Euler's characteristic is a topological invariant that describes the shape of a topological space regardless of bending. It was originally defined for polyhedra as V - E + F, where V is the number of vertices, E is the number of edges, and F is the number of faces. Euler proved that for any convex polyhedron, this formula equals 2. This Euler characteristic of 2 also applies to the sphere. The formula is used to classify shapes in fields like chemistry, architecture, and as a basic concept in topology.

3. 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
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. Name Image
Vertices
V
Edges
E
Faces
F
Euler
characteristic:
V − E + F
Tetrahedron 4 6 4 2
Hexahedron /
Cube
8 12 6 2
Octahedron 6 12 8 2
Dodecahedron 20 30 12 2
Cube octahedron 12 24 14 2

6. 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