The Mystery of Polyhedra
I have always been fascinated by Polyhedra.
I've stared at these objects for hours on end, read about them and their role in philosophies and the development of mathematics, searched for them in Escher's images, modelled them with computers, made them from paper and clay, build them with magnets, explored their symmetries in crystals, counted their faces and edges, meditated on them and I am still nowhere nearer to grasping their meaning then I was at the beginning of this journey.
I'm not alone in this fascination. Many have gazed at these shapes over time wondering about their significance, their numbers, ratio's and inherent properties. You'll find them all on the Net in different formats, part of collections of photos or VRML objects. But the best way to get to know them is by holding them in your hands and moving them about. I have wooden models, glass, plastic dice, metal, marble, paper and clay ones.
You'll find that you will be drawn to a specific one much dependant on your mood, as if a specific shape seems to fit best with your disposition at that time. When you pick them up they seem so basic, so simple, so atomic. Yet our names for most of them are astoundingly complex. If these shapes are so basic then why does our language not have a simple word for each of them ?
To me they represent small complex puzzle pieces which whisper secrets to me about multi-dimensional space, symmetry and it's possibilities, tantalizingly just beyond my understanding. The Platonic solids have now become well known friends.
Most people see them as part of a series but to me each has a distinct and separate personality. Each representing a unique and different aspect on how ideas and thoughts can be expressed in our 3 dimensional space. It's like holding the keys of the material universe in your hands without any clue to where the lock is ..
But then they are so completely useless at the same time! You cannot build with them or use them in volume. They seem to be successful only in isolation as a focus point, like they do not really belong in our world but symbolize a world of perfect order which could never be expressed in our material world filled with dirty fractal chaos.
There are quite a lot of hidden relations between the various numbers of faces, edges and vertices between the different solids. But here is one not many people have noticed.
The Number of Faces of the Platonic and Aristotelian Solids = 10 x Number of Faces of the Platonic Solids.