COLLABORATIVE RESEARCH PROJECTS



Research Topics  Descriptions 
Quadray, or simplicial coordinates address points in Euclidean space using 4tuples (a,b,c,d), which may be normalized such that a+b+c+d = some number (e.g. 0), or such that a,b,c,d are all nonnegative, with at least one of them equal to zero. The vectors (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) point from the center to the four corners of a regular tetrahedron (or simplex). Operations of vector addition and scalar multiplication have the usual meanings. David Chako first introduced me to this apparatus, while Tom Ace has developed the algebra's rotation methods. Many others have contributed their thinking. 

Jim Lehman is an artist with a fascination for matters geometric. His studies have focused on a lattice of spacefilling concave and convex dodecahedra, which juxtaposes with the facecenteredcubic lattice (FCC). The FCC was also the site of R. Buckminster Fuller's pioneering geometric explorations, and in particular his concentric hierarchy, a sculpture of nested polyhedra. In this work, I investigate these two sculptures in tandem. 

Steve Waterman has a longstanding interest in the CCP packing, and in particular those polyhedra defined by all CCP spheres equidistant from a central sphere, plus any closer ones which do not interrupt the convexity of the whole. This focus proved an ideal application for the freely available Qhull software package, which develops convex hulls given a listing of points in a space of however many dimensions. The polyhedra defined by Steve's criteria are quite aesthetically pleasing, given their quasisphericity and rotational symmetry. 

Here we have some additional research on more random sphere packings than the CCP using the Qhull package's ability to trace the voronoi cells around each point (ball center). This investigation was taken up in response to John Brawley's fifteen year interest in such ball packings, in the context of a broader philosophy. 

The standard ball packings used to study crystal lattices may be developed in relation to the CCP. More specifically, four CCP lattices, turned on or off in combination, create the SCP and BCC patterns, plus define the points of the HCP. Russell Chu did a lot of independent research in this area, as well as Scott Childs, who applied quadray coordinates in his studies (see above). 

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