Constructing Finite Geometries
Left: The Fano plane.†††† Right: A Four Point Geometry
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 arenít 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, Iím also open to research in these areas, e.g., searching for other unknown integer sequences within this geometry.
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.
∑ 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 Geometry, 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.