Welcome to the Polyhedra Appreciation Society!
Updated: May 21, 2021
Polyhedra Appreciation Society welcomes everyone (especially people who have no clue what a polyhedron is) to witness the beauty that is inherent in the mathematical objects we call Polyhedrons.
In Geometry, a Polygon is a two dimensional shape made up of straight lines that connect together to form a closed figure (for e.g. a Square, a Triangle, a Pentagon, etc.).
A Polyhedron (plural, Polyhedra or Polyhedrons) is a three-dimensional solid with flat polygonal faces, straight edges and sharp corners (or vertices).
I make all of these above Polyhedrons out of various materials, some of which include Paper, Plaster of Paris (POP), Cement, Melted Glue. I will give a walkthrough of exactly how I do this in a separate blog post.
The idea behind exploring these Polyhedra, is to derive from them (at the least), a sense of calm in their aesthetic beauty. As we learn and dive deep into these Geometrical marvels, we will learn about their structure, history, applications in solving real world problems (among other things).
If there is one thing, that I would like you takeaway from all of this, it is the following: a sense of awe in learning about these objects. For me, making these polyhedra from scratch and learning and writing about them, helps me break the otherwise inescapable monotony of daily life. I thoroughly enjoy making them, writing about them, and this is one of the few things that keeps me going.
Some of these polyhedra have caught the public's imagination, e.g. the Platonic solids, due to their past connection with notable historical figures. These solids are constructed by congruent (identical in shape and size), regular (all angles equal and all sides equal) polygonal faces with the same number of faces meeting at each vertex.
You might notice the dents and scratches and small irregularities in the above Platonic polyhedra. This is because, with every new model I build, I get better at making it. With practice I will get better and better at doing this.
The ancient Greeks discovered and studied Platonic Solids extensively. (From left to right in the above figure) 4-sided regular Tetrahedron, 6-sided regular Hexahedron or more popularly known as a Cube, 8-sided regular Octahedron, 12-sided regular Dodecahedron, 20-sided regular Icosahedron. They also discovered the 13 Archimedean Solids which includes the Truncated Icosahedron (see the image below for a model of the Truncated Icosahedron that I made out of POP), one which resembles soccer balls and the famous Carbon molecule Buckminsterfullerene. Astronomer and Mathematician Johannes Kepler added 2 Rhombic polyhedra.
All in all these were the only classes of polyhedron that were equilateral (edges of equal length), convex (meaning that they do not have concave dents on their surfaces) and with the high symmetry displayed by the Platonic polyhedra.
Now researchers Stan Schein and James Maurice Gayed at the University of California, Los Angeles have found a fourth class of convex equilateral polyhedra with polyhedral symmetry (polyhedron terminology will be discussed in greater detail in one of the upcoming blog posts). There rules, they claim can lead to developing other classes of convex polyhedra.
Interestingly, the new rules that create these polyhedra have structures that are similar to viruses. The fact that there has been no cure against viruses such as Influenza or Common flu (or even limited cure in case of Coronavirus), might be due to the limitations we face in Mathematically describing the structure of these viruses. With advances like the ones done by Stan and James, once we are able to Mathematically describe the structure of a virus more accurately, maybe we will be in a better position to fight it.
Also please note that the above mentioned classes of Polyhedra (Convex equilateral polyhedra) are just a subset of all possible polyhedra that exist. Some examples are, Concave Polyhedra, Johnson Solids (a category of Polyhedra that contains Polyhedrons that are strictly convex, each face of which is a regular polygon, but which is not uniform), named after the Mathematician Norman Johnson. There are few other categories all of which we will discuss in greater detail in a separate blog post.
In our exploration of these Mathematical objects, one important thing to realize is that: It is NOT the real world applications that justify the importance of Polyhedrons. The importance of Polyhedrons is justified by its existence alone, the fact that these objects exist and we can make and observe them, derive aesthetical pleasure from them, study their Mathematical structure is in and itself rewarding enough to justify their importance.
The Utility v/s Aesthetics debate has been going on for a very long time. It will continue because there will always be things, whose importance would be justified by their aesthetic presence alone, unaffected by the question of whether they are actually categorized as useful (given the world's definition of utility: being applied to some real world scenario to help solve some real world problem).
If topics and discussions like these spark your interest (even a little bit), I promise that there is lots more interesting content come.
I encourage you to sign up and join the society. Click here to sign up. If you have any questions, suggestions, some crazy Polyhedron story, I would love to hear it! Submit your message in this form and I will get back to you!