Longitude Hemispheres

[PatrickPowers]

In 4D we all know about latitude tori, how about longitude? What we get are longitudinal 2D hemispheres.

Start with a 4D sphere with radius one. Take two perpendicular great circles. The latitude is the arctan of the ratio of the distances between them. For longitude let's select one of the two great circles, the one in the wx plane. Select some arbitrary point on that circle to be zero. Then measure degrees around the circle from that point. For each point on the circle, all points on the surface of the sphere that are closest to the point share its longitude. As an example take point [1,0,0,0] and set its longitude to zero. Then the set of points closer to that point than to any other point in the circle is [w,0,y,z] with 0 < w and w²+y²+z² =1. This is a hemisphere with two degrees of freedom which extends over half of the planet. There is an antipodal hemisphere with w < 0 that corresponds to the point [-1,0,0,0] with longitude of 180 degrees.

In general for any point on the circle [W,X,0,0] with W²+X²=1, the points in the longitude hemisphere are in the set [cW,cX,y,z] with 0 < c and c²W²+c²X²+y²+z² = 1.

[quickfur]

The surface of a 3-sphere can be divided toroidally: take two mutually orthogonal great circles, these are your "toroidal ring-poles". Start from one ring-pole and draw a torus a unit distance away. Then walk two units away from the ring-pole and draw another torus, which will be concentric around the first torus. Keep going, and eventually the concentric tori will eventually converge onto the orthogonal ring-pole. The torus that equidistant from both ring-poles will be in the shape of the "latitude torus" of the duocylinder. This is the "toroid-equator" of the planet, i.e., the set of points on the surface equidistant from both ring-poles.

Now given these concentric tori, map a square grid onto each one and draw lines between grid intersections. This divides the surface of the 3-sphere into cubes (well, not exact cubes but close), except they are triangular prisms where they interface with the ring-poles. Now you have a system of latitudes, longitudes, and ... somethingelse-itudes on the surface of your planet. You can cut up the planet's surface into two equal halves along the toroid-equator. This produces two halves of the surface that are perpendicular to each other.

Assume the planet orbits the sun in the plane of one of the ring-poles. Then the half of the surface that contains this ring-pole will have the sun overhead all year round, so it essentially behaves like the "tropics"; the other half will see the sun overhead at a low angle (at most 45° above the horizon) so most of it will be "temperate", exhibiting yearly seasons, whereas the part around its ring-pole will be in perpetual winter ("permafrost"). So one ring-pole behaves like the analogue of the equator in a 3D planet, and the orthogonal ring-pole behaves like the poles of a 3D planet, except that it's not isolated poles but a connected ring.

4D planets are lots of fun.