# James Lawrence

lawrence@gang.umass.edu

Here's some software that I've written for GANG over the years at UMass while working with Rob Kusner. To Download Page.

 Hyperman is a four-dimensional viewer I developed here at GANG. I am interested in finding ways for people to reach an intuitive understanding of four dimensions. The 4D viewers I tried didn't do nearly what I wanted, so I built my own. This is a picture of three interlocked Clifford tori which are the basis for one of the new species of mouse control used by Hyperman. Hyperman runs as an external module to Geomview . (click picture to animate, again to stop)

 An envelope is the boundry of an infinite set of spheres (a surface). A while ago I wrote a Mathematica package to draw these surfaces. The surface here is the boundry of a set of spheres whose centers lie on a torus knot and whose radii are function of position on the knot. You can also make 2D envelopes (curves which are the boundry of a set of circles). Here is the package.
 How can you get a sphere from a paraboloid? The envelope of spheres whose center lie on the paraboloid z=x^2+y^2 and are also tangent to the x-y plane is a sphere.

 I thought up and implemented a fast symmetry detection algorithm for points in n-dimensional space. The program tells you the explicit permutation group acting on the vertices as well as the space representation, order, eigenvectors, and eigenvalues of the group elements. Click here for documentation and download.
The stereogram above is a coulombic minimum-energy configuration of twenty-four points in real 4-space constrained to a 3-sphere (as computed by Evolver). The order of the group is 72. Here is the group in more detail.

 I wrote a similar program which detects the symmetry group of a graph (the graph may possibly be weighted, colored, and directed). In the graph shown here, vertices 0, 1, and 2 may be permuted, in that order, without disturbing the nature of the graph; vertices 3 and 4 may also be permuted. The graph's symmetry group is the dihedral group of order 6. Click here for documentaion and download.

This is a picture of a plane and three coordinate axes on S^3 (a four-dimensional sphere), a snapshot from my conformal motion viewer. The yellow dot is the origin; the light blue dot is the point at infinity.