MATHSEARCH 2000
Senior Project Prepared by Geoff Ball and Humphrey Gastineau-Hills, University of Sydney.

Pedan Polygons and Polyhedra

Welcome to the 2000 MATHSEARCH Project which is organised by The Mathematical Association of N.S. W. to help its committee select Year 11 NSW students to be invited to participate in the prestigious National Mathematics Summer School to be held i Canberra in January 2001.

Project2000 is concerned with integer geometry. the geometry of points that are pairwise integer distances apart. In particular we will confine this project to some integer polygons and polyhedra.

Definition 1. An integer polygon is a is a convex polygon with the property that the distance between any two of its vertices is an integer. Recall that a convex polygon is one in which the interval joining any two points in the polygon also lies entirely within the polygon.
For example, the rectangle with adjacent side lengths 3 and 4 units respectively is an integer polygon with diagonals of length 5 units. On the other hand you might verify that no square can be an integer polygon.

Definition 2. A polyhedron is a three dimensional solid in which each face is a polygon.
An integer polyhedron is a convex polyhedron with the property that the distance between any two of its vertices is an integer.

In 1999, Blake Peterson of Brigham Young University and James Jordan of Washington State University published an article on integer polygons and polyhedra. This project investigates some aspects of their work. In recognition of their inspiration for the project we will call such figures Pedan polygons and Pedan polyhedra.

There will be times in the project when it is useful to compare different Pedan figures. To facilitate this we will refer the sum of all the integer distances as the perimeter plus and denote it by p⁺.

Final Note. It transpires that cyclic quadrilaterals will play an important role in your investigations. Cyclic quadrilaterals have all four vertices on the same circle. It is easy to show that a quadrilateral is cyclic if and only if the opposite angles are supplementary.

There are many important geometric facts associated with cyclic quadrilaterals. One which will be of particular significance for this project is due to great Greek geometer Ptolemy.

Ptolemy's Theorem. A polygon is cyclic if and only if the sum of the products of the lengths of the opposite sides is equal to the product of the lengths of the diagonals.

Lots of quadrilaterals are cyclic, for instance rectangles, and so Ptolemy's theorem is a useful tool in the search for Pedan polygons, and later, for Pedan polyhedra.

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