What the heck is a polytope?
Here's a 3D polytope. A polytope is a multi-faceted solid
About This Page
This is my (Stephen Schiller's) web site home page. One of my interests is the beauty of some mathematically generated patterns.
Penrose Rulings
For example, below are some images generated by ammann bars of a penrose tiling. These patterns differ from the usual one I have seen in that I have arranged the five different rulings to intersect at a point, instead of always being offset as small amount.
This image came from my curiosity about what the zero-offset ammann bars would look like if I colored the resulting tiles according to their shape. For a link to a larger, higher quality (vector graphics) version you download the PDF file
About one inch up from the bottom and a 1/2 inche from the left edge is the "center" of the pattern: a point of 5-fold symmetry for the whole thing. One problem that came up in generating this pattern was how to extend the one dimensional pattern used to generate each ruling of ammann bars in the negative direction. Normally this pattern consists of long and short intervals (long intervals having a distance of 1 and short intervals having a distance of 1/phi = 0.618) and is generated by the recurrence L -> LS, S -> L. However I use the Fibonacci number system instead as it yields the same results and is easier to deal with incrementally. (See "Concrete Mathematics" by Graham, Knuth and Patashnik). Negative one in this number system can be represented, in a two's complement sort of way, by either ...01010101. (A) or by ...10101010 (B). The above pattern uses (A). The following pattern (with slightly different coloring and no rulings shown) uses (B). Note how the center of the pattern is totally different in the (A) and (B) case.
Again, here is a link to higher quality PDF picture of the same thing. I can generate larger versions of these patterns. e-mail me if you are interested. e-mail is my last name at polytope dot org.
Parabolas and Cubics and Evolutes
Below is a set of perlendiculars to a parabola. These are aslo the tangents to a curve of the form {(t^3, t^2 | t real}. Click here for a PDF version.
More patters to come later. Whenever I get the time.
Electronic Test Equipment
Also, for the huge number of people out there who devote their free time to buying, selling and refurbishing electronic test & measurement equipment .
Trail Map of Redwood Regional Park with Topographic Lines!
I scanned in the topo map of this park (next to where I live in Oakland CA) and then put the PDF of the park district's map on top of it. Needless to say the trails didn't match exactly so it took a lot warping an fiddling to get them to line up. (I still haven't bothered dot fix the Skyline Blvd and some of the roads.) Note that there are trails on the topo map (from 1960's) that are no longer maintained but still there. Like the one that led me into the back of the archery range!