GVSU REU SUMMER 2016 Constructing Finite
Geometries Steven Schlicker |

Left:
The Fano plane. Right: A Four Point
Geometry

**Finite Geometries**

Geometry was originally envisioned as
a way to cope with measurements of land or the earth. From a mathematical
standpoint, a geometric system (or geometry) is any collection of objects
(called points) and lines that relate the points. Finite geometries (ones in
which there are only a finite number of points) are especially interesting to
me. If you have ever used a computer, you are familiar with a finite geometry.
A computer screen
is made of a finite collection of pixels, so in the computer world there are
only finitely many points. The geometry of the screen is then a finite
geometry. A famous finite geometry is the Fano plane, shown above at left. In
this geometry there are exactly 7 points and 7 lines, with each line containing
exactly 3 points. The Fano plane is an interesting object: among other things
it completely describes the algebra structure of the octonions, and is the
smallest projective plane. (You aren’t expected to know these things, but you
can look them up if you are interested.)

**Summer 2016 REU Project**

The
proposed project for this summer is to investigate what finite geometries can
be created by measuring distances in an unusual way. For example, what would
you say if someone told you that the distance between the integers 5 and 6 is
3? This may seem odd, but it is true in one geometry on the set of integers
where we define the distance from 5 to 6 as follows: find the largest of all
the shortest distances from the multiples of 6 to the multiples of 5 -
this distance is 2 (from 12 to 10). Then determine the largest of all the
shortest distances from the multiples of 5 to the multiples of 6 -
this distance is 3 (from 15 to either 12 or 18). The distance between 5 and 6
is the maximum of these two numbers, or 3. We will label this distance *h*, so *h*(5,6) = 3. Similarly, *h*(6,13)
= 6 and *h*(5,13) = 6. Thus, the
distance between 5 and 13 is not the sum of the distances from 5 to 6 and 6 to
13, which implies that 5, 6, and 13 do not lie on the same line (this is quite
different than what happens in standard Euclidean geometry). This summer we
will explore the kinds of geometries that arise from this type of distance
function in a finite setting. We will see how we can create some finite
geometries (an example of such a geometry is the 4-point geometry shown above
at right), or ask if some famous finite geometries (the Fano plane, for
example) can be created through this approach.

**What Will
This Research Involve?**

The
function *h *described above is called
the Hausdorff metric. We will use this metric (just like we use the standard
Euclidean distance for Euclidean geometry) to create our geometry from points
and lines, only the points will be certain types of sets and the lines will be
defined using the Hausdorff metric defined on those sets. The sets in this case
will be subgroups of finite groups.

In
past REUs we have studied the geometry of the Hausdorff metric on the compact
subsets of *n*-dimensional real space,
and have learned a lot about this geometry. For example, we have completely classified
circles in this geometry, have made connections to number theory (including
discovering a previously unknown infinite three-parameter family of integer
sequences), found that lines in this geometry can just stop, and proved
fascinating facts about the primes 19 and 37. Most recently, we have been
investigating the concept of orthogonality in this space with a goal of trying
to define a trigonometry of sets. Papers describing some of these ideas can be
found at the end of this article. If any of these topics intrigue you, I’m also
open to research in these areas, e.g., searching for other unknown integer
sequences within this geometry.

**My Background**

I started conducting research with undergraduate students in 1990 with a project involving fractal geometry and regular polygons. I continued this type or research when I came to Grand Valley, ultimately deciding to apply to the National Science Foundation to form a Research Experiences for Undergraduates (REU) site at GVSU. We hosted our first REU site in 2000 and our program has been running strong ever since. My current interests are in understanding different types of geometries that can be created using the Hausdorff metric. I have been engaged in this area since 2000. My groups have published several papers (listed in the references) and presented their work at local, regional, and national conferences.

**Desirable
Experiences for Applicants**

Applicants should
be willing to work collaborative with peers and faculty, but also be capable of
independent work. An interest in geometry is important, but it is not necessary
to have more than a basic familiarity with Euclidean geometry. The proposed project also deals with groups,
but we can learn the basics that we need in the first couple of days of the
REU. Most desirable is a positive attitude toward geometric and algebraic
ideas, and the willingness to just muck around with stuff.

**What
this Project is NOT...**

Although the Hausdorff
metric has many real life applications, I know almost nothing about them and
they are not related to this research. The same is true of finite geometries
(which, as I have heard, have applications in experimental design, information
security, particle physics and coding theory). This is a project in PURE
MATHEMATICS.

**How
to Apply**

For application
information and instructions, please visit the GVSU
Summer Mathematics REU home page.

**References**

· Pythagorean
Orthogonality of Compact Sets, Pallavi Aggarwal*, Steven Schlicker, and Ryan
Swartzentruber*. Submitted to *Involve*.

· Missing
edge coverings of bipartite graphs and the geometry of the Hausdorff metric,
Katrina
Honigs*. *Journal of Ge**ometry*, April 2013, Volume 104, Issue 1,
107-125.

· A Missing
Prime Configuration in the Hausdorff Metric Geometry, Chantel Blackburn*,
Kristina Lund*, Steven Schlicker, Patrick Sigmon*, and Alex Zupan*. *Journal
of Geometry*, (2009), 92, Numbers 1-2, 28-59.

· Polygonal
chain sequences in the space of compact sets, Steven Schlicker, Lisa
Morales* and Dan Schultheis*. *Journal of
Integer Sequences*, Vol. 12 (2009), Article 09.1.7.

· Fibonacci
sequences in the space of compact sets, Kris Lund*, Steven Schlicker, and
Patrick Sigmon*. *Involve*, Vol. 1
(2008), No. 2, 197-215,

· A
Singular Introduction to the Hausdorff Metric Geometry, Dominic Braun*, John
Mayberry*, Audrey Malagon*, and Steven Schlicker. *Pi Mu Epsilon Journal*, Vol. 12, No. 3, p. 129-138, 2005.

· When
Lines Go Bad In Hyperspace, Christopher Bay*, Amber Lembcke*, and Steven Schlicker.
*Demonstratio Mathematica*, No. 3,
Volume XXXVIII (2005), p. 689-701.

· Nine
entries in the On-Line Encyclopedia of Integer
Sequences: A152927, A152928, A152929, A152930, A152931, A152932, A152933,
A152934, A152939.

*Indicates an REU student author.