# CS798 Assignment 3

Due date: Monday, 13 March.

You can submit this assignment by emailing me a PDF or by handing it to me on paper. You can write it up by hand as long as it's tidy.

If you're auditing the course, the implementation question is optional.

## Question 1: Incidence geometry

In Hilbert's formulation of Euclidean geometry, the following axioms constrain the behaviour of the undefined relation of incidence:
• I1: For every two distinct points, there is a unique line incident with both of them.
• I2: Every line is incident with at least two distinct points.
• I3: There exist at least three points with the property that no line is incident with all of them.
1. Prove that there exist at least three lines that don't all meet in a single point.
2. Prove that every point lies on at least two lines.
3. Prove that the parallel postulate is independent of the axioms of incidence.
These are all "quickies" -- they follow from the axioms in a couple of steps. In fact, you don't even need I2 to prove any of them.

## Question 2: Hyperbolic models

Let four points, A, B, C and D be given in the Poincaré disk model of the hyperbolic plane. Give a simple algorithm for computing the intersection of the two line segments AB and CD.

## Question 3: Islamic star patterns

Write a program that draws Islamic star patterns using the "polygons-in-contact" approach. For more information on this technique, see my paper from GI 2005.

Given a tiling and a contact angle as input, and produce a postscript file containing the corresponding star pattern. Your program need not have an interactive user interface, though having an interface is a great way to explore the design space of star patterns. If you want to be fully general you can read specifications for tilings in from files, but it's alright to hard-code the tilings you want to use into the program.

Features
At a minimum, your program should do the following:
• You should support at least the Archimedean tilings 3.12.12, 4.6.12, 4.8.8, and 6.6.6, and the "rosette dual" tilings 10RD and 8RD. Specifications for these tilings (and others) are provided in two formats: In any case, you can either parse the specifications (in which case you'll get other tilings for free) or hard-code the appropriate tilings into your program.
• You should have a rendering mode that starts with a two-coloured fill of the interiors of the polygons in the star pattern, and covers that with a thickened drawing of the edges as ribbons. The ribbons need to have an interior colour and a border colour. You are not required to implement interlacing. The following image gives an example. You can click on it for a PDF version.

Non-trivial extension
You must implement at least one non-trivial extension to your program beyond what's decribed here. This part of the assignment is open-ended. Some ideas for extensions are given on a separate page.

This part of the assignment is not intended to be overwhelming; it's just a way to get you thinking about what ideas might follow on from the basic implementation. Don't feel you have to wear your fingers down to nubs trying to implement your extension. A proof-of-concept will suffice.

You must produce a short write-up describing your implementation. You can structure your submission as you wish, but should include at least the following:

• Describe your implementation. What set of languages, tools, and libraries did you use? What is the interface? Include a few select screen shots to show the various features of your system in operation.
• Mention any bugs or limitations, and say what you would do to overcome them.
• Describe your extension. Explain briefly how you modified the core implementation to accommodate the extension. Comment on how successful you feel it was.
• Include some polished samples produced by your program. At least one sample must clearly demonstrate the effect of your extension, if possible.
I don't plan to examine your source code, but I reserve the right to request it for marking purposes. I also reserve the right to request a demonstration (a demo might be the best way to show off some extensions).

## Bonus: Polyhedral Sculpture

Produce a sculpture of a polyhedron. Points will be awarded based on the originality and difficulty of the method of construction, and the aesthetic appeal of the chosen polyhedron. Some recommendations:
• Choose a polyhedron with lots of symmetries (i.e., cuboctohedral or icosidodecahedral symmetry).
• Choose one that's not convex. Very good choices would be some of the fancier uniform polyhedra, stellations of the icosahedron, compounds, or projections of 4D polytopes. If you're feeling very ambitious, you can copy Tom Lechner and build a room-sized wooden small inverted retrosnub icosicosidodecahedron. I would be suitably impressed.
• If the weather cooperates, sculpt your polyhedron out of snow (this was my original idea for the bonus question; convex polyhedra would be advisable in this case). If that's not possible, consider also looking up Rona Gurkewitz's books on modular origami polyhedra. See also the lovely and delicate paper polyhedra by Ulrich Mikloweit. Tensegrity is cool too.
• You actually have to build it. No kits like zometool or rods-with-magnets.
Submit a photograph and description of your sculpture. If possible, bring it to class.
 Craig S. Kaplan Last updated: