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A Generalization of Euler’s Formula and its Connection to Fibonacci Numbers

In: Applications of Fibonacci Numbers

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  • Jonathan F. Mason
  • Richard H. Hudson

Abstract

This paper began as a simple proof generalizing Euler’s well-known formula for the vertices, faces, and edges of a cube in 3 dimensions, to a tessaract, and to higher dimensions. Let an n-cube with n-dimensional volume 1 consist of all n-tuples (x 1, x 2, ..., x n ) where each x i , i = 1, ..., n satisfies 0 ≤ x i ≤ 1. The boundary points of the n-cube are the vertices, which we will call 0-cubes to indicate that they are 0-dimensional. For each such vertex, we clearly have x i fixed to be 0 or 1. A 1-cube will be an edge of the n-cube. For an edge, we have exactly one of the x i free to take on values between 0 and 1 (inclusive) and the other x i fixed to be 0 or 1 for each i = 1, ..., n. Similarly, a k-cube, k ≤ n, will have exactly k of the x i free to take on values between 0 and 1 (inclusive) and n - k fixed to be 0 or 1.

Suggested Citation

  • Jonathan F. Mason & Richard H. Hudson, 2004. "A Generalization of Euler’s Formula and its Connection to Fibonacci Numbers," Springer Books, in: Frederic T. Howard (ed.), Applications of Fibonacci Numbers, pages 177-185, Springer.
    • Handle: RePEc:spr:sprchp:978-0-306-48517-6_18
    DOI: 10.1007/978-0-306-48517-6_18
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