Maybe this will give you sime ideas on how to define a $3 Tensor? See Tensor Calculus:
To get a vague idea of the plan, note that the Brachistochrone is a Cycloid, so it enjoys the property that its involute is the same curve as itself. Now note that if we express the "squishing action" at any point of the derivative along the real line, as illustrated in this video (at 9 mins 20 secs) then the rate of change of the involute at that point ought to have some amusing properties, because it sounds to me like a basis for a notion of curvature of some kind. This might turn out to be nicely behaved in some sense, if the straight line along which the circle rolls to generate the brachistochrone were approximated from above and below by a circle rolling along the equator of a sphere as the sphere rotates through 5-space in the quaternion representation. At that point it is a sphere of infinite radius, so you get a straight line at any angle on the surface and it will look like a complex plane, I guess. See Quaternions Visualised for a nice video.
The English poet William Blake wrote somewhere about this finite world being error, "wheels without wheels" and he associated the divine with "wheels within wheels", a point which seems to ave passed by some analysts of Blakes work: Blake's Transformations of Ezekiel's Cheribum Vision in "Jerusalem". These two diagrams seem suggestive of some similar idea, to me, ...
This association is one I read 30 years or so ago, and I can't remember in which of Blakes works it was, nor can Google find it, but I am pretty confident I read it somewhere, maybe "The Marriage of Heaven and Earth". There is a suggestion of this in "Jerusalem", from Edward J. Rose's 'Wheels within Wheels in Blake's "Jerusalem"'
Anyway, to get some ideas for how to calculate derivatives on this weird revolving globe, which Blake thought of as the boundary between the human and the divine, see this video, which explains the Euclidean notion of "comensurable in square only":
Now note that the products of consecutive Fibonacci triples have this +1/-1 property, related to the continued fraction expansion of the Golden mean. See this video at 3 mins 42 secs from Pythagoras' Theorem and Dijkstra's Banker's Algorithm
Now recall that a solar year and a siderial year differ by one and some fraction of the order of 4/365 of one day, ... why? Because the earth rotates in the same sense, whilst it orbits the sun. But rotating a circle along a line with positive curvature inside the huge circle and rotating it in the other way along the outside, with negative curvature will do what after one full (albeit huge) orbit? ... so that gives the intuition, vague as it is, that I had, about how rotations in space could be used to give a rational model for tensor calcus of some kind, ... But for all I know, that's how Gauss did this stuff in the first place, ... I dunno, I never understood tensor calculus. 🤦
The reason I mention the Brachistochrone is its connection with variational mechanics. See also Tautochrone curve, which is also a cycloid, and so shares this property of being self-involute. This is also the curve used to produce an isochronous pendulum, if you happen to have a gravitational field handy, that is, .... But if you're in free-fall? ... well, I dunno 'bout that. Ask an experimental scientist, ... or Ricci?
So anyway, I love this song!
Because it's about the swarms of little guys you see here, caught up in the Incredibly Big Machine, .... wheels without wheels, ...
... choreographed by some divine comedy, but without individual significance, ... In Spanish we say personas particulares.
To get a vague idea of the plan, note that the Brachistochrone is a Cycloid, so it enjoys the property that its involute is the same curve as itself. Now note that if we express the "squishing action" at any point of the derivative along the real line, as illustrated in this video (at 9 mins 20 secs) then the rate of change of the involute at that point ought to have some amusing properties, because it sounds to me like a basis for a notion of curvature of some kind. This might turn out to be nicely behaved in some sense, if the straight line along which the circle rolls to generate the brachistochrone were approximated from above and below by a circle rolling along the equator of a sphere as the sphere rotates through 5-space in the quaternion representation. At that point it is a sphere of infinite radius, so you get a straight line at any angle on the surface and it will look like a complex plane, I guess. See Quaternions Visualised for a nice video.
The English poet William Blake wrote somewhere about this finite world being error, "wheels without wheels" and he associated the divine with "wheels within wheels", a point which seems to ave passed by some analysts of Blakes work: Blake's Transformations of Ezekiel's Cheribum Vision in "Jerusalem". These two diagrams seem suggestive of some similar idea, to me, ...
This association is one I read 30 years or so ago, and I can't remember in which of Blakes works it was, nor can Google find it, but I am pretty confident I read it somewhere, maybe "The Marriage of Heaven and Earth". There is a suggestion of this in "Jerusalem", from Edward J. Rose's 'Wheels within Wheels in Blake's "Jerusalem"'
Anyway, to get some ideas for how to calculate derivatives on this weird revolving globe, which Blake thought of as the boundary between the human and the divine, see this video, which explains the Euclidean notion of "comensurable in square only":
Now note that the products of consecutive Fibonacci triples have this +1/-1 property, related to the continued fraction expansion of the Golden mean. See this video at 3 mins 42 secs from Pythagoras' Theorem and Dijkstra's Banker's Algorithm
Now recall that a solar year and a siderial year differ by one and some fraction of the order of 4/365 of one day, ... why? Because the earth rotates in the same sense, whilst it orbits the sun. But rotating a circle along a line with positive curvature inside the huge circle and rotating it in the other way along the outside, with negative curvature will do what after one full (albeit huge) orbit? ... so that gives the intuition, vague as it is, that I had, about how rotations in space could be used to give a rational model for tensor calcus of some kind, ... But for all I know, that's how Gauss did this stuff in the first place, ... I dunno, I never understood tensor calculus. 🤦
The reason I mention the Brachistochrone is its connection with variational mechanics. See also Tautochrone curve, which is also a cycloid, and so shares this property of being self-involute. This is also the curve used to produce an isochronous pendulum, if you happen to have a gravitational field handy, that is, .... But if you're in free-fall? ... well, I dunno 'bout that. Ask an experimental scientist, ... or Ricci?
So anyway, I love this song!
Because it's about the swarms of little guys you see here, caught up in the Incredibly Big Machine, .... wheels without wheels, ...
... choreographed by some divine comedy, but without individual significance, ... In Spanish we say personas particulares.
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