Don't You Hate it When... 🤬

... you work out an elaborate reply to a post and when you click "Submit" you'll be notified that the post no longer exists... 🤬☠️🤬
That post was titled "If you were to start an OnlyFans, what kind of content would you post? Lol." and my well thought-out and carefully worded reply would have been:
My OnlyFans would deal predominantly with the history of conic sections, which can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C. Modern aspects would have to be taken into account as well, such as the theory of limits according to Bolzano and Cauchy which clearly show that a straight line is the limit of a parabola a + b*x + c* x*x for c –> 0. 😑

💀💀💀 Eat Flaming Death!!! 💀💀💀

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ElwoodBlues · M

@helenS In the spirit of the deleted question and your conic answer, I think it would be fun to start an OnlyFans on representations of rotations & orientations in 3D and in higher and lower dimensions.

In 3D (with 3 rotational degrees of freedom) we have Euler angles, 3X3 rotation matrices, axis & angle, and quaternions to name the major ones.

Interestingly, in 2D there is only one rotational degree of freedom, while in 4D there are six rotational degree of freedom! Weird!!

helenS · 36-40, F

@ElwoodBlues Here's an interesting discussion of the topic:
https://math.stackexchange.com/questions/1281182/how-many-degrees-of-freedom-would-a-rotation-matrix-in-r5-have

ElwoodBlues · M

@helenS Yeah, looking at constraints on an NxN rotation matrix and degrees of freedom of the matrix definitely works. It also makes me think that axes of rotation don't generalize.

Not so coincidentally, the number of rotational degrees of freedom in a space of N dimensions is the number of mutually orthogonal 2D planes the space can accommodate, which is also the number of pairs of axes. This leads me to believe that it makes more sense to think of rotation as happening in a plane rather than about an axis. The axis concept (perpendicular to plane of rotation) only works in 3D.

helenS · 36-40, F

@ElwoodBlues From a physics point of view, the number of degrees of freedom is irrelevant as long as the degree of degeneracy isn't taken into account. A non-linear molecule containing N atoms, for example, has 3N-6 vibrational degrees of freedom, in principle, but many of them may belong to degenerate point groups, so in fact you will have much less. Octahedral and tetrahedral molecules are good examples. Same is true for space groups in crystals.

ElwoodBlues · M

@helenS The 1 DoF of a 2D rotation matrix is real and true. And, although I don't know how to prove it, I very strongly suspect that the 10 DoF of a 5x5 rotation matrix is real and true. I don't think those higher dimensional DoFs are degenerate, but I don't know how to prove it unless counting DoFs in orthogonal unit vectors of a rotation matrix is a proof.