can Euler’s formula go beyond 3D?

[AntonioLao]

Topology as a branch of higher mathematics is essentially free of the many axiomatic restrictions of Euclidean geometry. And since Euler solved the Konigsberg bridge problem, a new branch of topology known as network theory has emerged. Its modern day applications can be found in the analysis of communications networks, the design of computer circuits, and the study of neural automata for the science of artificial intelligence and robotic technologies.

In 2D network theory, Euler’s formula (discovered in 1751) is given by V-E+F=1, where V is the number of vertices, E is the number of edges, and F is the number of faces. In 3D, Euler’s polyhedral formula is given by V-E+F=2. If this formula can be extended to other dimensions within Euclidean n-space then the generalized formula is given by V-E+F=n-1, where n is the dimension of the E-space. In E1, Euler’s formula is given as V-E+F=0, E2: V-E+F=1, E3: V-E+F=2, E4: V-E+F=3, E5: V-E+F=4, E6: V-E+F=5, E7: V-E+F=6, E8: V-E+F=7, E9: V-E+F=8, E10: V-E+F=9, E11: V-E+F=10, E12: V-E+F=11, E13: V-E+F=12, E14: V-E+F=13, E15: V-E+F=14, E16: V-E+F=15, E17: V-E+F=16.

[Guille]

Well, Euler discovered his formula to fight the birdges of konigsberg problem. The problem was impossible to give a soluteion, because it was in 2d. If there would have been undergrounf tunels, it may have been possible. So Euler's formula should be needed in other higher dimensions.

[AntonioLao]

How do we check the validity of Euler's formula beyond the 3rd dimension?

[Guille]

How do we check the validity of Euler's formula beyond the 3rd dimension?

good question.

I guess that all we can do is base our selves on pure math.