Robbins' formulas, the Bellows conjecture + polyhedra volumes|Rational Geometry Math Foundations 128
4,060 views · Published 11 February 2015 · 34:25 · Indexed 20 September 2026
Channel: Insights into Mathematics · 2015 · Education
We discuss modern developments in the direction of our latest videos, namely formulas for areas of polygons in terms of the quadrances of the sides. We discuss work of Moebius, Bowman and Robbins on the areas of cyclic pentagons. There is also a rich story about 3 dimensional generalizations of Heron's formula or Archimedes' theorem to polyhedra: going back to Tartaglia for the case of the tetrahedron, connecting with a rigidity theorem of Cauchy, a remarkable example of R. Connelly, the so-called Bellows Conjecture, and its lovely resolution by I. Sabitov. We will see throughout that it is really the quadrances of sides, not the lengths of sides, which these formulas are based on. The intrinsic algebraic aspect of this kind of geometry comes to the fore. Video Content: 00:00 Intro to area formulas for polygons 2:30 Cyclic quadrilateral quadrea theorem 4:46 Brahmagupta's formula of cyclic pentagon 7:13 F. Bowman 1952 15:56 Tartaglia's Formula 20:43 Euler rediscovered Tartaglia's formula 25:18 Flexible polyhedron 30:37 Octahedra ************ 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) Wild Egg Maths Playlists: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5m3XKK0OPc31l-GTqDiAjs (4 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz4CVqTYS1fwInPOZxBxDaQf (64 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7PpsRFHuGb3EhwZtEOdRjV (45 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz6HqaFPB4HTZOQ8jgDAVnuz (20 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz71VHhAIUN-7Y6-JVcwF-xA (8 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz75-t4UaYvHwPe5AHGA_EWe (9 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5W25vaXSreTMTwpPhWs1w2 (52 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz4DCenuIAr9R2kXLq4ywRAm (9 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7OynfQubQrYkxYurauO-HD (14 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz7cyg5JdKBpzB4fTQT7f-Xl (8 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5Bcq1VN_R7zYDZH3u49FH8 (25 videos) https://www.youtube.com/playlist?list=PLzdiPTrEWyz5flY1TFd7cekRMZRvPGkzM (30 videos)