tetraplanetary coordinates

Higher-dimensional geometry (previously "Polyshapes").

tetraplanetary coordinates

Postby Aale de Winkel » Mon Dec 01, 2003 1:29 pm

According to the glossary we have next to our regular trionian longitude and latitude the laptitude in the marp, garp direction. A short derivation:
tetra sphere: x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = R<sup>2</sup>
north / south: x<sup>2</sup> + y<sup>2</sup> = R<sup>2</sup> ==> R [ cos(α) , sin(α) ]
east / west (at y = y<sub>0</sub>): x<sup>2</sup> + z<sup>2</sup> = R<sup>2</sup> - y<sub>0</sub><sup>2</sup> = r<sup>2</sup> ==> r [ cos(β) , sin(β) ]
marp / garp (at (y,z) = (y<sub>0</sub>,z<sub>0</sub>): x<sup>2</sup> + w<sup>2</sup> = R<sup>2</sup> - y<sub>0</sub><sup>2</sup> - z<sub>0</sub><sup>2</sup> = r<sup>2</sup> ==> r [ cos(γ) , sin(γ) ]

Thus the position is given by the triplet ( α , β , γ ). zero for α is given by the planetary equator, for β it is given by some chosen meridian, zero for γ is given by? (the polar equator(?))
if not mistaken ( α , β , γ ) correspond then to the tetrasphere point: R [ sin(α) , cos(α) cos(β) , cos(α) sin(β) cos(γ) , cos(α) sin(β) sin(γ) ]
With γ = 0 this mimics the situation on a regular trionian sphere (for other γ it might well be that some modification is needed, probably adding cos(γ) to the y slot)
{ note: x in the south - north direction, y toward the 0-meridian, z toward the 0-???, w perpendicular to all of them (???) }

Just trying to make sense.

Assuming that there would be a polar and solar equator would mean that a tetronian planet behaves like the rotatope duocylinder { x<sup>2</sup> + y<sup>2</sup> = R<sup>2</sup>, z<sup>2</sup> + w<sup>2</sup> = R<sup>2</sup> } whose surface points are given by: R [ cos(ρ) , sin(ρ) , cos(θ) , sin(θ) ] and thus those planets are dually rotating through tetraspace.
I've no idea how to map the positions here in longitude, latitude and laptitude.


The tetrasphere point of view only needs some "greenwich meridian", and some "what sil we callit" to pinpoint a zero laptitude.
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Postby wendy » Wed Jan 19, 2005 6:22 am

This is frightfully different to the one i came up with. The thing requires on the assumption there is an equator and poles.

Assuming of course, the laws of physics work, we should imagine that there is a transfer of energy so that the modes of rotation have the same energy. This means, in 4D, that the thing rotates in 'clifford-parallel' style, with equal rotations in the wx and yz axies.

Every point, then goes around the centre of the planet, in the sort of swirl-motion that makes swirl-prisms.

The same lattitude represents the points where the same star comes to a zenith (or overhead). There is a great-circle of these stars in the heavens, representing the daily track of the zenith-point.

A map of the rising of stars, will for all parts of the world, be the same, and take the nature of a 3d sphere. This is the same 3d sphere that forms the basis of swirl-prisms, btw.

If the sun does not move along one of these tracts (ie the world is tilted), then its zenith-point crosses the zenith-circles of a circle of places on the lattitude sphere, but this does not have to be the 'equator' of the lattitude-sphere.

Assume the tilt is 23.5 degrees.

On the lattitude sphere, we have at 47 degrees from the south-pole a circle representing the locations where the sun becomes overhead. This is the tropic torus. Everything 'south' of that is the tropics, with at the south 'pole' a ring representing the location where the sun is always 23.5 degrees off the zenith.

Going the other way, we have a circle (artic torus) represnting the places where the sun is always for mid-winter, a horizon-hugger. This is 47 degrees from the N pole on the lattitude-sphere, and the sun never gets more than 23.5 degrees into the sky.

The sun then moves around the zodiac, the "tropic line", once a year. As it does, mid-summer (and all the seasons) follow it. So where we have a world where somewhere it's 10pm, and somewhere it's noon, the 4D world has this, and somewhere it's spring, and somewhere it's autumn and somewhere it's summer.

This depends on whether the 'longitude on the latitude sphere' happens to be sunward at the time.

The 'latitude on the lattitude sphere' is the climata, or lattitude of climate. This ranges from 0 (tropic ring) to 90 (artic ring), passing the tropic torus at 23.5 degrees, and the artic torus at 66.5 degrees.

If there was a star whose zenith-point was in the artic-ring (so its rising point is in the centre o fthe zodiac-circle), then the number of degrees the cumulation of it from the horizon would designate the climate conditions. The higher the star rises, the colder the climate is. Kind of like polaris.

The 'sesonalita' or 'longitude on the latitude sphere' represents the current season. One has 'month-zones' like time-zones. So if we want christmas in december to be cold, then we would have different months happening everywhere. The distance on the ground of a month is something like 2000 miles, so you shan't expect Jack-Frost in the middle of summer or for a drive down to the corner-shop.

The longitude is the length along the horizon, measures the eastings and westings. In other words, the great circles on the 4-sphere are the same as ours, the time-zones being set by the half-3sphere gimble that sweeps around the world.
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Re: tetraplanetary coordinates

Postby Keiji » Wed Jan 19, 2005 5:09 pm

Aale de Winkel wrote:According to the glossary we have next to our regular trionian longitude and latitude the laptitude in the marp, garp direction. A short derivation:
tetra sphere: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] = R[sup]2[/sup]
north / south: x[sup]2[/sup] + y[sup]2[/sup] = R[sup]2[/sup] ==> R [ cos(α) , sin(α) ]
east / west (at y = y[sub]0[/sub]): x[sup]2[/sup] + z[sup]2[/sup] = R[sup]2[/sup] - y[sub]0[/sub][sup]2[/sup] = r[sup]2[/sup] ==> r [ cos(β) , sin(β) ]
marp / garp (at (y,z) = (y[sub]0[/sub],z[sub]0[/sub]): x[sup]2[/sup] + w[sup]2[/sup] = R[sup]2[/sup] - y[sub]0[/sub][sup]2[/sup] - z[sub]0[/sub][sup]2[/sup] = r[sup]2[/sup] ==> r [ cos(γ) , sin(γ) ]


Erm, that's kind of unreadable. I don't think the sup tags worked since the board was reinstalled :\
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Postby Paul » Wed Jan 19, 2005 8:32 pm

Hello all,

You can just use the HTML 'sup' tags (and 'sub' tags):

a<sup>3</sup>, a<sub>2</sub>

Which is, in code:

Code: Select all
a<sup>3</sup>, a<sub>2</sub>
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Postby Keiji » Wed Jan 19, 2005 9:04 pm

Thanks, I've edited his post now ;)
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