GEOMETRY VS TOPOLOGY

    x = V - E + F
Euler characteristic = Vertices - Edges + Faces.

Draw triangle on a sheet of paper.  It has 3 vertices, 3 edges and 1 face.
Hence the
Euler char. = 3 -3 + 1 = 1
For a square
Euler char. = 4 - 4 + 1 = 1
For a circle, imagine two points or vertices on the circle.
So, Euler char. 2 - 2 + 1 = 1.
The circle, square and triangle all have same Euler characteristic 'one'.  They are geometrically different, but topologically same.  That means, all the shapes can be converted into one another, if they are made of rubber sheet.

"Topology is concerned with the properties of space that are preserved under continuous deformations; such as stretching, crumpling and bending, but not tearing or gluing" (Wikipedia).

     Now we get the idea of topology.  Let us see one more example.
    Take a 'tetrahedron' solid (prism).  It has 4 vertices, 6 edges, 4 faces.
    Euler char. = 4 - 6 +4 = 2
    for a cube
    Euler char. = 8 - 12 + 6 = 2
   A sphere has also Euler char.  2.
    Hence  prism, sphere, cube, all belongs to same topological family.  But have different geometrical properties.

   The shape of coffee cup and Doughnut have zero as Euler characteristic.  They are one and the same.  Euler's formula helps us to find the topologically invariant shapes.  ( you might have noticed one point in the above cases, that is, all the vertices meet with same number of edges).

   DNA has a special shape called double helix (a twisted staircase).  This Euler formula helped us to understand the DNA even better.