∞, 5, 6, 3, 3, 3, …
Watch me and Plato grapple with 4D polytopes (the best shapes in the best dimension) by making them out of wire and cardboard. Bonus existence proof at the end for the meticulous!
Special thanks to Plato, Athena, Alicia Boole Stott, Ludwig Schläfli, and Aristotle (for letting Plato find the polytopes first.)
REFERENCES
Drake Thomas taught me most of this!
Rich and concise written polytope expo: https://math.ucr.edu/home/baez/platonic.html
Beautiful visuals: https://www.polyhedra-world.nc/polytop_.htm.
https://4dtoys.com/ (The stunning app)
Polytope foldouts: https://researchmap.jp/read0178984/presentations/29298678/attachment_file.pdf.
About Alicia Boole Stott: https://link.springer.com/content/pdf/10.1007/s40329-014-0065-x.pdf.
(Images from here shown at 11:48)
Branko Grünbaum’s paper I use to impress Plato: https://faculty.washington.edu/moishe/branko/Your%20polyhedra,%20my%20polyhedra/grun.pdf.
Existence proof by Christian Blatter on Stack Exchange: https://math.stackexchange.com/q/632967.
Dali’s “Corpus Hypercubus”: https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus)
4D weirdness list: https://en.wikipedia.org/wiki/4-manifold#Special_phenomena_in_4_dimensions.
Polytope wiki: https://polytope.miraheze.org/wiki/Regular_polytope.
Tom Mrowka’s quote from a curious Quanta article: https://www.quantamagazine.org/mathematicians-marvel-at-crazy-cuts-through-four-dimensions-20240422/.
I show snapshots from these great videos:
Perfect Shapes in Higher Dimensions - Numberphile: https://youtu.be/2s4TqVAbfz4?si=eQj4BUjINo1bhx2L.
Meeting Salvador Dali in the Fourth Dimension - Thomas Banchoff: https://youtu.be/GPt9VxbrBIw?si=hI5dn8Oy30LeqOyJ.
Thinking outside the 10-dimensional box - 3Blue1Brown
https://youtu.be/zwAD6dRSVyI?si=trK08WMBzS8_mAto.
H. S. M. Coxeter wrote a whole book “Regular Polytopes,” which I have not read (yet?)
TIMESTAMPS
0:00 - Building polytopes from the vertex
0:39 - Plato’s new quest
2:15 - Platonic solids from regular polygons
4:55 - What is a vertex figure?
5:30 - Defining polytopes by analogy
7:45 - Assembling 4D vertices
11:52 - Schläfli symbols
13:53 - Justifying existence
17:11 - 4D is the place to be