User talk:Tzstar

Doing a Report
I'm in a advanced middle school geometry class and they made us do a report based on all of geometry. We eahc had to make a little origami figure of one that the teacher chose for us and i got a cuboctahedron. My friend is so mad because he got a snub dodecahedron. Anyways, read my report an i hope you enjoy it...

I truly do not know how to start this off. Polyhedrons are geometric figures with flat faces and straight edges. The word polyhedron goes back to its Greek roots, poly meaning many and hedron meaning faces or bases. From there you go to an unlimited world, platonic solids (regular) to Archimedean solids to Kepler–Poinsot polyhedrons to Catalan solids, truncated tetrahedron to cuboctahedron (I do not know even if I spelled it right.) Then there comes great mathematicians like Euclid, Kepler, Poinsot, Cauchy, Euler, etc. I hope that after you read this you will be immersed into the world of geometry. (Most likely not, most of you will be asleep.)

Let’s start with Archimedean solids and Kepler-Poinsot polyhedrons. Archimedean solids are symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. For example, my example, a cuboctahedron was an Archimedean solid. Plato, a Greek philosopher, was the first one to mention semiregular polyhedrons, but Archimedes, another Greek philosopher, was the first one to do some serious studying. That is why they are known as Archimedean solids. Archimedes’s work was lost and here came in Johannes Kepler. Kepler reconstructed all the semiregular polyhedra. For example, he formed the one I am assigned, the cuboctahedron. Kepler started with a cube then cut off all the edges with planes that started at the midpoints of the old edges, thus forming twelve new vertices, but still ending up with original twenty-four edges. Each of the old six square sides had been trimmed down a smaller square side. This operation also formed eight new triangles at the truncated corners. Kepler-Poinsot polyhedrons on the other hand, are regular star shaped and there are only four. They go from a small stellated dodecahedron to the great dodecahedron. What differs from them are their faces, some having triangles, others with pentagons and pentagrams. Their history goes back to the 1400’s with a small stellated dodecahedron and Paolo Uccelo. Uccelo, an Italian painter, had engraved a small stellated dodecahedron on the floor of Saint Mark’s Basilica in Venice, Italy. Most star polyhedra appeared in Renaissance era paintings. Some more modern ones include M.C. Escher’s Gravitation, using a small stellated dodecahedron.

Next, we go to Euler’s (Pronounced Oiler) Theorem on Polyhedra. Euler wasn’t the first one to know about this formula. Descartes in 1639 had known about it and also to Leibniz in 1675 through his incomplete manuscript. It may also be possible for Archimedes to have known of it. Even though all these people have known of it, it was Euler who published it. His theorem stated: v – e + f = 2. V = number of faces, E = number of edges, and F = number of polygonal faces. Euler’s formula can be used to prove that there exist just five types of regular polyhedral (simplified versions).

As a close up, I would like to talk a bit about the cuboctahedron, the polyhedron that I am assigned. The cuboctahedron is an Archimedean solid with eight triangular faces and six square faces. It has twelve identical vertices, with two triangles and two squares meeting at each. The cuboctahedron has twenty-four identical edges, each separating a triangle from a square. It is semiregular, convex, and quasiregular, meaning an Archimedean solid. It is also known as a heptaparallelohedron or dymaxion (according to Buckminster Fuller; Rawles 1997). Tzstar (talk) 05:35, 8 November 2009 (UTC)