[16] Aristotle & the Geometric Gap

Foundational Questions Institute: Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020) Support materials for the submission from Bruce Camber in April 2020.

Determinant becomes unpredictable, uncomputable, and undecidable (PDF)

[1] Decidability
[2] Computability
[3] Predictability
[4] Undecidability
[5] Uncomputability
[6] Unpredictability
[7] First units
[8] Grand reductionism
[9] Triangulation
[10] Fourier
[11] Lorentz
[12] Poincaré spheres
[13] Planckspheres
One second: 299,792± km
[14] Automorphic forms
[16] Aristotle’s Mistake
[17] Fuzzy Universe
[18] Scholars
pentastarBackground: The FQXi call for papers brings into focus this geometric gap created by five tetrahedrons sharing a common edge. It opens a gap of over seven degrees. Aristotle did not know that gap existed. It was one of his key mistakes. And, the failure of scholarship to catch it for over 1800 years was an even greater mistake. The key evocative question is, “How has his mistake impeded progress over all these years?” We are just now learning, so of course, there will be much more to come.

[16] Geometric gap. There was just a bit of arrogance among the Greek scholars. So self-assured, they teach us that such arrogance opens mistakes and then enshrines them. That Aristotle (384–322 BCE) had perfect models of the tetrahedron seems unlikely. That scholars never corrected his mistake for 1800 years is beyond belief. That is reverence taken to an extreme. For that reason alone, I am ever so grateful for the work of Jeffrey C. Lagarias and Chuanming Zong  (Mysteries in Packing Regular Tetrahedra (PDF)).

We all have a lot of work to do here. We need to grasp F.C. Frank and J.S. Kasper work (Complex Alloy Structures Regarded as Sphere Packings) and how it all relates to work of Daniel Shechtman and Roger Penrose.  Stay tuned! We’ll be back!

Wikipedia references: In the 1960s the first concepts around aperiodic tilings were introduced. In 1976 Roger Penrose introduced his unique tilings and Alan Mackay followed up experimentally to show how a two-dimensional Fourier transform (with rather sharp Dirac delta peaks) manifests a fivefold symmetry. For more, see quasicrystals and notes.  In 1982 Daniel Shechtman began his public-struggle to open the exploration of quasicrystals.