Visualizing Waterman Polyhedra with MuPAD

Part 4: More explorations with Waterman library

by M. Majewski

Zayed University, United Arab Emirates

Waterman polyhedra with large radii are like big colorful jewels. They may have hundereds of faces. I did mention already that creating such polyhedra could be a serious computational problem. Another problem could be displaying a few such examples on a single web page using JavaView applets. Due to computer memory limtation it may happen that a web page with too many examples or too large examples will not show all examples or the web browser will crash. For this reason I created a separate page with some larger examples of Waterman polyhedra.

Four examples of large Waterman polyhedra

We will start with two examples of Waterman polyhedra from Newbold's sequences, w_3,45 and w_7,77. Here the number of faces is still not very large. However, the shapes of such polyhdera are getting quite unusual.

 plot(WatermanPoly(3,45), Scaling=Constrained, Axes=Boxed):

 plot(WatermanPoly(7,77), Scaling=Constrained, Axes=Boxed):

Another natural question would be to investigate symmetries of Waterman polyhedra and show the one with the largest number of symmetries of a particular type. For example the polyhedron obtained for P(0,0,0) and R=21 shows a number of rotational and reflexion symmetries,

 plot(WatermanExplore([0,0,0],21), Scaling=Constrained, Axes=None)

For very large radii we obtain a polyhedra looking like a big jewels or mysterious crystals. Below we show polyhedron w_2,1000 obtained in MuPAD. Analyzing its structure could be quite challenging job.

 plot(WatermanPoly(2,1000), Scaling=Constrained, Axes=None):

We will leave explorations of Waterman polyhedra for another opportunity. I suggest to MuPAD users to install the Waterman library and continue these interesting investigations on their own.

mathPAD Online

The Magazine of the MuPAD Research Group, Vol. 15, Year 2006

CURRENT ISSUE

SECTIONS

RESOURCES

Four examples of large Waterman polyhedra