# Diamond Twin

## TL;DR

This paper explores the relationship between the diamond crystal and a hypothetical crystal called the diamond twin, highlighting their shared symmetry properties and introducing the concept of orthogonally symmetric lattices.

## Contribution

It establishes a mutual relationship between diamond and its twin using building blocks and period lattices, and introduces the concept of orthogonally symmetric lattices.

## Key findings

- Diamond and its twin share a remarkable symmetric property.
- Introduction of orthogonally symmetric lattices as a generalization.
- Distinctiveness of diamond and its twin among crystal structures.

## Abstract

As noticed in 2006 by the author of the present article, the hypothetical crystal---described by crystallographer F. Laves (1932) for the first time and designated ``Laves' graph of girth ten" by geometer H. S. M. Coxeter (1955)---is a unique crystal net sharing a remarkable symmetric property with the diamond crystal, thus deserving to be called the diamond twin although their shapes look quite a bit different at first sight. In this short note, we shall provide an interesting mutual relationship between them, expressed in terms of ``building blocks" and ``period lattices." This may give further justification to employ the word ``twin." What is more, our discussion brings us to the notion of ``orthogonally symmetric lattice," a generalization of irreducible root lattices, which makes the diamond and its twin very distinct among all crystal structures.