Latin bitrades, dissections of equilateral triangles, and abelian groups
✍ Scribed by Aleš Drápal; Carlo Hämäläinen; Vítězslav Kala
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 264 KB
- Volume
- 18
- Category
- Article
- ISSN
- 1063-8539
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
Let T=(T^*^, T^▵^) be a spherical latin bitrade. With each a=(a~1~, a~2~, a~3~)∈T^*^ associate a set of linear equations Eq(T, a) of the form b~1~+b~2~=b~3~, where b=(b~1~, b~2~, b~3~) runs through T^*^{a}. Assume a~1~=0=a~2~ and a~3~=1. Then Eq(T,a) has in rational numbers a unique solution \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $b_{i}=\bar{b}_{i}$\end{document}.
Suppose that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document} for all b, c∈T^*^ such that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document} and i∈{1, 2, 3}. We prove that then T^▵^ can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T^*^ can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010