WT46: Red geometry (I)

[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??

[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. 

[njwildberger]

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