**Truth,
Geometry, Algebra**

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'."

– Coxeter, 1987,
introduction to
Trudeau's *The
Non-Euclidean
Revolution*

1.1 Trudeau's *Diamond
Theory* of Truth

1.2 Trudeau's *Story
Theory* of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 *Coxeter
Theory* according to Borovik

2.1.1 The Geometry–

*Mirror
Systems* in *Coxeter
Theory*

2.1.2 The Algebra–

*Coxeter
Languages* in *Coxeter
Theory*

2.2 *Diamond Theory*
according
to Cullinane

2.2.1 The Geometry–

Examples: *Eightfold
Cube*
and *Solomon's
Cube*

2.2.2 The Algebra–

Examples: Cullinane
and
(rather indirectly related) Gerhard
Grams

**Summary
of the story thus
far:**

Update of Feb. 21, 2010--

**Reflections**

From the Wikipedia article "**Reflection
Group**"
that
I
created
on Aug.
10,
2005– as
revised
on
Nov.
25,
2009–

Historically, (Coxeter
1934)
proved that every reflection group [Euclidean, by the current Wikipedia
definition] is a Coxeter group (i.e., has a presentation where all
relations are of the form r_{i}r_{j})^{k}),
and
indeed
this
paper
introduced
the
notion
of
a
Coxeter
group,
while
(Coxeter
1935)
proved that every finite Coxeter group had a representation as a
reflection group [again, Euclidean], and classified finite Coxeter
groups.
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981). |

**Related material:**

This note, "A Simple Reflection Group of Order 168," and

by Ascher Wagner, U. of Birmingham, received 27 July 1977

Journal | Geometriae Dedicata |

Publisher | Springer Netherlands |

Issue | Volume 9, Number 2 / June, 1980 |

[A
*primitive*
permuation group preserves

no nontrivial partition of the set it acts upon.]

Clearly the eightfold cube is a counterexample.

Update of Feb. 24, 2010--

**Transvections**

A topic related to the above material on reflection groups over GF(2)--

*Transvection*groups over GF(2). See, for instance...

- Binary
Coordinate
Systems, by Steven H. Cullinane, 1984

- Classification
of
the
Finite
N-Generator
Transvection
Groups
Over
Z
_{2}, by Jizhu Nan and Jing Zhao, 2009,*Advances in Applied Mathematics*Vol. 44 Issue 3 (March 2010), 185–202

- Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"

Coxeter vs. Fano

Excerpts from Coxeter's Projective Geometry sketch his attitude toward geometry in characteristic two.