Some help needed plz,?

Question

ok, plz explian Eulers law, and how he derived the formula.


for those not falmiliar with it, here's the formula
edges = (vertices + faces) - 2

Answer 1

Remark
The Theorem is due to Descartes (1639). It was rediscovered by Euler who published its proof in 1751. The proof below is due to Cauchy (1811).

Proof of Theorem 1
Remove one face of the polyhedron. By pulling the edges of the missing face away from each other, deform all the rest into a planar network of points and curves. With no loss of generality it's possible to assume that the deformed edges remained straight line segments. Regular faces cease to be regular polygons if of course they were regular to start with. However, the number of vertices, edges and faces remained the same as those of the given polyhedron (the removed face corresponds to the exterior of the network.)

I am going to apply repeatedly a series of additional transformations that would simplify the network without changing its Euler's number (also Euler's characteristic) F-E+V.

If there is a polygonal face with more than three sides, we draw a diagonal. This adds one edge and one face. Continue adding edges until all the faces are triangular.
Remove (one at a time) all the triangles with two edges shared by the exterior of the network. This removes a vertex, two edges and one face.
Remove a triangle with only one edge adjacent to the exterior. This decreases the number of edges and faces by one each and does not change the number of vertices.

Carry out steps 2 and 3 repeatedly one after another until only one triangle is left. For a single triangle F = 2 (counting the exterior), E = 3, V = 3. Therefore F - E + V = 2. Which proves the theorem.

P.S. I copied this directly from the website below.

Answer 2

Euler's law states that the number of vertices plus the number of faces in every system will always equal the number of edges plus two.

For example:
Cube 8 vertices + 6 faces = 12 then add 2 for a total of 14

How to derived it??....well not so easy it is advance mathematical vocabulary....any way please check this web page....http://muweb.millersville.edu/~jdooley/macro/derive/euler/presgrad/presgrd.htm


Good Luck!

Answer 3

Not much to explain. Do you understand edges, vertices and faces? I don't know if Euler DERIVED the law or just observed it and expressed that in a formula.

Answer 4

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