by wendy » Thu Aug 09, 2012 8:41 am
A good deal of thought has gone into this, even here. Here is my take on it. It pretty much follows the geocentric idea (rotating earth, sun goes around the earth, stars fixed). You can make it more complex by moving the earth around the sun, but it makes the maths harder.
A planet in four dimensions will, under the laws of physics (specifically, tidal equalisation of energy in different modes of rotation) will tend to adopt a 'clifford rotation', where every point goes around the centre.
The effect of this, starwise and sunwise, is that all of the heavens will be visible from all parts of the planets. In our world, venturing into the northern hemisphere will reveal entirely new constellations not previously known. The sun (essentially stationary or slow-moving in the stars), will rise and set exactly 12 hours apart, so the day never lengthens or shortens according to the seasons. (Yes, seasons do occur).
Now, we look at the instruments of navigation. Longitude as in 3D, requires accurate clocks. The stars give siderial time, and the time of year gives the drift from siderial time to solar time.
The lattitude is the selection of great circle. This is most interesting, because lattitude corresponds to points on a sphere. Consider a person in 4D, you have a horizon (which gives a 3sphere). There is an 'east' pole and a 'west' pole, and the line between these gives the line of skimming. Stars that lie on the line of skimming never rise or set. The number of degrees the star rises eastwards of the skimming-line, would cumulate the same number of degrees, and set exactly antipodally, in the western half of the sky. A star rising at the east-pole, will cumulate at the zenith, and set at the west pole.
We now construct the 'lattitude 3-sphere'. Between the east-pole and west-pole, there is a 3-space which contains all of the cumulation-points. In this 3-space, we draw a 3-sphere with a diameter between the observer and the zenith point. Now any star that cumulates does so in this 3-space, and we plot its lattitude, by drawing a line from the star to the observer. Where it strike the lattitude-sphere is the 'lattitude'.
What's interesting is that every place on the 4-sphere uses the same lattitude circle, but according to where the place is, the zenith stars are different. We might, for example, have for every star, a town where this star reaches zenith.
The zodiac is the trace of where the sun moves through the year. The actual track in the sky is a circle, but not necessarily corresponding to any of the zenith-arcs that the rotating earth makes on the sky. In practice, it doesn't, and the angle between these (when they cross), is the 4D version of the tilted earth. The sun, for moving in front of stars not on the same zenith-traces, will track a cicrle (like a lattitude in 3D sphere, not necessarily the equator).
The points on the longitude circle that the sun falls directly over is the 'tropics'. The points directly opposite form the 'artics', the axis which makes these points as 'lines of lattitude' in 3D correspond to the 'Equator' and 'Polar' circles. If you take, say the northern hemisphere of the earth, and then reduce it in the manner of the lattitude-sphere above (from an observer at the centre of the earth, the reduced sphere runs diametrically from the centre of the earth to the north pole). The lattitudes are effectively doubled (so that SP = 0 degrees, NP = 90 degrees). You have at 90-23.5×2 = 43 S, the tropics, at 43 north the artics.
Most of the planet would be temperate climates, with bands of really hot areas and really cold areas. The sun skims the horizon for one day (mid-winter), on the artics (only), and is zenith only in the tropics (only).
Now. The motion of the sun on the zodiac circle is to pass through the 12 signs over a year, and therefore to move around the tropics once a year. You can then mark the tropics with the twelve signs of the zodiac (eg tropic of cancer, tropic of leo, tropic of virgo etc), as points on the 43 S line. This line of zodiacs is similar in function to when one might see a zodiac line drawn across a map of the earth (representing the zenith-lattitudes of the sun).
When you take a particular day, say 5 May, you can place the sun at a specific point on the zodiac point. We put the lattitude sphere with this point at the top. The sun is maximum at this point, and rises only as high as the the angle between the sun-point and the locale. Taken at a given localle, the sun rises at different points of the horizon, (in the shape of a circle), where it rises east-most, it corresponds to the highest rise of the sun (mid-summer). Where the sun rises west-most (mid-winter), the sun rises lowest in the sky, and the land recieves less warmth from the sun.
What this means, is while our planet only has a 'tropic of cancer' and 'tropic of capricorn', the 4-planet has tropics of all signs, and therefore there is a place at every season of the year. In 3d, we have diverse regions have N = season, S = season + 6 months, in 4D, there are season-zones like time zones. The circle having 360 degrees, has a change of season (at month level), at a distance of 30 degrees or 1800 n. miles.
You could, for example, have sports-teams who literally 'follow the sun'. Cricket for here, is something that internationally, is played here around late december to janruary etc. Once this season has moved on, you could then decamp the teams to the next part of the world, and let it go all again. The autumn teams move in for their spin, and the the winter teams follow, and so forth.