How Do You Spell COXETER PLANE?

1. A Coxeter plane is a mathematical concept that refers to a two-dimensional surface with specific geometric properties. It is named after the mathematician H.S.M. Coxeter, who made significant contributions to the field of geometry.

   In a Coxeter plane, the main focus is on the study of regular polygons and their symmetries. A regular polygon is a polygon with equal sides and equal angles, such as a square or a equilateral triangle. Symmetry, in this context, refers to the process of mapping an object onto itself through certain transformations without altering its shape or size.

   Coxeter planes are characterized by their reflectional and rotational symmetries. Reflectional symmetry refers to the property of a figure being identical to its mirror image, while rotational symmetry indicates that a figure can be rotated by a fixed angle and still appear the same.

   Within a Coxeter plane, regular polygons can be combined in various ways to form larger structures, such as tessellations or wallpaper patterns. Tessellations are arrangements of regular polygons that completely cover a plane without overlapping or leaving any gaps. Wallpaper patterns, on the other hand, are repeated arrangements of regular polygons that can fill an infinite plane.

   The study of Coxeter planes is essential in various branches of mathematics, including geometry, group theory, and crystallography. Coxeter's extensive research in this area has greatly contributed to our understanding of the symmetries and patterns that can be found in two-dimensional space.

The term "Coxeter plane" is named after the mathematician Harold Scott MacDonald Coxeter. Harold Coxeter was a renowned British-Canadian mathematician who made significant contributions in the field of geometry, especially in the study of polytopes, symmetry, and higher-dimensional Euclidean geometry.

The concept of the "Coxeter plane" emerged from Coxeter's work on reflection groups and higher-dimensional geometry. His research on regular polytopes and symmetry groups led him to study how these geometric objects could be projected onto a two-dimensional plane while still preserving certain symmetries and properties.

It is worth noting that Coxeter did not explicitly use the term "Coxeter plane", but the term was later coined by other mathematicians to honor his contributions to the field. The Coxeter plane refers to a 2-dimensional plane where symmetric patterns related to reflection groups can be visualized and explored.