And here resumes the tale of how it wove.
When dreams obsessive haunt the questing mind
and darkling, daunt the reach to find
a chance to carve our names upon the Tree –
for those who will come after, strive to Be.
— MRK
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After I had survived three Quarters at UF my parents decided I needed to take a little break and wake up and smell the Styrofoam coffee. In other words, I was encouraged to try working for a living.
They were concerned about me, no doubt. I had few friends, no job, no plan. Although I had been continuing my college education majoring in Physics, the only significant change in my life was my current obsession with hypergeometry that would distract me away from any remaining chance of rekindling my relationship with Beverly.
I can think of no greater folly than to neglect a chance for love, in order to follow a chance for money, or fame. That is the classic Scrooge mistake, to value money and things over people. But this bit of wisdom I learned the hard way. There is no excuse for it. This was 20 years before the Internet, and my worldview then was that long-distance relationships do not last. I therefore considered my romance with Beverly doomed, since in my present circumstances I would see her even less often than before. Unlike my father, who had met my mother in his senior year and married her 8 months later, I met Beverly in my Youngster (sophomore) year at USNA. There would have been a minimum of two more years before we could have been married, and now that I had resigned from USNA and moved back south to Gainesville, I had no idea when I would graduate, have a decent job, and be able to pull off supporting a wife and starting a family.
Beverly Ann of Philadelphia, this is my open apology to you before the world. I have tried to tell myself that my distraction was understandable, but I keep coming around to the fact that one real love is worth a thousand inventions. Confronted with the Mysterious, I wandered away into the world of people and ideas and technology. I hope you had and have a wonderful life without me, for you always deserved the best. If this binary message in a bottle reaches you, I wish I could assure you that the fault was all mine, that I did not miss out on the great adventure of your life because I found you lacking; quite the contrary — you deserved a better man than I was then. I know that in some of the time lines in the Multiverse you and I did marry and have children. Maybe some of them had my eyes and your smile, or vice versa.
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Here is the folding pattern that started it all. I realized that I wanted to make an inside-out model of a tesseract. The cube-within-a-cube could be seen as one of two ways to view the 4D geometry from the perspective of 3-space.
In one version, the usual perspective, a tiny cube is apparently “inside” a larger cube and the corresponding corners are connected by inward-pointing lines that make every point the juncture of 4 edges, just as all corners of an ordinary 3D cube are the junctures of 3 edges.
In the other version the inner cube seems to have been pulled apart. The inward-pointing truncated pyramids (whose square tops came together to make the little cube apparently floating at the center of the large outer cube) are now pointing away from the larger cube. Each of the six truncated pyramids points by itself from a face of the large cube outwards along one of the +x. -x, +y, -y, +z, and -z directions. The little inner cube has thus become the six little outer squares at the tops of the outward-pointing pyramids.
I had been making and contemplating these cardboard models since before the Spring quarter ended. I had sometimes referred to them as “tunnel crystals”, since meditation (which I had taught myself at USNA using Benson’s The Relaxation Response) often allowed the visual illusion that the shape had folded back in on itself and the outward-pointing pyramids had become, instead, corridors leading to the center of the tesseract, dwindling in cross section because of parallax perspective.
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I had shown this peculiar inside-out shape to my friend Thomas Weiss. He was immediately struck by its unique symmetry, and simplified the design by extending the sub-pyramids to points instead of truncating them. This eliminates all the little squares in the folding pattern and makes it much simpler. Now it consists only of twelve rhombi. When they come together you get a shape known as the Rhombic Dodecahedron. It was known to Kepler (who had a positive fetish for nested crystalline geometries, who had seen the peculiarity that you can slice a cube apart into pyramids, turn them around, and re-assemble them to form a rhombic dodecahedron with the points on the outside and a cube-shaped empty hollow in the middle.
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Here is a picture taken of me holding a miniature wooden Rhombic Dodecahedron. Tom and I had never seen this shape before. Now we saw it everywhere we looked. It is a shape that appears in the crystals of Copper, Magnetite, Lapis Lazuli, and a chemical used to sterilize swimming pools. Matter uses geometry; since this is a fairly simple and symmetrical geometry we ought to find it represented in nature, and we do.
After long argument, many now accept that honeybees use a stretched form of this as the basic cell of the honeycomb. We might think the honeycomb is made of mere hexagonal cylinders. But it surns out that when you make a layer of them on one side and another layer butting up against them on the other side, the ends of the comb cells where they bump into the ends of the other cells meet in three rhombi — there may be a flat or hemispherical cap on the access end of these wax food/egg storage modules, but where their bottoms butt together it is more space- and wax-efficient for them to close-pack with Rhombic Dodecahedra. So they do. Yes, that’s right. Honeybees, mere insects with only a million brain cells, can not only find nectar-bearing flowers and return to give directions to the food to their fellow bees, but they also, as bees, secrete some of the carbs they ingest as wax. And they are programmed to use this wax to make stretched rhombic dodecahedra, which make perfect unit cells for a close-packing space-filling wax-efficient design. Nature uses what works, and necessity streamlines the design to find the moist efficient incarnation.
At the end of the Spring quarter my brother James and I left our dorm room at Reid Cop-op and applied for and received jobs in the Magic Kingdom. it was time to try working for a living, and we tried it in the wonderful wacky world of Walt Disney World.
We thought as college students we might find some kind of internship or reasonably decent temporary positions. What we were offered and accepted was positions as Custodial Hosts. All workers at Disney World are reminded to think of themselves as Hosts, and the customers as Guests. So we were NOT janitors, darn it, any more than the young people in airliners were stews any more, but were now to be called “Flight Attendants”. NO, of course not. We were Custodial Hosts rather than janitors, we served Guests rather than customers, and we wore Costumes rather than uniforms. I’m serious. I work a work uniform that consisted of a khaki shirt and dark pants, but I resported to Costuming to pick up laundered replacement. And I had been hired in a building that was NOT the Personnel or Employment office — it was, we were reminded seriously, known as Central Casting, as if we were being given roles in a Disney movie instead of applying for jobs cleaning and maintaining the theme park.
My brother James and I had received positions as Graveyard Shift Custodial Hosts. This meant an extra 25 cents “shift differential”, meaning that instead of the minimum wage of $2.35 per hour we would be receiving the staggering sum of $2.60 per hour. We found an inexpensive apartment in Orlando and moved in.
This was another strange summer. The summer before, I had learned about naval guns near Virginia Beach, flown back seat in a T-28 Trojan (not making this up!) training jet, nearly drowned upside-down in the Dilbert Dunker in Pensacola (has nothing to do with the cartoon techie by the same name), wandered through the bushes of Quantico with an M16, and gone underwater from New London, CT in a nuclear Attack sub.
That had been eventful. This summer was different. Star Wars played at the same local multiplex all summer, and my brother and I went to see it like 5 times.
Shortly after we began working at WDW, James and I drove our ancient gray 1966 Dodge Dart over to Gainesville to see how Tom and his brother Rob were doing there. There we discovered that Tom had found a practical use for the geometry. He had emphatically suggested the shape’s utility as a loudspeaker cabinet, and when his father poo-poohed the idea Tom went ahead and build a prototype. It worked even better than Tom had expected, astounding his father, who among other things had been an amateur loudspeaker builder in his past. He and Tom could hear right away that the woofer they mounted on the shape was putting out purer and fuller sound than a normal implementation.
It looked like we had found something useful and important. Tom informed me of his discovery, as I had showed him my cardboard models, and we began making plans to apply for a United States Utility patent. This would not be mere “design” patent, like a novel shape for a door knob. It would be a utility patent, and add to mankind’s catalog of useful and improved technological geometries like the lens and the parabolic dish.
Next: Blue Skies and Radial Arm Saws in 1978: the first attempt to mass-manufacture hypercubes.
–MRK
It had been almost as decade since I looked this way in 1970; now it was late 1979. I had interrupted my pursuit for a Physics degree to try to get into business constructing and selling hypercube loudspeakers. When our trip to NY didn’t turn out as well as we had hoped, we were broke again; the corporation more or less went dormant, waiting for the eventual issuance of the patent which happened 12 months later in Nov 4, 1980 as Reagan was being elected President.
I returned to school by entering the University of Central Florida in Orlando (UCF) majoring in Physics. On the way to my BS Physics degree I was gluttonous and took electives like Medical Physics, Numerical Integration Methods (Physics simulations using FORTRAN to calculate trajectories and orbits), Quantum Physics, The Physics of Science Fiction, Electronics, and Laser Physics (a Graduate-level course using Amnon Yariv’s Quantum Electronics as the text).
And now for something slightly different. For w while now I have obsessed on rectilinear geometry. When I think of measuring area, 3-volume, or 4-volume I think of square feet, cubic feet and hypercubic feet, so when I think of hypersolids, surfaces in 3+ space, I think of hypercubes with straight lines and sharp edges.
The teeth that do not smile bite through
I cannot believe that the Bill of Rights is only a pile of sand castles, to be swept away on a wave of paranoia.