Original scientific paper
On cyclic characterizations of regular pentagons and heptagons: Two approaches
D. Svrtan
D. Šterc
I. Urbiha
Abstract
In this paper we present two different proofs of an algebraic
characterization of regular pentagons and regular pentagrams in
terms of two cyclic (complex) algebraic equations on a
five--dimensional torus (Theorem1 and Theorem2). The problem arose in functional analysis (as communicated to one of the authors by A.\ Björner some twenty years ago). No published proof has appeared so far.
Apparently a proof was given by L. Lovász (unpublished and not
known to the authors). Here we give two different proofs, both
somewhat tricky. The first one relies upon discrete Fourier
transform and the second one is more direct. Also several
generalizations to heptagons are presented including an explicit
description of some new irregular heptagrams. Some additional
conjectures on general polygons are stated.
Keywords
pentagon; heptagon; cyclic equations; discrete Fourier transform; bi-unimodular sequences
Hrčak ID:
783
URI
Publication date:
15.6.2002.
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