Yair Shenfeld, MIT, Cambridge, Massachusetts USA
First email: September 7, 2022 at 7:38 PM
Dear Dr. Yair Shenfeld:
Might a most simple geometry be involved with the Brownian transport map? My intuition says, “Maybe. It’s worth checking out.”
1. Five tetrahedrons. Are you aware of the 7.35610+ degree gap created by five tetrahedrons sharing a common center point? Aristotle missed it. Mysteries in Packing Regular Tetrahedra is an unusual history (PDF), but it is usually forgotten and is relatively unrecognized even though in 2015 the AMS gave the Conant award to the authors.*
2. Five octahedrons. More recently in May 2022 we became aware of the same gap using five octahedrons: https://81018.com/2022/05/19/five/#Gap. This simple geometric figure consists of five octahedrons, all sharing a centerpoint (three sharing two faces with another octahedron and two sharing only one face). It is relatively unknown and undocumented.
3. A Stack. An interesting image is when the five-tetrahedrons are added on the top and bottom. That stack has 15 objects sharing the centerpoint. Interactive geometry software (IGS) with their dynamic geometry environments (DGEs) can not re-create this geometry. Nor can any of the kits like Zometool. This simple geometry requires true physical models.
I took the picture of those pages in June but to date it appears that there is no scholarship about it. Have you seen any scholarly analysis of it? Do you think it is important? My intuition is telling me it is related to quantum fluctuations. If so, then it is huge and it will become a major part of scholarship. Thank you.
* Key parts of this history came from an MIT professor of mathematics, Dirk Jan Struik from his work, Het Probleem ‘De impletione loci’ (Dutch), Nieuw Archief voor Wiskunde, Series 2, 15 (1925–1928), no. 3, 121–137.