Map of a 4D planet

[Hornbein]

A map of a four-dimensional planet is three dimensional, so such can exist in our Universe. I made one and posted a video to the Internet.

This is all based on William Kingdon Clifford's math from the 19th century.

It works like this. A 4D planet has two perpendicular planes of rotation. The intersection of such a plane with the surface of the planet is a great circle. We can define latitude as the arctan( distance from one plane/distance from the other plane). The set of all points with the same latitude is a latitude torus. Partition the planet into latitude tori. Each of these is a Cartesian product of two perpendicular circles. In this map 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. These correspond to the north and south poles here on Earth. 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 while a great circle is 360 degrees. This is why the map is (theoretically) four times as wide as it is tall.

Each rectangle is free of distortion. 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 in the map. 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 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.

 

[jambaugh]

The surface of a 4-ball is a 3-sphere. Here's a narrative to help you "visualize" it.

Imagine you are on a tiny smooth planet (2-sphere). You set down a wading pool made of super-elastic bubble plastic on the north poll. You begin filling it with water and instead of overflowing its floor and walls stretch. Eventually your pool fills up the northern hemisphere, at which time the circumference of the wall (a 1-sphere) does not grow any more and now begins to shrink. Eventually your pool covers the entire planet except a small circle around the south pole.

Now suppose your little planet is floating around inside a 3-sphere. Your pool now looks like a balloon encasing the planet and covered with water. You pump the water out (and through the South pole nozzle into the interior of the balloon/under the plastic pool.)

You heat the water into steam and your balloon will grow around your planet. Call the center of this planet the "Here pole" and as the balloon grows it will (in 3-sphere space) eventually get to the half-way point forming an equatorial 2-sphere of maximum size. It then starts shrinking until it collapse to a small inside-out balloon encasing a point opposite your planet (the "There pole" ).

 

[DaveC426913]

A map of a four-dimensional planet is three dimensional, so such can exist in our Universe. I made one and posted a video to the Internet.

This is all based on William Kingdon Clifford's math from the 19th century.

It works like this. A 4D planet has two perpendicular planes of rotation. The intersection of such a plane with the surface of the planet is a great circle. We can define latitude as the arctan( distance from one plane/distance from the other plane). The set of all points with the same latitude is a latitude torus. Partition the planet into latitude tori. Each of these is a Cartesian product of two perpendicular circles. In this map 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. These correspond to the north and south poles here on Earth. 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 while a great circle is 360 degrees. This is why the map is (theoretically) four times as wide as it is tall.

Each rectangle is free of distortion. 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 in the map. 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 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.

I've read Flatland and other similar books, and I occasionally dabble in multi-dimensional real-world shapes, such as tesseracts, but the above description of a 4D hypersphere is beyond my ken.

 

[Hornbein]

I too was unable to visualize it. The map is 3D so it can exist in our Universe. I had a model made of glass. Here's a video of it.



I don't know why the glass is green. Lead? Another clue is that the glass is denser than one would think. This model weighs maybe 40 pounds.

It's not possible to touch the inside directly but other than that it's realistic.

 

[DaveC426913]

No I saw the video from your post 1. I saw it, but I dont get it.

I guess that's not quite true. I've seen 3D slices of 4D objects before, and they are always chopped up like this, because we can't visualize the extra parts in one view with our 3D eyes.

So the fact that I can't quite reassemble this particular 3D projection into its 3D shape doesn’t mean I don't get the idea.

One can step into Flatland and try to show the citizens what a 3D sphere looks like and they get a similar chopped up projection.

 

[Hornbein]

No I saw the video from your post 1. I saw it, but I dont get it.

I guess that's not quite true. I've seen 3D slices of 4D objects before, and they are always chopped up like this, because we can't visualize the extra parts in one view with our 3D eyes.

So the fact that I can't quite reassemble this particular 3D projection into its 3D shape doesn’t mean I don't get the idea.

One can step into Flatland and try to show the citizens what a 3D sphere looks like and they get a similar chopped up projection.

It's layered like that because we used sheets of glass. If the thickness goes to infinitely small then you get a smooth 3D object. All maps are like this. You can think of a Mercator map of our Earth as the assemblage of an infinite number of lines.

I could get a glass blower to make a map but that's too much trouble. Having a wood carver make one would be doable, just leave the oceans as empty space. Can't then have islands though.

If you wanted to show Flatland people a map of our Earth you could take any map you like, make it out of glass instead of paper so that the interior is visible to a 2D person, and put it down there.

 

[sbrothy]

A map of a four-dimensional planet is three dimensional, so such can exist in our Universe. I made one and posted a video to the Internet.

This is all based on William Kingdon Clifford's math from the 19th century.

It works like this. A 4D planet has two perpendicular planes of rotation. The intersection of such a plane with the surface of the planet is a great circle. We can define latitude as the arctan( distance from one plane/distance from the other plane). The set of all points with the same latitude is a latitude torus. Partition the planet into latitude tori. Each of these is a Cartesian product of two perpendicular circles. In this map 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. These correspond to the north and south poles here on Earth. 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 while a great circle is 360 degrees. This is why the map is (theoretically) four times as wide as it is tall.

Each rectangle is free of distortion. 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 in the map. 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 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.

I humbly wonder who your audience is supposed to be?

EDIT: Oh I was thinking novel but I suddenly realize that’s not necessarily the case.

 

[Hornbein]

I humbly wonder who your audience is supposed to be?

EDIT: Oh I was thinking novel but I suddenly realize that’s not necessarily the case.

Interest in applied 4D geometry is quite small but greater than zero. A possibly larger audience might be those who care not of the math but find the resulting shapes intriguing for their own sake.

 

[sbrothy]

Interest in applied 4D geometry is quite small but greater than zero. A possibly larger audience might be those who care not of the math but find the resulting shapes intriguing for their own sake.

Indeed. I found the idea intriguing sure, I’m just skeptic cf the “average” reader (which I’m aware sounds arrogant and patronizing). Then again the “average” reader was never your intended audience. “Because I like it” is justification enough. Not that you need one mind!