Monday, September 28, 2009
The summary below is something I tossed out on the Internet recently. A fun way to test your skills, should you accept this challenge, is to find where I put it. Hint: filed yesterday.
[ posting to some list ]
Just thought I'd continue some earlier threads.
The 4D geometry I've been talking about takes the tetrahedron as topologically more minimal than the cube (with good reason: it is) and therefore switches to a different model of 3rd and 2nd powering, introducing growing/shrinking tetrahedra and triangles in place of cubes and squares.
Instead of XYZ (cubist), the matrix or scaffold is developed from sphere packing in the well known CCP pattern (same as FCC) with all edges the same length.
In the CCP, every unit radius sphere is surrounded by 12 others tangent to it, in a cuboctahedral conformation. Successive layers of balls, packing around a nuclear ball, define a progression of 1, 12, 42, 92, 162... balls, always in the same cuboctahedral conformation. That's just how CCP is defined.
Four unit radius balls, inter-tangent to one another, define our unit of volume in this system, our model of 3rd powering. The cube defined by two such tetrahedra intersecting at mid-edges has a volume of 3. The octahedron dual to this cube, the other void in the CCP complementing the tetrahedra, has a volume of 4. The rhombic dodecahedra that encase each of the CCP balls in a space-filling manner, such that edges between adjacent ball centers penetrate their diamond faces at 90 degrees, have a volume of 6.
These easy whole number volumes in conjunction with a sphere packing lattice sets the stage for a bevy of geometric concepts, tightly organized and accessible to grade schoolers, not just adults.
Linking the 4D tetrahedron to Karl Menger's "geometry of lumps" provides a definitional context that certifies this as a non-Euclidean geometry, but not in the sense of jiggering with the fifth postulate, although there's more we could say on that topic as well.
All of this material was published in the 1970s by a famous architect and later Medal of Freedom winner. Since that time, students of this geometry have spawned several new areas of investigation, including elastic interval geometry which adds dynamism to the edges. EIG was also inspired by the work of Kenneth Snelson, the internationally recognized sculptor who pioneered the tensegrity genre. Gerald de Jong was an early developer of EIG. You'll find several free tools on the Internet, such as Tim Tyler's work at springie.com. Alan Ferguson's SpringDance, a Delphi application, seems to be no longer on-line.
The cuboctahedron defined by 12 balls around a nuclear ball has a volume of 20 tetravolumes. We also have a bridge to the five-fold symmetric family and a modular system for dissecting these shapes, including the A, B and T modules, all of equal volume 1/24. Two As and 1 B combine to give a space-filling irregular tetrahedron called the MITE in our namespace, and depicted on page 71 of Coxeter's Regular Polytopes. There's reason to bill this the minimum space-filler in the sense that it's tetrahedral (simplest polyhedron) and comes without the need for a complement. The dissection into As and Bs is specific to this 4D geometry (4D in the sense of "four directional").
Given the links to architecture, art and computer graphics ("geometry of lumps" makes sense in ray tracing), it's not surprising that students looking at careers in these areas are boning up on the related syllabus, much of it free and on-line. You'll find lots on YouTube as well.
The T module, also of volume 1/24, comes in left and right handed versions, as do the A and B modules. 120 of them (60 left and 60 right) define a rhombic triacontahedron of volume 5. The radius of this shape is just a tad less than unity i.e. it almost shrink wraps the CCP ball. Expanding this radius by the 3rd root of 3/2 takes the volume from 5 to 7.5, where the radius turns out to be phi/sqrt(2). This larger rhombic triacontahedron intersects the edges of the volume 6 rhombic dodecahedron, which is tangent to the CCP ball at its 12 diamond face centers.
To make a little chart:
A, B, T vol 1/24
Mite (space-filling) vol 1/8
Tetrahedron vol 1
Cube vol 3
Octahedron vol 4
Rh Triacontahedron vol 5
Rh Dodecahedron vol 6
Rh Triacontahedron vol 7.5
Icosahedron vol ~18.51
Cuboctahedron vol 20
A few animations communicate this information fairly succinctly. Clocktet by Richard Hawkins, which premiered at the Fuller Centennial in 1995 in Balboa Park in San Diego, was one of the first in this genre. We've seen several since. Getting more in the pipeline, from such shops as Disney / Pixar, is a priority of my working group in Portland. We also do a lot with Python, showing students how to do their own ray tracings and interactive geometry cartoons (POV-Ray and VPython), plus how to use generators to yield such sequences as 1, 12, 42, 92... (above), an easy entre to programming, which by now has an integral role in the digital math track we're developing in the state of Oregon.
All of the above is well documented, including write-ups of the various classroom pilots, open source software, animations etc. There's an institute (bfi.org) along with affiliated think tanks all supporting one another in various ways. I've personally been flown to Sweden and Lithuania to brief my peers.
Universities have been slow to catch on however, which is why the initiatives have been mostly in the private sector. Dr. Arthur Loeb (MIT, Harvard) helped us quite a bit. Dr. John Belt with SUNY has also been of considerable assistance.
Fuller's chief collaborator on the magnum opus in question, available on the web for some years, was a career intelligence officer and author of Washington Itself, Cosmic Fishing and Paradise Mislaid. I've included his picture on my Myspace page.
Posted by Kirby Urner at 7:58 PM