Drawing a 4D Sphere

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Drawing a 4D Sphere

Postby PatrickPowers » Thu Sep 26, 2019 8:29 am

I've found that the more symmetrical the object, the harder to draw the 4D version. The 4D sphere is the worst. I seem to recall reading here that light reflected from it (a "highlight") would be ring-shaped instead centered around a bright spot. Does anyone known?
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Re: Drawing a 4D Sphere

Postby gonegahgah » Thu Oct 03, 2019 5:50 pm

I seem to recall reading here that light reflected from it (a "highlight") would be ring-shaped instead centered around a bright spot. Does anyone known?

The "highlight" for a 4D sphere equivalent (Glome) will be a sphere.

You might be referring to the surrounding mid-tone?
For an ordinary 3D sphere, the surrounding mid-tone appears to us as a semi-flat ring.
Likewise for a 4D glome, the surrounding mid-tone appears to the 4D person as a semi-flat (4D flat) ring.
I say 'semi-flat' because each lies on the respective surface of the sphere or glome.
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Re: Drawing a 4D Sphere

Postby quickfur » Tue Oct 15, 2019 6:54 pm

The difficulty with drawing a hypersphere is that it's so featureless. :D With other shapes you have edges, holes, different curvatures, that can project into lower dimensional features that can be used to try to understand the original shape. With the n-sphere, all projections to lower dimensions are the respective lower-dimensional spheres, so there's really nothing you could latch on to, in order to "see" the original n-sphere.

Drawing highlights don't necessarily help, either, since an n-dimensional specular spot generated by shining an n-dimensional light on the n-sphere would just be some kind of (n-1)-ellipsoid, which projects just to a lower-dimensional ellipsoid. I suppose the shape and orientation of the ellipsoid might help you visualize what's going on, but there's really not much to go on there.

The other way is to do what we do in 3D when drawing an otherwise completely-featureless sphere: draw longitude and latitude lines to make the sphere not-so-featureless, then project that into 2D where the lines become curves that you can then infer curvature from by the density of the lines, and their orientations, and so on. This is a pretty good approach, but here you run into a possibly unexpected complication: there are multiple ways of drawing a grid of lines on an n-sphere, and not all of them correspond with how we understand grid lines on a 3D sphere.

In 4D, there are at least two generalizations of longitude/latitude lines:

1) Polar interpretation: just like in a 3D globe where you have an axis, then latitude lines where the globe intersects with parallel planes perpendicular to the axis, then longitude lines when you have a plane rotating around the axis generating longitude lines where they intersect, so in 4D you can have a polar axis (intersect the 4-sphere with a line passing through its center), then compute the intersection of the 4-sphere with parallel hyperplanes that lie perpendicular to this axis. This produces a series of parallel 3D sphere intersections. Here's where the complication arises: if you want lines rather than spherical surfaces, you then have to repeat the process in 3D to create grid lines on these parallel 3D spheres. This implies picking a second axis, and generating latitude/longitude lines for each sphere that way. So the result you get will change depending on how you pick this 2nd axis. Furthermore, if you go this route, then if you segment the surface of the 4-sphere by tracing out squares and triangles outlined by the grid lines, you'll get various cube-like sectors along with some triangular wedge pieces, as well as some tetrahedral pieces: similar to how in the 3D globe grid lines you have mostly squarish sectors but around the poles you'll have triangular sectors. In the 4D case, not only you'll get tetrahedral pieces around the poles, but you'll also get triangular prisms running down the sides of the sphere where the grid lines cluster around the 2nd axis you chose. The result is not bad, but it does show non-uniformity (triangular wedges and tetrahedral sectors in addition to cube-like sectors).

2) Toroidal interpretation: another way of generalizing the 3D longitude and latitude lines is to increase the dimension of the poles: instead of using points as poles, we take advantage of the Hopf fibration to find two mutually-orthogonal great circles that are equidistant to each other on the surface of the 3-sphere. Then imagine expanding each of these great circles into concentric toroidal tubes. Believe it or not, eventually the concentric tubes from either starting great circle will coincide in an "equatorial" toroid. Each tube corresponds with a uniformly-distorted torus with a 2D surface that can be segmented into congruent squares. If you then join up these squares with corresponding squares from adjacent concentric tubes, then you'll get a grid on the 4D sphere that consists mostly of cube-like sectors, with triangular wedge sectors clustered around either of the 2 starting great circles. Here you get a nice analogy with the 3D case where you get triangular sectors clustering around the poles and square-like sectors everywhere else. This division of the 4D sphere's surface is directly related to the "ring poles" idea of a 4D planet undergoing Clifford rotation, posted in that "famous" thread on this forum about 4D planets.

The bad thing about (2) is that you get a bunch of toroidal shapes, which are not as easy to understand as in the (1) with the parallel spheres. But it makes up for that by being a lot more uniform in terms of sector shapes. And it also reveals the unique 4D Hopf fibration structure of the 4D sphere that isn't at all obvious when you generate grid lines according to method (1).
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