More Martian Math
As of this writing, it’s one day after my completing a five day gig as a Martian Math instructor under the auspices of Saturday Academy. Several instructors pool their talents and offer campers opportunities to explore various “worlds” which is a fancy philosophical term for “namespaces” or even “shop talks”. Martian Math is a shop talk, involving a Concentric Hierarchy, modules, lots of Greek metaphysics, some concepts from topology, machine learning.
Another instructor was sharing Origami skills, which dovetailed nicely with my class, where I gave a mix of “high and low diving board” challenges on my “public swimming pool” model (take risks when you’re ready, with others witnessing success or failure).
Another instructor, using the same lab as myself, was sharing about Artificial Intelligence in the Real World. Campers were not all on the same program or sequence, however there was lots of overlap. The Reed College campus in Portland, Oregon was our venue. My classroom was on the 2nd floor of the ETC building (Google up a Reed College campus map if interested in more geographic information).
The Concentric Hierarchy includes the Platonic Five (tetrahedron, octahedron, icosahedron, cube, pentagonal dodecahedron) and then some.
I’d begin each meetup with a short lecture, making full use of the A/V equipment available and then offer various jobs and experiences, with the default being to enjoy collaborative recreational pastimes using well-known internet venues.
A dedicated C6XTY team emerged, under the leadership of my assistant. A three-frequency tetrahedron, octahedron, and cuboctahedron took shape, using plastic materials supplied by Flextegrity Inc. (a Sam Lanahan company).
The course material uses Science Fiction to anchor itself. Learn vector addition against a backdrop of “the Martians have landed” or “terraforming Mars” or some other scenario involving whole planets and their relationships.
To establish a role for Science Fiction, I like to revisit the famous Halloween CBS radio broadcast by Orson Wells, which caused a panic among its listeners, given a paucity of disclaimers reminding the audience of the program’s fictional fare. The impersonation of FDR was quite persuasive, as were many other sound effects. Radio had evolved many techniques for playing upon one’s imagination.
C6XTY is named for C60, or Carbon-60 (Bf), or buckminsterfullerene. On the last day, I remarked on how similar the modern day conception of a virus, and some of those War of the Worlds artist conceptions of Martians in their tripods turned out to be. That Earth’s microbial army proved the only effective defense against the alien invaders underlines the importance of the Nano-World then beginning to emerge in the science of the 1890s.
By the 1960s, the details of virus architecture were starting to come to light thanks to X-Ray diffraction patterns, and in 1963 Buckminster Fuller imagined his “synergetic geometry” was soon to become famous, given his page one New York Herald Tribune treatment as a contributor to our emergent knowledge of the virus. Using the contemporary shoptalk of memetics: Fuller expected his contribution to the virus story to help Synergetics go viral.
What happened instead is an alternative mathematical equation discovered by Michael Goldberg, covered several classes of geodesic sphere Fuller’s formula didn’t. H.S.M. Coxeter, a champion academic geometer, could publicly lecture Fuller, the maverick outsider, on what he didn’t know, even about geodesic spheres.
Coxeter did later credit Fuller, when providing background information to the New Yorker magazine’s fact checker, for discovering a cuboctahedral (also icosahedral) number series generator: 10 times Frequency to the second power, plus two. That formula, along with V + F == E + 2, figured in a few of my morning warmup lectures (this June 2018 class featured five such meetups).
In light of C60’s later discovery in the 1980s, professor Harold Kroto, later knighted by the Queen of England, thought “buckminsterfullerene” would be an appropriate posthumous tribute to the Geodesic Dome King. The science community went along, partly because “buckyballs” and “buckytubes” have a friendly journalistic tone that helps popularize carbon chemistry, much as “machine learning and data science” have helped rescue “statistics” by making it sound sexier. Also true is that Fuller (RBF) was as keen about whole spheres as about the dome-shaped parts of same, that might be (have often been) used for domiciles.
C60 (a compound made of sixty carbon atoms) was registering in Texas, on Rice University mass spectrometers, as a “thing” having first registered on spectrometers pointed at interstellar space, where Sir Harold noticed them as a signature wavelength. A Nobel Prize was in the making, shared three ways. A new allotrope of carbon had been discovered, and needed a name.
By this time Fuller had received the Medal of Freedom from US president Ronald Reagan, so earlier qualms Scientific American might have had, about injecting Fullers name directly into the official literature, could be considered allayed.
C6XTY uses six identical panels to construct a “soccer ball” shell (the classic Adidas “hexapent” pattern), which may be then interconnected in a three-way weave pattern of mutually perpendicular connectors. The World Cup (soccer) was going on the the background throughout our meetups, while closer to home (Silicon Forest), the OSU Beavers were winning the World Cup Series in college baseball.
This blending of XYZ with IVM (i.e. CCP or FCC) is just what the Martians needed for their outreach program. We’re recruiting among the youth (not exclusively) to find Earthlings who might work with Martians at Collaboration Mesa (translation). We’re building a hydro-power dam together, ala Glen Canyon in Arizona.
However the Martians and Earthlings have different Units of Volume when it comes to concrete. A conversion constant must be established. Sphere packing is employed to build the bridge.
The Martian tetrahedron T of edges D has a specific volumetric relationship with the Earthling cube C of edges D/2 or R, where R and D are respectively the radius and diameter of some arbitrarily chosen canonical sphere (perhaps a C60 buckyball, or one of higher frequency). I polled the campers on whether they thought C > T or T > C in terms of volume. One guessed C == T. In fact, C > T (most guessed it right), even though R < D. The C:T ratio is the 2nd root of 9/8, or S3, the sought-for conversion constant.
Interestingly, S3 is likewise the scale factor taking the cuboctahedron (lets say of volume 20) to the corresponding larger volume of a rhombic triacontahedron (RT), where the latter is formed from the icosahedron and its dual pentagonal dodecahedron, where the icosahedron is formed from the cuboctahedron by Jitterbugging (definitely a shoptalk). Said RT has a volume of 20 times S3 where, again, S3 is the 2nd (Earthling: “square”) root of 9/8.
When a Martian pours one unit of concrete, in their bookkeeping system, they’re actually pouring a smaller-than-unit amount of concrete in the Earthlings’ bookkeeping. However neither side accuses the other of being imprecise on the basis of this ~6% difference. Martians and Earthlings may have their misunderstandings, but this is not one of them.
The “modules” of which I speak are again more of the Fuller-Sadao-Noguchi shoptalk, the shop in question once based in Long Island City, part of Queens. The Noguchi Museum is to this day across the street from the place where Fuller and Sadao used to plot world domination for their world map, first published using the trademark “Dymaxion” as in “Dymaxion World Map”.
I had such a Dymaxion Projection in our classroom, against the backdrop of early 21st Century Digital Global Graphics Systems (DGGS), which make “hexapent” tessellations of a planet, any planet (real or fictional) pretty easy.
These modules are: A, B, T (a volume category), E (same shape as T), and an S.
S > E > B = A = T in terms of volume, with the latter three each equal to 1/24. A + A + B = MITE, one of the few tetrahedron space-fillers that doesn’t require a chiral counterpart. Duncan Sommerville, Michael Goldberg and quite possibly Aristotle studied these space-filling tetrahedrons as well. Fuller used his own Synergetics shop talk (Mites, Sytes, Kites…), with its tetrahedron unit volume (the Coupler of eight Mites is likewise unit volume).
S:E is what we’re calling the S factor, however the summer camp edition of the material did not go into much depth with these modules. I had plane nets of A modules on hand, a completed tetrahedron made of 24 of them, and a Github page giving more context.
The core skill I was teaching, and we got it out of the way early, as the one “everyone do this high dive”, was cloning my MartianMath Github site and booting a Jupyter Notebook server in order to edit the contents. I walked them through cutting and pasting a Flickr photo (e.g. from my Storyboards) into a Jupyter Notebook. I did not have them push any changes back to Github nor coach them in writing Python.
I had sample Python and bread crumb trails in every direction, such as a Quadray Coordinates implementation, but no “forced march” plan to visit every exhibit. My opening Show & Tell lectures provided some overview.
They’ll need months and years, not mere days and hours, to gradually acquire the full skill set in question. Like we’ll have more time once they get to Collaboration Mesa. I told them the bus would be waiting outside. They knew I was talking science fiction by this point. No one panicked.