Articles: The Best Homework Ever, 1996
ArXiv: The Gauss map of polyhedral vertex stars (PDF), 2019
• Beyond The Third Dimension investigates ways of picturing and understanding dimensions below and above our own. Ranging from Egyptian pyramids to the nineteenth-century satire Flatland to the paintings of Salvador Dali, Scientific American Library, Freeman 1990 (Review)
• Differential Geometry of Curves and Surfaces, Thomas F. Banchoff, Stephen Lovett, Aug 5, 2022
• Flatland: The Movie Edition, 2008
Homepage(s): Brown University, Beyond The Third Dimension, MAA
Twitter: @BanchoffF Congrats, PBS-Lanier, Dalí-Vassilieva, TinyGeometry-Third Dimension
YouTube: Meeting Salvador Dali in the Fourth Dimension, 2019
(born April 7, 1938)
Second email: July 16, 2022 at 5:05 PM
Dear Prof. Dr. Thomas F. Banchoff,
Again, let me say a profound congratulations on such a diverse and full life (as evidenced above).
I got stuck in elementary things and didn’t move very far forward. During a visit with John Conway in the Spring 2001, he accused me of being hung up on the octahedron-and-tetrahedron and he was right. Finally, it was your colleague at Brown, Phil Davis, who dislodged me from the tetrahedron to the sphere in 2012, but It didn’t last long. When I found the work with cubic-close packing of equal spheres, I was back into octet things. When I learned about Aristotle’s 1800-year old mistake, it was an eye-opener. I thanked Lagarias and Zong profusely for their work, but both were reluctant to speculate on the nature of that five-tetrahedral gap of 7.35610317+ degrees.
Work within a high school geometry class also complicated matters. My nephew was the head of the math department in his high school and the geometry teacher. On a couple occasions he had me take over to introduce the platonic solids. I had boxes filled with clear-plastics models of octahedrons and tetrahedrons that I had manufactured (a silly idea). We were able to make clusters of that tetrahedral gap and then went on to make a twenty-tetrahedral icosahedron and then even the Pentakis dodecahedron. We called it squishy geometry, imperfect geometry and quantum geometry. Of course, I wanted to associate it with quantum fluctuations and the interstitial area between the imperfect geometries with gaps and perfect geometries without gaps.
To complicate matters in 2011, these classes learned about the Planck scale and the idea that Planck had calculated the smallest possible unit of time and length. On December 19, 2011, the students followed the embedded geometries of the tetrahedron and octahedron back to Planck’s length. Much later we went back to Planck Time. It was 45 steps down into particle physics and another 67 steps to the Planck units. Going out, doubling all the way, there were an additional 90 steps for a map of the universe of just 202 base-2 steps. Empowered by it, I asked Freeman Dyson and Frank Wilczek about its efficacy. We were encouraged to continue our explorations. We didn’t get very far. It was labelled idiosyncratic by John Baez. It was a non-sequitur and Hawking ruled, so we backed down.
Just recently, I put the five octahedrons together and found that it, too, has a gap. Confirming with geometers that it is an identical gap, I began studying the two together. Salvatore Torquarto agreed with me that it is simple and that it has not been studied much beyond what I did within my initial overview.
Might you add your thoughts? Do you think it could be important? …within the infinitesimal? Thank you.
PS. Please excuse my long email. I only wish I could say it all more succinctly. I was especially impressed to see that photo of you and Dali, I copied it out and attached it here. May I use it on our overview page about you and your work? -BEC
First email: Jul 7, 2022, 3:28 PM
Dear Prof. Dr. Thomas Banchoff,
We have been working with the Platonic geometries in our high school and cannot find any references online to a very simple geometric figure of five octahedrons, all sharing a centerpoint (and three sharing two faces with another octahedron and two sharing only one face). It is a very interesting image when the five-tetrahedrons are added on the top and bottom. That stack has 15 objects sharing the centerpoint. I took the picture below just a few weeks ago but, to date, it appears that there is no scholarship about it.
Have you seen any scholarly analysis of it? Thank you.
File was once /a34/ and this information is part of /a31/
Every prime number has its own flavor and personality. There are over 101 different types identified by Wikipedia editors. Are there mathematical experts within the studies of the functionality of prime who could help guide our thinking? Is it possible that each prime introduces a new mathematical set? Is this set properly described by set theory? Is each notation, within itself, defined by set theory and each transformation to the next notation defined by group theory?
To begin the process of answering these questions. We’ve made guesses regarding the ordering of the emergence of numbers, forms and functions within each prime. This listing comes from line 11 of the chart and an article about numbers.
2 • Euclidean geometries, starting with pi and cubic-close packing of equal spheres and lattice generation
3 • Bifurcation theory, including the Feigenbaum constant, and the various manifestations of the theory
5 • Golden ratio (Phi), the Fibonacci sequence and the nature of addition; five-fold symmetries, indeterminacy, the imperfect, fluctuation theory, and ratio analysis
7 • Computer automata theory with John Conway and Stephen Wolfram (this may be a special application of bifurcation theory)
11 • Group theory and projective geometry
13 • Algebraic geometries
17 • Langlands groups, Langlands correspondence, Langlands program, Langlands conjectures
19 • Zermelo–Fraenkel set theory (ZFC) and quantum gravity
23 • S-matrix theory, unitarity equations, Hermitian analyticity, connectedness
29 • Mandelbrot set, Julia set, Möbius transformations, Kleinian group