WT48: Red geometry (III)

Remarkably, the main laws of trigonometry in the relativistic (red) setting are exactly the same as those in the Euclidean setting. To establish this, we show first that the Cross law---the most powerful law---holds in red geometry. Then the others, such as Pythagoras, the Triple quad formula, the Spread law and the Triple spread formula, follow much as they do in ordinary (blue) geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

Category:

Science & Technology

Tags:

• Geometry
• rational
• trigonometry
• red
• relativistic
• Cross
• law
• Wildberger

[benthurston27]

So all those theorems you've proven in the last 45 videos should be equally valid in red geometry. That is really impressive. What I'm wondering is if that is a striking coincidence or not, I mean, would the same be the case no matter how you defined perpendicularity? Then I think of the affine geometry that didn't even have a notion of perpendicular. I guess it's going to take a while to solidify in my mind.

[njwildberger]

@benthurston27 Correct, a lot of the theorems also hold in red geometry, once you state them correctly in that framework. It is an aspect of the universality of rational trigonometry.

[benthurston27]

@njwildberger My thinking was everything you'd proven up to this point you had used the cross rule and the rules derived from it to prove it, so the cross rule being the same in red geometry meant all of those proofs still held, but now I'm thinking maybe somewhere along the way you might have used something like just the definition of the spread which is different between the two. Anyway thanks for all your replies!