International Journal of Combinatorics
Volume 2010 (2010), Article ID 803210, 30 pages
doi:10.1155/2010/803210
Research Article

On Isosceles Sets in the 4-Dimensional Euclidean Space

Hiroaki Kido

Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Received 22 July 2010; Accepted 4 November 2010

Academic Editor: Gerard Jennhwa Chang

Copyright © 2010 Hiroaki Kido. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A subset 𝑋 in the 𝑘 -dimensional Euclidean space ℝ 𝑘 that contains 𝑛 points (elements) is called an 𝑛 -point isosceles set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly two 11-point isosceles sets in ℝ 4 up to isomorphisms and that the maximum cardinality of isosceles sets in ℝ 4 is 11.