SOME ESSAY TOPICS
MTH 338

Below are some possible topics for your writing project. This list is not intended to be exhaustive -- feel free to suggest alternatives. Nor is it absolutely guaranteed that each of the topics will automatically lead to a decent project! Ultimately, each topic depends on what you make of it. It is essential that your essay include some mathematics, not just words.

Topics in taxicab geometry:

• Add more features such as mass transit routes, lakes, mountains, 1-way streets.
• Generalize to 3 dimensions in one of several ways. For instance, you could assume walkways between nearby buildings or require the use of elevators.
• Use a triangular layout of streets (Chinese checker geometry).
• Use an octagonal/square tiling to add "roundabouts".
• Use the taxicab unit circle to define taxicab trig functions and discuss their properties.
• Given 3 points, either prove that the 3 midsets meet in a point or find a counterexample.

Topics in hyperbolic geometry:

• Contrast the Poincaré disk model of hyperbolic geometry with the Beltrami-Klein model, in which the lines are cords of circles.
• Show how to use Euclidean geometry to construct lines between 2 points, lines perpendicular to given lines, etc., in the Poincaré disk and/or in the Beltrami-Klein models of hyperbolic geometry.
• Focus on one or more features such as SAS, the angle of parallelism, Saccheri quadrilaterals, equivalence and area, etc., and discuss them using a particular model such as the Poincaré disk.

Topics in elliptic and spherical geometry:

• Show how to use Euclidean geometry to construct lines between 2 points, lines perpendicular to given lines, etc., in the Klein and/or spherical models of elliptic geometry.
• Focus on one or more features such as SAS, Saccheri quadrilaterals, equivalence and area, etc., and discuss them using a particular model such as the Klein model.
• Discuss the significance of the non-orientability of (single) elliptic geometry.
• Contrast the Klein model of (single) elliptic geometry with spherical geometry (also called double elliptic geometry).
• Consider (some of) the results in §3 of the text, derived in the context of neutral geometry, and determine whether they hold in elliptic geometry.
• Discuss polygons in elliptic geometry, along the lines of the treatment in §6.4 of the text for hyperbolic geometry.
• Discuss area in elliptic geometry, along the lines of the treatment in §6.5 of the text for hyperbolic geometry.
• Discuss the ratio of circumference to diameter for circles on the sphere.

Topics in neutral geometry:

• Using Geometer's Sketchpad (or otherwise), attempt to construct Figure 6.4.5 in the Poincaré disk. Discuss the implications of your success or failure.
• Fix the proof in the text (before the statement of Theorem 6.4.7) of the equivalence of triangles and their associated Saccheri quadrilaterals.

Topics in finite geometries:

• Discuss the properties of the affine geometries and/or the projective geometries such as the number of lines and points.
• Describe and compare several examples of finite geometries.
• Use the 7-point projective plane (Fano's geometry) to discuss the multiplication properties of octonions.

Topics involving more than one geometry:

• Relate the Klein disk model of spherical geometry to the unit sphere in Euclidean space. Analogously, relate the Poincaré disk model of hyperbolic geometry to the unit hyperboloid in Minkowski space (the space of special relativity).
• Some of the above topics can be discussed in more than one setting, so that your essay could then compare the results for different geometries.
• Consider a hybrid geometry which has a combination of hyperbolic, elliptic, and/or Euclidean regions.