John Hiigli

As a student of Synergetic Geometry I became interested in drawing and painting high frequency tetrahedrons and other polyhedrons. In order to draw these complex polyhedrons I worked very large using a six foot sliding architect ruler mounted on the wall. Later a friend, Stephen Weil drew a few of these using Mathematica. I had Stephen's Mathematica drawings blown up at a graphic arts studio and transferred them from paper to canvas stretched on a wall in my studio. The lowest-frequency tetrahedron to contain a cube octahedron is the four-frequency tetrahedron. The sixteen-frequency tetrahedron in Chrome 175 contains three distinct cuboctahedrons, with each consecutive iteration eight times the volume of the previous one, i.e. 20 tetrahedrons, 160 tetrahedrons, 1280 tetrahedrons. In addition many other polyhedrons are revealed. We might conclude that the series of high-frequency tetrahedrons manifests scale change, similar to the octave in music.

I met Janos Saxon-Zsasz in Budapest at the International Symmetry Festival in 2003. Since then I have admired his work and collaborated with him whenever possible. In 2009 I decided to extend this collaboration in respect to the Saxon-Szasz 2000 work Poly-dimensional Space. In this computer animation Janos replaced the plane figure with a corresponding cube, to which he attached more cubes at the vertices in 1:3 proportion. Working together Stephen Weil and I produced two versions of Saxon's brilliant drawing, each with a different color system but the same size. I took the 'red version' to Hungary in 2010 and presented it to Janos Saxon-Szasz. He seemed pleased to see this Mathematica version of his original drawing and we exhibited it at Bridges10 in Pecs. The 'black and blue' version shown above contains one more iteration than the Saxon-Szasz original. This work is a virtual scale-change machine and can also be regarded as a 3-dimensional fractal; I like it very much!

This work models the Isotropic Vector Matrix invented by the American architect/inventor Richard Buckminster Fuller. Inter-diagonalizing vertices and mid-face points of the cube locates other structures which have a common edge length: cube-octahedron volume 20; duo-tetrahedron volume 12; dodecahedron volume 6, octahedron volume 4, the regular tetrahedron with a volume of 1 and a smaller octahedron in the very center with a volume of .5 or one-half of the regular tetrahedron. The octahedron of four tetrahedrons and the octahedron of .5 tetrahedrons is another instance of scale change. These polyhedrons form a family-of-relationships united by a common edge length and this system is inter-transformable. It inculcates a cube-octahedron phase and a duo-tetrahedron phase and these two phases can cycle back and forth freely. This is the 'miracle of space' that Fuller, and of course many others, only led us…..to discover for ourselves!