by wendy » Mon Nov 19, 2018 10:21 am
Suppose you are lieing on your back, watching the night sky in fast motion.
In 3d, some stars will rise on the east, and make only a small arc before they set. If you are in the northern hemisphere, these will be around the south part of the horizon. Some stars in the north will never set. Their full path in the night sky remains visible at all times. Stars too far south are never seen in the north.
In 4d, all stars rise and set. You see exactly half the trace in the sky, of every star. Everything still rises on the east half of the sky, and sets in the west, but there is now a second motion. The sky turns around the east-west axis as well. This is not north/south, but two new directions, say 'forward' and 'reverse'. In the course of a star being visible, it will move 180 degrees forward, The sky is acting like a giant screw.
The stars cumulate, or come to their highest point, at 90 degrees forward of their rising. This always happens on the circle dividing East (or rising) and West (setting). If you imagine a globe, you could have it spinning once a day, and all stars rise in the north (at various lattitudes), and set in the south (at the same lattitude S). Because it takes 12 hours from rise to set, a star that rose at 150 deg E, will travel to 30 W (180 degrees), and set there. This is how the horizon looks in 4d. Stars that rise at the E pole, go straight overhead and set at the W pole. They still spiral, but it's very tiny. Likewise stars that 'rise' on the equator, always set on the equator, and cling to the horizon.
So stars always set on the diametric opposite point of the horizon.
The Gimbel
The curved bracket that holds a globe in 3d, is called the gimbel. In essence, the world spins and the points of the gimbel trace the lines of lattitude. The lines of longitude are broken into the parts of the day. In 3d, the gimbel runs from 90 S to 90 N. The 4d gimble is an entirely different thing, but still each point represents all of the places that have the same zenith-star.
Before the southern hemisphere was known, you could think of the year as a circle, and the north-pointer pointing to the season concerned. So December is in winter, and June is in summer. With the southern hemisphere, you have a second pointer on the opposite side, so December is summer and June is winter. If you replace this pointer with a full disk, you get what happens in 4D. You might have December at the beginning of spring, or late autumn. You would have season-zones like 3d time zones. (The time zones still exist, but we have rendered the whole circle to a point).
North and South still define the climate, but we move south to the equator, so on the circle above, S is the equator, N is the north pole, and the lattitude from 0 to 90 defines the climate. This becomes the climata direction. The more north you get, the colder it gets, sort of.
So our disk with the seasons, we can take one hemisphere, and shrink the equator to a point. This is the 'lattitude sphere' in 4d. The actual derivation is to suppose you are standing at a point (the nadir), and the lattitude sphere stretches from the nadir to the zenith. You draw a line from the nadir to some point in the sky, and this maps onto the lattitude sphere. Points opposite represent orthogonal great circles.
This sphere is the same for everyone. The main difference is which point is touching the ground. It's the same NS sphere as above, but your zenith point is a certian point on this sphere, and the forward/backwards match the cumulation of the stars, is the 2-plane between the zenith and nadir. This sphere stands so that the E-W axis is perpendicular.
Here comes the Sun
The sun's motion across the sky is the elliptic, or zodiac. In 3d and 4d, we take these as circles as the first approximation. It will cross some of the EW circles at some angle, as it is not parallel with the EW circles. There are two circles for which it is equidistant, one is the South-Polar, and one is the North-Polar. The trace of circles that allow the sun to be overhead, is the tropic torus. This is the margin on a duocylinder, or all points some specific distance from the south-polar. The path of the sun is on this torus, is to travel around both the long and short axis once a day, and slowly move across the torus. If you were to unfold the surface to a rectangle, the sun-path would run as a diagonal of the rectangle, and slowly edge to cover it.
The diagonals of all toruses of a sphere are in fact great circles, and as such are the same length.
In terms of our 4d gimbel, we have the N pole, as cold. Some distance down is the artic circle (where the sun might hover on the horizon for a day), the tropics (where the sun comes overhead), and the S pole (where the sun always gets to some same 'highest' point every day of the year).
To locate the sun, we should take this gimble, and set it up in the manner described (between the E and W hemiglobes), and note that the sun is a star that moves around the tropic once a year. It travels on a 94-degree circle on this, and sometimes it rises high in the sky, and other times it doesn't. The inclination to the ground changes through the year, rather as our sun does. It is this, rather than the shortening of the days etc, that make the summer and winter happen.