Viz Lab Summer Grant 2006

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Creating a program that will print out 2-dimensional "nets" of patterned polygon faces for polyhedra of positive genus, which can be folded up and glued together, forming the whole polyhedron

Douglas Dunham Computer Science

The original proposed research was to create a program to print out 2-dimensional "nets" of patterned polygon faces for polyhedra of positive genus, which can be folded up and glued together, forming the whole polyhedron. To lay out the polygon faces in a connected net is an unsolved problem even for a genus 0 polyhedron (no holes in it), so we plan to use an approximate algorithm. The patterns on the polygons will have color symmetry and the program will also allow the user to visualize the polyhedron before printing.

Unfortunately, between the time the proposal was submitted and the time the project was to begin, the graduate student who was to work on this project decided to return to India instead. As a result, work was done on several other projects (listed below) instead.

Even then, a small bit of progress has been made on this project. A specialized program has been written that creates a net for one pattern on a polyhedron of genus 3. The polyhedron is shown below:

and the net is at:
http://www.d.umn.edu/~dduhham/polynet.pdf (PDF - 148kb)

Other projects that were worked on included animation of hyperbolic pattern-drawing algorithms, drawing semi-regular hyperbolic tessellations, work on "Circle Limit III" patterns, and hyperbolic rug patterns.

The work on animation of hyperbolic pattern-drawing algorithms was done by Joshua Jacobs, another VDIL researcher, in conjunction with my graduate student Ajit Datar. The work done by Joshua Jacobs was the animation of three algorithms that I implemented some time ago. The first one is a"Hamiltonian path" algorithm that iterates through each copy of a motif, following a Hamiltonian path in the symmetry group, and is shown at:

http://www.d.umn.edu/~jjacobs2/Research/Hyperbole/HamiltonianC.htm

The third algorithm recursively traces a spanning tree of cosets in the symmetry group, and is shown at:

http://www.d.umn.edu/~jjacobs2/Research/Hyperbole/CosetSpanning.htm

Some work was done by another one of my graduate students, Ravi Bharadia, on a Java program to generate semi-regular tessellations of the hyperbolic plane. Two sample tessellations are shown at:

http://www.d.umn.edu/~ddunham/t33337.pdf (PDF - 1.3Mb)

and at:

http://www.d.umn.edu/~ddunham/t33337.pdf (PDF - 356kb)

I made the most progress on an ongoing research project to generalize the hyperbolic "Circle Limit III" pattern by the Dutch artist M.C. Escher. This is usually considered to be his most attractive hyperbolic pattern. But, in my view, it is also the most mathematically interesting. I have submitted a paper to the Bridges 2006 Conference, which has been accepted; the paper will be presented at the conference in August in London. It can be viewed at:

http://www.d.umn.edu/~ddunham/dunbrid06.pdf (PDF - 2.1Mb)

Perhaps the most interesting pattern in the paper is at:

http://www.d.umn.edu/~ddunham/figure12.pdf (PDF - 220kb)

I also prepared my talk, "H.S.M. Coxeter and Tony Bomford’s Colored Hyperbolic Rugs" for the Bridges 2005 Conference, which was held in at the end of July 2005 in Banff, Canada. The talk slides may be viewed at:

http://www.d.umn.edu/~ddunham/br05talk.pdf (PDF - 4.2Mb)

There is a new proposed hyperbolic rug pattern at:

http://www.d.umn.edu/~ddunham/figrug.pdf (PDF - 416kb)