Sociology and Pure Maths - Regular Polytopes and the math vs physics issue | Sociology and Pure Maths | N J Wildberger

The notions of "space" in mathematics and physics are in some ways parallel, but in other ways divergent. Here we consider a remarkable aspect of this that ought to reorient you to a fundamental tenet of mathematical knowledge going back thousands of years: that there are exactly five regular, or Platonic, solids: the tetrahedron, cube, octahedron, icosahedron and dodecahedron. Could this completely well accepted and universally held notion actually be wrong? There are two separate issues here: the physical one about the realization of the Platonic solids in physical 3D space: a fact which is hard to deny since we have all played around with models of these things at one time or another. But what is the actual mathematical reality here? Are we able to honestly confront the fact that the mathematical reality appears to diverge with the physical reality? We start the discussion with the important prior question about regular polygons in the plane. Remarkably, I offer a Conjecture here which is completely at odds with the prevailing mathematical dogmas. I also connect this topic with Rational Trigonometry, where we attempt to understand metrical geometry purely algebraically, hence over the rational numbers, or over a finite field, and the properties of the "Spread Polynomials" which appear there. After the 3D version, we consider Ludwig Schlafli's beautiful classification of regular solids in higher dimensions, and make contact with three remarkable 4D solids. Or perhaps we should say... make contact with the one truly remarkable 4D solid, and mention also two others that can be obtained by a more complicated process involving first creating a prior extension field containing a "square root of 5". Video Contents: 00:00 – Introduction: Rethinking Regular Polytopes 02:30 – Regular Polygons in the Plane 05:00 – The Five Platonic Solids: A Closer Look 08:15 – Schläfli's Classification in Higher Dimensions 11:00 – Rational Trigonometry and Spread Polynomials 14:30 – The Math vs Physics Perspective 17:45 – Extension Fields and the Square Root of 5 21:00 – Sociological Influences on Mathematical Acceptance 24:30 – Conclusion: Towards a More Grounded Mathematics ******* Research Gate page: https://www.researchgate.net/profile/Norman_Wildberger Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon: https://www.patreon.com/njwildberger Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriFv3G22iOUidUhkIGXuh Insights into Mathematics Playlists: https://www.youtube.com/playlist?list=PLIljB45xT85CL0-5wO4P-OwuH7SzBUyC0 (31 videos) https://www.youtube.com/playlist?list=PLIljB45xT85B0aMG-G9oqj-NPIuBMnq8z (18 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DHcleY93bd2WlywQYxHv7H (6 videos) https://www.youtube.com/playlist?list=PLIljB45xT85AZjh2GUdwsMuAWTEHXCLb- (4 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DeX148Q2T2elycnIYHTxdP (44 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A6D0eylgAM4Xc1b_dDTgHW (8 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DGxj1x_dyaSggbauAgrB6R (226 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A_VZD126sND3XECDSI-qGs (26 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DfokRNdx5eQ_qA43BwyM-_ (45 videos) https://www.youtube.com/playlist?list=PLIljB45xT85CvrEUnhS4dkTm9Lci5JTYS (43 videos) https://www.youtube.com/playlist?list=PLIljB45xT85D7wczwyUQdwDe2duZ7wPTf (40 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Ar_tuHmRDI2Xr5tWOTI1oW (55 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DPSrQpExd2L870ggxd2KEB (34 videos) https://www.youtube.com/playlist?list=PLIljB45xT85C3-5f6-qQS3YhgygZgnGYs (8 videos) https://www.youtube.com/playlist?list=PLIljB45xT85D7hfVLKyEJbQxOJ4yYm1-9 (9 videos) https://www.youtube.com/playlist?list=PLIljB45xT85BtiJIcDF2uuFc7_ndd1UL6 ( 46 videos) https://www.youtube.com/playlist?list=PLIljB45xT85B6MYBrWldlJ3w-Q1LaEdoy (40 videos) https://www.youtube.com/playlist?list=PLIljB45xT85DIpzaWggqJDIK8m2u09dQL (7 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Ag6ragdmHaxUOKcQsIsrXZ (7 videos) https://www.youtube.com/playlist?list=PLIljB45xT85A5w_vZ5BB1IQAxAMhz5o5S (19 videos) https://www.youtube.com/playlist?list=PLIljB45xT85BjhYV7QswANMF3gKPx9BqT (21 videos) https://www.youtube.com/playlist?list=PLIljB45xT85Bi7PvrTkAaAswTwsfsR5HV (10 videos) https://www.youtube.com/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh (94 videos)