Regular polygons in higher dimensional Euclidean spaces

Abstract

Basic properties of polygons in Euclidean space and some related regularity questions were explored in the first part of the Nineteen century. Systematic investigations of polygons and their degree of regularity in a higher dimensional Euclidean space E r began in the 1970s, in the vein of Blumenthal’s fundamental work (Blumenthal et al. Theory and applications of distance geometry. Chelsea Publishing Co., New York, 1970) on distance preserving maps of E r . Such investigations were also stimulated by a practical question from organic chemistry posed to van der Waerden, see van der Waerden (Elem Math 25:73–78, 1970), and the subsequent discussion around it. An useful indicator of degree of regularity was introduced by Grunbaum (The geometry of metric and linear spaces. Springer, Berlin, 1975) who generalized the concept of an equilateral polygon in a higher dimensional Euclidean space E r : An n-gon \({P_1P_2\dots P_n}\) spanning E r is called at least k-equilateral, if $${\overline{{P_1}{P_j}}=\overline{P_{1+h}P_{j+h}},\quad h=1,2,\ldots,n-1,\quad\quad\quad(0.1)}$$ holds for every \({1 < j \leq k+1}\) where the indices are taken mod n. If \({P_1P_2\dots P_n}\) is at least k-equilateral with \({k\geq [n/2]}\), then (0.1) holds for every \({1\le j\le n-1}\). In this case, \({P_1P_2\dots P_n}\) is called a totally equilateral n-gon since any two chords (or sides) P i P j and P k P l have equal length if the same holds for the chords (or sides) Q i Q j and Q k Q l of the planar regular n-gon \({Q_1Q_2\cdots Q_n}\). Grunbaum (The geometry of metric and linear spaces. Springer, Berlin, 1975) conjectured and Lawrence (k-equilateral (2k + 1)-gons span only even-dimensional spaces. Springer, Berlin, 1975) [independently van der Blij (Linear Algebra Appl 226/228: 345–352, 1995)] proved that totally equilateral polygons with odd number of vertices span always an even-dimensional space. In this paper, we prove that every at least (r + 1)-equilateral n-gon with \({n\geq r+1}\) is totally equilateral. This generalizes a previous result in three dimensional Euclidean space, see Korchmaros (Riv Mat Univ Parma (4) 1:45–50, 1975).