## At A Glance

### UnivHypGeom23: The Triple quad formula in Universal Hyperbolic Geometry

The Triple quad formula is the second most important theorem in hyperbolic geometry (just as it is in Euclidean geometry!) It gives the relation between the three quadrances formed by three collinear points. It is a quite challenging theorem to prove: relying on a remarkable polynomial identity. It is a deformation of the Euclidean Triple quad formula, and happens to agree in form with the Euclidean Triple spread formula. We sketch an argument for this seeming coincidence. This is one of the more algebraically challenging of the videos in this series.CONTENT SUMMARY: pg 1: @00:11 Triple quad formula; example; suggested exercise @07:12pg 2: @07:31 Understanding the Triple quad formula; comparing the corresponding formulas in affine/projective geometry; notice the direction of the arrows @11:10pg 3: @11:39 Triple spread formula from affine RT; spread in vector notation ; spread in vector notation @13:50pg 4: @15:35 Euclidean dot? products; Relativistic dot productspg 5: @19:40 Why the Triple quad formula holds; note? on 4 main laws of hyperbolic trigonometry @22:06pg 6: @23:38 Triple quad formula; proofpg 7: @27:44 Triple quad formula; proof continued; a small miracle @30:40 ; remark about proof @32:25 ; encouragement to do algebra @33:18pg 8: @35:28 The Triple spread function is defined; exercises 21.1,2pg 9: @36:21 exercises 23.3,4 ; Complimentary quadrances theorem; Equal quadrances theorem (THANKS to EmptySpaceEnterprise)
Length: 39:11