#### Date of Award

6-1998

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Arthur T. White

#### Second Advisor

Dr. Allen J. Schwenk

#### Third Advisor

Dr. Clifton Ealy Jr.

#### Fourth Advisor

Dr. Michael Raines

#### Abstract

The standard non-Euclidean geometries, hyperbolic geometry and elliptical geometry, both arise by negating the parallel postulate of Euclid. Both these geometries share with Euclidean geometries an infinitude of points and lines. But also possible are many finite geometries. Among these are the class of projective geometries *PG(m,q) *of projective dimension *m* (*m* > 2) and the *k*-configurations. These mathematical objects, although primarily geometric in nature, provide related structures of combinatorial interest: block designs. These have applications in scheduling problems and the design of experiments for statistical analysis. Recently, A. T. White has added a topological flavor to the study of the geometries *PG{m,q).* In particular, when *m* = 2 he found models, as imbeddings of Cayley graphs on surfaces and pseudosurfaces, using voltage graphs (obtained from the classical field construction of *PG(m,q)). *In this dissertation we have extended his work for the case *m + I* a prime number in a similar manner. We have also used this approach (and *ad hoc *methods at times) to successfully provide concise voltage graph constructions for 3-configurations (a -configuration is a finite geometry such that exactly *k *points lie on any line, exactly *m *lines intersect at any given point, where* m* is a constant, and any two points determine at most one line). The class of 3-configurations is interesting since it contains some classical finite geometries, e.g., the *Fano plane *and the *Desargues *and *Pappus configurations.*

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Figueroa-Centeno, Ramon Manuel, "Surface Models of Finite Geometries" (1998). *Dissertations*. 1573.

http://scholarworks.wmich.edu/dissertations/1573