of viruses are built from identical repeated sub-units of protein, and it just so happens that the icosahedron is the easiest shape to do this by conserving energy.

I discovered whilst writing this article that scientists have made DNA tetrahedron nano-structures for applications is biology, that's pretty cool. If you're interested check the paper HERE

One Dimension further..

I wasn't going to include this but I thought it was cool so i'll quickly introduce it. The platonic solids are 3-D right? Now take the principles and rules of them and apply it to 4-D. I can't even begin to imagine a 4-D shape so don't worry, but let's do our best. These shapes are referred to as Polytopes, you can read a bit further in the link at the bottom. It's completely impossible for our feeble human brain to imagine such a shape, but we can map out some representation. Below is a link to a short youtube video to help you visualise the 4th Dimension.

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References:-
Platonic Solids
Carved stone balls
Theaetetus
Great Red Spot
Polytopes

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