WT46: Red geometry (I)

Red geometry is a two dimensional relativistic geometry in the spirit of rational trigonometry, using variants of the usual quadrance and spread. The usual grid plane is still the arena in which this geometry lives, but the notion of perpendicularity is new. A wonderful new world of geometry emerges from just a twist of the old definitions!

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:

• Mathematics
• geometry
• rational
• trigonometry
• Einstein
• special
• theory
• of
• relativity
• red
• Wildberger

[dreznik]

I'd also love to share with you that in my 25 years of writing complex scientifc code I have *never* needed to use angles (except when collecting user input) and/or compute actual trig functions of angles (except in the same situation). I have always felt they were redundant and inefficient. For example, an angle was always representable by a pair (e.g., its sine and cosine -- e.g., initialized to trivial values and then transformed via rotations). Square roots were occasionally needed though.

[njwildberger]

@dreznik Thanks for that, I am not surprised. There is a big gap between the fanciful idealizations of modern pure mathematics and what actually goes on in the computers of the world out there.

[dreznik]

Dear Norman, thanks for your response. I am loving your videos and the material, will buy your book on rational geometry! Can I define an angle as the length traversed along the circumference of a unit circle (CCW from [1,0]), and the sine as a ratio of leg / hypothenuse of the right triangle with vertices defined by the endpoint of said traversed length and its dropped projection on the x axis? Are these problematic?

[njwildberger]

@dreznik These notions are more problematic than is usually admitted. In my MathFoundations series I talk a little about the weaknesses of angles. For example, how does one define the length of a curve? That presupposes a prior theory of real numbers, also highly problematic. Sines and cosines as ratios don't connect them directly to angles, which is really necessary.

[dreznik]

In blue geometry is the notion of "spread" just the sine^2 of the angle between two vectors (cross product a x b = |a| |b| sin(th); likewise, is the notion of quadrance simply square of the euclidian distance between two points? why the introduction of these new terms? finally, in *red* geometry, the "-" signs which pop up seem to be consistent with blue geometry, except the second variable becomes imaginary, e.g., x + i y. is that what's going on?

[njwildberger]

@dreznik In blue geometry, ie ordinary Euclidean geometry, quadrance and spread are as you say, but they are defined in a much simpler more algebraic way, independent of transcendental notions. This makes things computationally faster, and also more general, as now the theory works over a general field. Also more logical: do you really have a proper definition for sine of an angle??

[njwildberger]

@benthurston27 It is just a different notion of perpendicular. Some mathematical concepts come in different flavours, this is an example.

[benthurston27]

So I'm guessing the fact that this notion of perpendicular doesn't look to form a right angle is because of what they call the curvature of spacetime? I figure it muse curve concave into the board if you can get a spread of more than 1? I'm thinking of something I saw where you can draw a triangle on a balloon then inflate it and end up with more than 180 degrees.