Video of Map of a 4D Planet

[PatrickPowers]

I made a map of a 4D planet. It turned out much better looking than I expected.

https://youtu.be/u5Cz9aIm954

[PatrickPowers]

It works like this. Partition the planet into latitude tori. Each of these is a Cartesian product of two perpendicular circles. In this case we have 15 equally spaced latitude tori. Such tori have the property they can be cut open and flattened into a rectangle without changing the distance between any two points. Stack the rectangles.

All of the rectangles have the same length of diagonal. This is because the diagonal corresponds to a great circle on the surface of the planet.

The map can be based on any great circle. On a planet a natural choice would be a plane of rotation. On a planet with two different periods of rotation these two planes could be identified and chosen as the top and bottom rectangles, though such rectangles would be infinitely thin and so degenerate into lines. Note that the top and bottom rectangles are perpendicular to one another.

A 4D planet has only 90 degrees of latitude between the two circles. This is why the map is four times as wide as it is tall.

Each rectangle is free of distortion (I think). The distortion occurs going from one rectangle to another. Two out of three dimensions undistorted : not bad.

The equator of the planet is a square. Topologists love this thing : they call it the "square flat torus".

Recall that each of these latitude tori is a Cartesian product of two perpendicular circles. Usually they are of different sizes. If the diagonal is of length one then the radii of the circles are such that r1^2+r2^2=1. When flattened we get a r1 x r2 rectangle.