# Week 3: Research on Geometry/October 22, 2018by Valzorra

### Polyhedrons and the Platonic Solids

Polyhedrons are rather fascinating as they are essentially constructed from two-dimensional faces, which meet at straight lines and vertices in three-dimensional space. There are a numerous polyhedrons out there in the world, but the most common type of polyhedrons are known as Irregular Polyhedrons. Irregular Polyhedrons do not tend to have much symmetry in comparison with the Platonic Solids, which I will be taking a look at shortly. A major group of Polyhedrons are the cuboids, which are Polyhedrons that have three different rectangular shapes for their sides. Some examples of cuboids include bricks, most PC cases, books, etc. Another major faction of Irregular Polyhedrons are pyramids, which tend to have irregular triangular sides, with the exception of square-based and pentagon-based pyramids. What’s most exciting about polyhedrons is their very nature of construction, featuring zero-dimensional (vertices), one-dimensional (edges), two-dimensional (faces), and three dimensional components (the whole polyhedron). These familiar and easily accessible shapes have a multitude of dimensions embedded in them, which I find absolutely fascinating.

Irregular Polyhedrons are quite common and standard, however, Polyhedrons that follow special rules and proportions are much more exciting. A group of such polyhedrons is the Platonic Solids. The Platonic Solids are all highly symmetrical and regular, which means that each face of the solid is a regular polygon. All edges and vertices of the Platonic Solids are indistinguishable from each other. There are a total of five Platonic Solids and no other solid in existence can possibly follow their symmetry and regularity. The five Platonic Solids are the Tetrahedron (built from four equilateral triangles), the Cube (built from six squares, all meeting at the right angles), the Octahedron (built from eight equilateral triangles), the Dodecahedron (built from twelve regular pentagons), the Icosahedron (built from twenty equilateral triangles, which meet at every corner in fives.)

The Platonic Solids were first explored by the philosopher Plato, who held them in exceptionally high regard. Pluto even believed that these five shapes were essentially the structure of the universe and assigned an element to each one. He believed that the tetrahedron corresponded to fire, the cube to earth, the octahedron to air, the icosahedron to water, and the dodecahedron as the constellations of the universe.

What I find incredibly exciting about the Platonic Solids is that they change their shape when moving between two-dimensional and three-dimensional space, yet they somehow maintain their mathematical properties and information. This is the transfer of the same data between two dimensions, this idea of information transcending the borders of reality through mathematics. Additionally, the Platonic Solids are the only ones that have perfect mathematical symmetry in three-dimensional space. This means that there is a form of perfection that can be obtained and visualised through mathematics, which could have a variety of applications in numerous fields, for example, Data Visualisation.

### Polychorons and the Hypercube

As exciting as Polyhedrons and the Platonic Solids are, they are limited to our most familiar dimension, the third dimension. Introducing Polychorons, the four-dimensional equivalents of Polyhedrons. Polychora are built constructed from three-dimensional cells, which meet at two-dimensional faces, one-dimensional edges, and zero-dimensional vertices. Much like with Polyhedrons, there is a series of regular and convex polychorons, not too dissimilar to the Platonic Solids. These shapes are as follows:

1. Pentachoron - built from five tetrahedrons.

2. Tesseract or Hypercube - built from eight cubes.

3. Hexadecachoron - built from sixteen tetrahedrons.

4. Icositetrachoron - built from twenty four octahedrons.

5. Hecatonicosachoron - built from one hundred and twenty dodecahedrons.

6. Hexacosichoron - built from six hundred tetrahedrons.

Polychorons and Regular Convex Polychorons are incredibly exciting for much of the same reasons Polyhedrons are. Polychorons can be represented in four separate dimensions, their shape and appearance changing through each one. However, the transfer of mathematical data and information regarding the number of vertices, edges, faces, is all constant regardless of dimension. What’s even more exciting here is that information goes into the fourth dimension, which humans oftentimes have difficulty even imagining. Therefore, Polychora surpass the capabilities of human perception and all we are left with to interpret it are representations of the fourth dimension in three-dimensional and two-dimensional space. An exciting research avenue here is how does one represent that which cannot be perceived.

Examples of regular Polychorons. It is important to note that the examples above are concave or starry Polychorons, which slightly differ from the concave ones described above.

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