The Cross Law in Universal Hyperbolic Geometry - Universal Hyperbolic Geometry - NJ Wildberger

Explore the intricacies of the Cross law in Universal Hyperbolic Geometry through this comprehensive 35-minute video lecture. Delve into the fourth and most complex of the main trigonometry laws in hyperbolic settings, learning its proof through a remarkable polynomial identity. Discover the law's application to the relationship between quadrance and spread in equilateral triangles. Follow along as the lecturer reviews the Euclidean Cross law, introduces the hyperbolic version, and provides a detailed proof with computational assistance. Engage with exercises and explore the pleasant consequences of this powerful law, including the Equilateral triangle theorem. This lecture offers a deep dive into advanced geometric concepts, suitable for those with a strong foundation in mathematics and an interest in hyperbolic geometry.

CONTENT SUMMARY: pg 1: @ Cross law in Euclidean R.T. review; Cross law in UHG; Cross Dual law
pg 2: @ Reluctant exposure to classical hyperbolic geometry; exercise 28.1
@09:25 ; observation - Euclidean cross law as a limiting case @
pg 3: @12:46 Cross law hyperbolic version; proof; heavenly assistance
pg 4: @16:21 proof continued; using a computer Scientific Workplace to verify an identity @21:41 ; proof complete @
pg 5: @28:22 A pleasant consequence of the cross law; Equilateral triangle theorem; proof; exercise 28-2 @
pg 6: @ exercises 28.3-5 THANKS to EmptySpaceEnterprise