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


![](https://steemitimages.com/DQmdrKWDNNr7dQAuSNbV3QW9ZE5DemXHfVSboWFxAYg8qA6/image.png)
Image Credit:- [wikimedia](https://commons.wikimedia.org/wiki/File:Platonic_solids.jpg)

![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)


# 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. 

![Stone balls](https://steemitimages.com/DQmYd9pzWn5TmvsoaaE9s3PZFo4gkvoGs64CMvXJWAR9WFA/image.png)
Image credit:- [Wikimedia](https://en.wikipedia.org/wiki/Carved_Stone_Balls#/media/File:Kelvingrove_Art_Gallery_and_MuseumDSCF0239_11.JPG)


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](https://en.wikipedia.org/wiki/Theaetetus_(mathematician)), who derived the mathematical representation.

![platonic solids.png](https://steemitimages.com/DQmUeKCM1qEMGGoK6JDYKaZmrfbUajS7GLNQwzFvqbn9va7/platonic%20solids.png)
Image credit:- Extracted from [Wikipedia]((https://en.wikipedia.org/wiki/Platonic_solid))



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_](https://en.wikipedia.org/wiki/Timaeus_(dialogue) "Timaeus (dialogue)") 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.

![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)


# 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.





![platonic table.png](https://steemitimages.com/DQmSEQgZcqBvKrQXqMC7Qzsz8jpDfSaYedqzfyZYrpDe2Tr/platonic%20table.png)
Image credit:- Extracted from [Wikipedia]((https://en.wikipedia.org/wiki/Platonic_solid))


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 

![dihedral angle eqn.png](https://steemitimages.com/DQmYsRc9d9T2JCyu7hZHVNs8yjFVdFJET8uZuoG1XDWkRpw/dihedral%20angle%20eqn.png)

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](https://en.wikipedia.org/wiki/Circumscribed_sphere "Circumscribed sphere") which intersects all the vertices.
-   A [midsphere](https://en.wikipedia.org/wiki/Midsphere "Midsphere") that's tangent to every edge, at the edge's midpoint.
-   An [inscribed sphere](https://en.wikipedia.org/wiki/Inscribed_sphere "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

![radius eqn.png](https://steemitimages.com/DQmeXYFhjYyek7kT1CqXZhHesxa6gjJjVnEZPhtS1xSJwUa/radius%20eqn.png)

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

![radius eqn 2.png](https://steemitimages.com/DQmPYRtxqgur8VvRHozqVxX24ZVJKV7zJyw4fR667NaGFJX/radius%20eqn%202.png)

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.

![platonic table 2.png](https://steemitimages.com/DQmfBiDq2osf7uEsjJgDwnvqT42muaaEvM8nFWsUCv1xugk/platonic%20table%202.png)



![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)

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


### Tetrahedron 

![Tetrahedron.gif](https://steemitimages.com/DQmegDVwWjkyeu8BazKA73Lm27FHW5Hz2yV74S5gRoH3vwH/Tetrahedron.gif)

### Hexahedron

![Hexahedron.gif](https://steemitimages.com/DQmcL5d7wkk96o5FHCgYhz4uYgqPAUDHYR853cih8pwDUML/Hexahedron.gif)


### Octahedron

![Octahedron.gif](https://steemitimages.com/DQmYELrsao9PR2geBMH6QcmiV7vcH6VfR4kgD6PppiYrBLh/Octahedron.gif)


### Dodecahedron

![Dodecahedron.gif](https://steemitimages.com/DQmPdh38UyQP6RH3syYFMVt5ez85iCjdpKLVEfbiVAjvSss/Dodecahedron.gif)


### Icosahedron

![Icosahedron.gif](https://steemitimages.com/DQmSKcMAbPUweVQJYfGXBak4EZbEzFnA68DdaeHY9v9dkqf/Icosahedron.gif)


*These GIFs were obtained from wikimedia, sourced in the **platonic solid** reference below.*
![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)

# 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.

![790106-0203_Voyager_58M_to_31M_reduced.gif](https://steemitimages.com/DQmWC7q6m3vofyeHRWS9TSzEDAQrZXSzKqufc9WsmkNqMaa/790106-0203_Voyager_58M_to_31M_reduced.gif)
Image credit:- [wikimedia](https://en.wikipedia.org/wiki/Great_Red_Spot#/media/File:790106-0203_Voyager_58M_to_31M_reduced.gif)


### 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.

![](https://steemitimages.com/DQmbzuE6AjkrTbKcPvfk1sc1CttSx5VT6C1TT91nwnaigsv/image.png)
Image credit:- [wikimedia](https://en.wikipedia.org/wiki/Platonic_solid)

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](https://www.ncbi.nlm.nih.gov/pubmed/28835943) 

![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)

# 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.

https://www.youtube.com/watch?v=eyuRLmCphHc
![](https://steemitimages.com/0x0/https://steemitimages.com/DQmPM2S3oWJDXa8j1f5M2MHyGgPxTejJZ9Lr3BbxgcsrVhu/image.png)

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References:-
[Platonic Solids](https://en.wikipedia.org/wiki/Platonic_solid)
[Carved stone balls](https://en.wikipedia.org/wiki/Carved_Stone_Balls)
[Theaetetus](https://en.wikipedia.org/wiki/Theaetetus_(mathematician))
[Great Red Spot](https://en.wikipedia.org/wiki/Great_Red_Spot)
[Polytopes](https://en.wikipedia.org/wiki/Regular_polytope)

_All images are Creative Commons or public domain, no copyright infringements have occurred._