Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms

Abstract

All geometrically distinct nets (edge-unfoldings) are explicitly constructed for the infinite family of polyhedra known as uniform antiprisms. The Fibonacci numbers are shown to count the number of nets of the n-antiprism that have point symmetry. (The n-antiprism refers to the antiprism whose surface consists of two regular n-gons separated by a band of 2n equilateral triangles.) Smaller evenly indexed Fibonacci numbers count certain subfamilies of these nets. Well-known formulas for sums of alternately indexed Fibonacci numbers motivate and become accessories to our constructions and proofs. For counting and constructing the symmetric nets, we first use a known result connecting compositions of integers and Fibonacci numbers. In an alternative proof, we illustrate a striking self-similarity among the counts of certain natural subfamilies of the symmetric nets, which is used to generate an efficient recursive construction of all the symmetric nets of the n-antiprism from those of the -antiprism. (This second proof is found in the online supplement to this article.) Finally, each nonsymmetric net of the n-antiprism is constructed by combining a pair of distinct symmetric nets of the n-antiprism.