Trying to Make a Sci-Fi Movie

I'm sure maths doesn't have to be this violent!

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The idea is to make a program that lets you explore the kinds of surface that appear when you use multiplication over finite rings to develop ruled surfaces between two parallel circles in 3-space:

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In principle it can't be that hard, but I don't really have much of an idea of how OpenGL represents surfaces: https://docs.gl/gl3/glBegin:

See https://www.glfw.org/.

Prompted to make this video by Daniel Tubbenhauer's video on the Hodge conjecture:

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I had this idea sixteen months ago: see Ian Grant's Weather Report 8/6/22 and Ian Grant's Weather Report August 07, 2022.

The Oloid is another kind of product of two circles: it has the same area as the sphere of the corresponding radius. 

A Sphericon is another developable surface that has half the volume of the corresponding sphere.

Part II on Ruled Surfaces: they are a sort of pairwise product of vectors from two paramaterised sets.

9:45 An interesting way to determine a developable surface by deforming it 'uniformly' so that a curved edge lies in a plane. Quite what 'uniformly' means, however, is another thing, ... 12:25 Euler's three types of developable ruled surfaces. 26:29 Projective view. see Freya Holmér - Why Can't You Multiply Vectors? and David Hestenes - Tutorial on Geometric Calculus. At 36:00 See https://en.wikipedia.org/wiki/Cubic_surface#27_lines_on_a_cubic_surface. 42:10 the triple-spread formula for the square sine of an angle.

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His spelling misstake is freaking me out:

Topological spaces and manifolds

44:00 "In practice, it is a lot simpler to build up spaces by relying on vector spaces and parts of vector spaces as building blocks."

See also Topological space and Kuratowski closure axioms which might let you develop a topology as a series of finite steps that construct a lattice.



See T1 space which has two totally bewildering lists of ten equivalent definitions each for T1 and R0 spaces. (A topological space is an R0 space if and only if its Kolmogorov quotient is a T1 space.)

See also Freya Holmér on Continuity of Splines December 15, 2022 which I updated after watching it a second time yesterday.

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