Polyhedron in Geometry

A polyhedron is a solid in Euclidean 3-dimensional space with flat faces and straight edges. More specifically, it is a 3-dimensional solid such that each of its faces is a polygon, and each of its vertices is incident to the same number of faces. A cube is an example of a polyhedron. In general, the polygons can be any shape, as long as they are not coplanar. The word “polyhedron” comes from the Greek words for “many” and “seat”.

Types of Polyhedra

There are two types of polyhedra: regular and irregular. A regular polyhedron has all faces the same size and shape, and all vertices the same number of faces incident to them. An irregular polyhedron does not have these properties. The Platonic solids are examples of regular polyhedra.

The five Platonic solids are shown below. From left to right: tetrahedron, hexahedron (cube), octahedron, dodecahedron, icosahedron.

Table 1: The Platonic Solids Figure 1: The Platonic Solids

Tetrahedron 4 Hexahedron (Cube) 6

Hexahedron (Cube) 8 Octahedron 12

Octahedron 6 Dodecahedron 20

Dodecahedron 12 Icosahedon 30

Icosahedon 8

There are also four Kepler-Poinsot polyhedral which are the regular star polyhedral. They are shown below in Table 2 and Figure 2.

Table 2: The Kepler-Poinsot Polyhedral Figure 2: The Kepler-Poinsot Polyhedral

Small stellated dodecahedon 12 Great dodecahedon 20

Great stellated dodecahedon 30 Great icosahedon 12

Great stellated icosahedon 20 Small icosahedon 8

Hypercube Torus Sphere Prism

8 vertices Endless No Flat Faces 5 faces

3 cube connected together Doughnut S=4∏r² Pentagon(5), Hexagon(6)… Torus Try to imagine Vertical cut through Infinite sides in But only finite up down but drawing it a a donut where one a cylinder where plane but goes on and going around sphere would be hole is big enough circle in all directions forever difficult for humans to pass like sphere but like a slice of pie like infinitely long so we use symbols through has flat sides line instead instead instead of actually instead of just connecting at connecting at being curved drawing it out one point two end points everywhere because it would like cylinder but like rectangle but go on forever be impossible has endless sides instead of just to draw connecting at going straight one point up and down across two end points instead becoming like never ending zigzag pattern loop line like surface of water where wave goes up and down but never ends or repeats itself like sine wave

In conclusion, a polyhedral is a three-dimensional figure with flat surfaces and straight edges. There are two types of polyhedral – regular and irregular - with the latter not having congruent faces or similar vertices. Furthermore, there exist five Platonic solids and four Kepler-Poinsot polyhedral which are the regular star polyhedral. Lastly, other notable mentions include the hypercube, torus, sphere, and prism.