A polyhedron is a three-dimensional solid made by a regular polygon. A regular polygon is a shape in which all sides have the same length and all of the angles are congruent. The word "polyhedron" comes from two Greek words —poly, meaning "many" and hedron meaning "surface". Polyhedra have been recognized for centuries, dating back to the Neolithic Ages. Around 2000 BC, hundreds of carved stone spheres were found during an excavation of an ancient settlement in Scotland. Some of the stones were carved with lines corresponding to the edges of regular polyhedra. One of these stones had twelve knobs just like a dodecahedron, marking the appearance of the dodecahedron in history before the Greeks wrote about it. The purpose of these stones is a mystery; it is only clear that they have similarities to the polyhedra we recognize today.
Egyptians (2650-500 BC) were intrigued by polyhedra and studied them in great detail. The way that they approached mathematical problems is different from the current customary methods. They used examples to get to the basic volume of a pyramid formula that is used today. They found a way to determine the volume of a truncated square pyramid using the formula \(\frac{1}{3}h(a^2+ab+b^2)\). Not much is known about how the Egyptians approached mathematical problems, but it is recognized that they were knowledgeable about areas, slopes, and calculating volumes of pyramids and other shapes. Babylonians (2000-500 BC) used similar pathways to calculate the volume of solids (Kiper).
The Ancient Greeks' approach to mathematical problems was completely different from those used before in other cultures. They believed that proofs were required for effective analyses. Plato (427-347 BC), "...is most well known for the comments on the five regular polyhedra, which are named after him"(Kiper). Plato was not the first person to discover the regularity of polyhedra, however. Pythagoreans identified the cube, tetrahedron (or pyramid), and the dodecahedron. Theaetetus, who was a friend of Plato, was the first to discover the icosahedron (a polyhedron with 20 faces that are equilateral triangles) and the octahedron. Plato's contribution to polyhedra was associating the regular polyhedra to the elements constructing the world. He assigned each polyhedra to a basic element that the Ancient Greeks believed made up the physical world: earth, water, fire, and air. He found the regular polyhedra intriguing and imagined the whole world consisting of them. He associated the tetrahedron with fire, the cube with earth, the octahedron with air, and the icosahedron with water. The fifth polyhedron, the dodecahedron, was named for the universe as ether. "These became known as the Platonic solids (for Plato). The Platonic Solids are convex polyhedra with each face congruent"(Anderson).
The first person to prove that there are only five regular polyhedra was Euclid (325-265 BC), an Alexandrian. "His claim also defines what a regular solid is: no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another"(Kiper). Euclid discussed the platonic solids in his work "The Elements". As complete as Euclid's definition is, it is missing a key point - the condition that each vertex should join an equal number of faces. He keeps his proof short and simple by proving how many regular polygons can meet at a vertex to create each regular polyhedron.
Archimedes (287-212 BC) was another Greek mathematician who studied polyhedra. He found that there are exactly thirteen semi-regular polyhedra. The thirteen semi-regular polyhedra are named after him as "The Archimedean solids". The Archimedean solids have the following characteristics: each of their faces is a regular polygon and around every vertex the same polygons appear in the same sequence. They can be made up of two or more types of regular polygons. "The first five Archimedean solids are created by truncating the original Platonic solids"(Anderson). Truncating is cutting off the corners of the Platonic solid to create the new solid. For example, a truncated cube is created by cutting off the corners of the cube to create eight equilateral triangles. At each vertex there are two octagons and one triangle which can be notated as (3,8,8). Johannes Kepler (1571-1630) related the then six known planets in the solar system to the five Platonic solids. Just as Euclid did, he assigned each planet to a regular solid. He was the second mathematician to discover the Archimedean solids. The way he constructed the solids was by using a vertex-based systematic method (Kiper).
The study of polyhedra had fallen out of vogue for about a century until Leonard Euler (1707-1783) drew attention to them again. He did not like the terminology that previous mathematicians had used in their definitions so he created his own terminology. He defined the characteristics of the geometry of polyhedra as solid angles, edges, and faces. He was the first mathematician to think about the edges of a polyhedron and how they compared to the solid. Today the terms "edges" and "faces" are used; "vertex" is substituted for "solid angle". Euler derived many relationships regarding polyhedra, but the vital one is known as Euler's formula: V+F-E=2, where "v " is the number of vertices, "e " is the number of edges, and "f " is the number of faces each polyhedron has (Kirk).
Paul Schatz (1898-1979) was one of the first mathematicians to see the importance of the turning inside out, or inversion, of solids. "Through the model of Schatz's cube, one can show very well the (projective) inversion of the entire space—the planes of the cube tip sideways through six mobile joints until they reach a level prior to reforming, this time to a hollow cube"(Heinz). There are twelve possible ways to invert regular polyhedra with a sixfold chain of joints. The inversion movement of polyhedra has been found to be a new kinematic kind of movement. The inversion is a looping movement in space that always returns back to its starting point.
Mathematicians have moved on from studying the geometry behind polyhedra. However, people have introduced ideas from combinatorics and graph theory into the study of polyhedra, which has made people interested in their geometry again. In the second part of the twentieth century the fact that mathematicians defined "polyhedron" in different ways to suit what they needed at the moment was brought to everyone's attention. The accepted definition of polyhedron was solidified and the process resulted in the accepted modern concept of an abstract polyhedron (Vidermanova & Klepancova).