The 1- skeleton of pyramid is a wheel graph In geometrya pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face.
Building polyhedra and polytopes with foldout patterns. Building higher-dimensional poly-objects from lower-dimensional instruction sets.
Exploring the relationship among the poly-objects. Other kinds of poly-objects. Introduction The regular polyhedra The pictures above are pictures of the five regular polyhedra in three-space.
There are no others. Click on any of them to be able to play with it. All of the regular polyhedra singular polyhedron are constructed from regular polygons.
A regular polygon is constructed from equal-length segments joined by equal angles. In the same way, a solid regular polyhedron is constructed using equal-sized regular polygons joined at their edges by equal angles. In most of the discussions here, the polygons must all be the same polygon all squares or all triangles or all pentagons, no mixing allowed.
The fourth dimension We humans find it easy to think of things in three perpendicular dimensions; for example, we look at the corner of a room where the ceiling meets two walls, we understand easily that we are seeing three meeting planes.
We are not so used to thinking in four dimensions. In four-space, there must be a fourth line, perpendicular to all three of the lines you see in the corner, that meets that corner. Constructing objects in this kind of space is much harder to imagine. We can use our powers of analogy and our knowledge of geometry to make that construction less difficult.
Polytopes It is possible to construct an answer to this question: The answer is a regular polytope.
A regular polytope is bounded by regular polyhedra joined at the faces by regular angles in four-space. If this doesn't seem to make sense, fear not, we will try to explain it more clearly. Even if our brilliant explication leaves you in the dark, you can still travel through our site to look at the pretty pictures.
For a good exploration of three-space polyhedra, including a walk-through of Euclid's proof of the existence of only five, try George Hart's Virtual Polyhedra Page at Hofstra University. Slicing Polyhedra The Flatland Perspective Most people know that we live in a three-dimensional world.
But have you ever wondered what it would be like to live in a world that wasn't three-dimensional? InEdwin A.Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.
 Abstract polyhedra also have duals, which satisfy in addition that they have the same Euler characteristic and orientability as the initial polyhedron. Matroid polytopes form an intermediate structure useful in searching for realizable convex spheres.
In this article we present a class of self-polar 3-spheres that motivated research in the inductive generation of matroid polytopes, along with two new methods of generation. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.
 Abstract polyhedra also have duals, which satisfy in addition that they have the same Euler characteristic and orientability as the initial polyhedron. So a polyhedral 2-manifold is a geometric model of an abstract 2- manifold, made up of convex polygons and, in this paper, always embedded in E3, so necessarily oriented.
Introduction to Graph Theory in the Context of Dual Polytopes. Feb. 11th, I just finished a presentation on graph theory for the math club I’m a part of, and I’ve wanted to write a post about it, but I’ve never actually put forth the effort to write it.
The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all Coxeter groups and given in the notation described in that article.
The number following the name of the group is the order of the group.