
Elements of Finite Geometry
by Steven H. Cullinane
A work under construction, last updated Jan. 16, 2016.
(Downloadable compressed folder last updated Jan. 9, 2016.)
- A
downloadable compressed folder of the source website
finitegeometry.org/sc was last updated on Jan. 9, 2016. Download at
http://finite-geometry.github.io/galois/finitegeometry-Jan-9-2016.zip .
- This is a work under construction using WorkFlowy. The Web address to read the preliminary version is https://workflowy.com/s/WStyDzkupQ . (Click "Help" for instruction on using folded text.)
- Introduction
This
material is intended as a course supplement or independent-study source
for college students (undergraduate and graduate) who have taken
courses in abstract algebra (linear algebra and group theory) and
modern geometry (affine, projective, finite).
It may also interest some professional mathematicians, since the
content has not been published previously except as follows.
A preliminary version was distributed at the 1976 summer AMS-MAA joint
summer meetings in Toronto. Excerpts were published in the
journal Computer Graphics and Art (Vol. 2, No. 1, February
1977, pp. 5-7).
Additional notes were distributed informally, and are available
as Notes on Groups and Geometry, 1978-1986 (pdf ). These and
later notes are, at this writing, on the Web at
http://finitegeometry.org/sc/map.html. This book is based on the notes
at that site.
- Prerequisites
"There is no royal road to geometry."
-- Saying attributed to Euclid
.
The reader is not expected to fully understand this book unless he or
she has had undergraduate instruction in linear algebra, abstract
algebra, and modern geometry. The following references may, however, be
helpful:
Books:
Symmetry, by Hermann Weyl, Contemporary Abstract Algebra, by Joseph A. Gallian, and
Geometry and Symmetry, by Paul B. Yale
- History
- Galois fields
In 1830, Galois introduced finite (or, now, Galois) fields, a new type of number systems that would later prove widely useful.
- Permutation groups
Galois also introduced permutation groups, a very basic and important concept with which we assume the reader is familiar.
- Froebel cubes
In 1838, educator Friedrich Fröbel introduced his "third gift," a boxed
set of eight identical cubical unmarked kindergarten blocks.
- Fano geometries
In 1892, Gino Fano introduced finite
geometries. These were abstract structures defined axiomatically. They
were unrelated, so far as anyone knew at the time, to any actual physical structures.
- Structures considered
- 1. The fourfold square — A 2x2 array of square unit cells
- 2. The eightfold cube — A 2x2x2 array of unit cubes
- 3. The 16fold square — A 4x4 array of square unit cells
- 4. The 24fold rectangle — A 4x6 array of square unit cells, split into three 4x2 "bricks"
- 5. The 64fold cube — A 4x4x4 array of unit cubes
- Coordinates
Coordinates
formed from the two-element Galois field GF(2) may be applied to each
of these in such a way that the resulting labeled structures have large
and interesting symmetry groups. In the case of the 4x6 array, it is
useful to view coordinates in each 4x2 brick not as 3-tuples formed
from GF(2), but rather as elements of the field GF(8).