**PrePrints of RESEARCH PAPERS**

** The structure of self-dual plane graphs with maximum degree 4** (with Elizabeth Hartung)

**Abstract:** Self-dual plane graphs have been studied extensively. C. A. B Smith and W. T. Tutte published *A class of self-dual maps* Cand. J. Math 2 179-196 (1950); in 1992, Archdeacon and Richter *Construction and classification of self-dual spherical polyhedra* J. Comb. Theo. B 54, 37-63, described a method for constructing all self-dual plane graphs; a second construction was produced by Servatius and Christopher *Construction of self-dual graphs*, Amer. Math. Monthly 99 No. 2, 153-158 (1992). Both constructions are inductive. In this paper, we produce four templates from which all self-dual plane graphs with maximum degree 4 (self-dual spherical grids) can be constructed. The self-dual spherical grids are further subdivided into 27 basic automorphism classes. Self-dual spherical grids in the same automorphism class have similar architecture. A smallest example of each class is constructed.

**PrePrints of EXPOSITORY PAPERS**

*Listing the Positive Rationals*

**Abstract.** The rational numbers are countable. The usual proof demonstrates that there exists a one-to-one function from the natural numbers onto the positive rational numbers or simply a list of all positive rationals (without repeats). But the list is not explicitly given. That is, there is no easy way to say which rational is 150th in the list or where 21/13 appears in the list. In this paper,we discuss a different listing of the positive rationals for which we can easily answer such questions.