Explore the hyperbolic version of Pythagoras' theorem in this 36-minute lecture on Universal Hyperbolic Geometry. Delve into the fundamental concepts of quadrance measurement between points, and understand how this theorem serves as a deformation of its Euclidean counterpart. Learn about point and line incidence, cross ratios, and the projection of 3D space onto a viewing plane. Examine the quadrance formula in planar coordinates, complete with illustrations and examples. Follow the step-by-step proof of the hyperbolic Pythagoras' theorem, witnessing a "small miracle" in the process. Engage with practical exercises to reinforce your understanding of this pivotal concept in hyperbolic geometry.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ Pythagoras' theorem in UHG; points, point/line incidence, quadrance/cross ratio
pg 2: @ projecting 3-dim onto 'viewing plane'
pg 3: @11:03 quadrance in planar coordinates; GSP illustrations of different quadrances in the plane @
pg 4: @ quadrance planar formula; note - null point restriction; zero denominator convention; example
pg 5: @17:22 Pythagoras' theorem hyperbolic version; the importance of the theorem @ ; example
pg 6: @ exercises 22.1,2
pg 7: @24:22 The proof of Pythagoras' theorem; a small miracle @27:04 ; suggested exercise @
pg 8: @29:03 The proof of Pythagoras' theorem continued from pg 7; "That's a proof" @
pg 9: @ exercises 22-3:5 THANKS to EmptySpaceEnterprise
Taught by
Insights into Mathematics
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