Euler's polyhedron theorem

A little description about the famous Euler's Polyhedron Theorem

1. Euler's Polyhedron Theorem A project by:- Harsh Bandhey Shreyash Mitra PriyamMukherjee Pranay Dutta

2. Polyhedrons In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek π ολύεδρον , as poly- (stem of π ολύς , "many") + - hedron (form of ἕδρ α, "base" or "seat"). A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface . Cubes , pyramids and some toroids are examples of polyhedra

3. Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics , the Euler characteristic (or Euler– Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by X (Greek letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work The Euler characteristic X was classically defined for the surfaces of polyhedra , according to the formula X=V-E+F where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic X= V - E + F = 2 . This result is known as Euler's polyhedron formula or theorem. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra . An illustration of the formula on some polyhedra is given below.

4. Euler's Formula Euler's Formula For any polyhedron that doesn't intersect itself, the Number of Faces + the Number of Vertices (corner points) - the Number of Edges always equals 2 This can be written: F + V − E = 2 Try it on the cube: A cube has 6 Faces, 8 Vertices, and 12 Edges, so: 6 + 8 − 12 = 2 Every polyhedron has Eulers Characteristic X=2

5. Examples Name Image Vertices V Edges E Faces F Euler characteristic: V  −  E  +  F Tetrahedron 4 6 4 2 Hexahedron   /   C ube 8 12 6 2 Octahedron 6 12 8 2 Dodecahedron 20 30 12 2 Cube octahedron 12 24 14 2

6. Applications Chemistry Euler’s characteristics is used in the field of chemistry specifically in organic chemistry to specify the rigidity and shape of an organic compound Architecture Euler’s formula is extensively used in architecture and designing structures and vehicles. Mathematics It is also important to note that Euler's characteristic is the basics of topology and is very crucial for understanding the concepts of topology