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

Jonathan F. Mason and Richard H. Hudson

A chapter in Applications of Fibonacci Numbers, 2004, pp 177-185 from Springer

Abstract: 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.

Keywords: 11B39; 11A99 (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-306-48517-6_18

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