The circle of fifths is a well-known device for showing relationships between keys, and shows the concept of close vs distant keys.
But is there any way of arranging keys in three dimensions rather than two, and would that give any insights that aren't available from the 2D circle of fifths?
A circle is actually a one-dimensional figure. The two-dimensional figure that we think of when we hear "circle" is a disk. A circle is one dimensional because, given a circle, you can construct a coordinate system that allows you to identify a point on the circle with a single value. For example, you can choose a point on the circle as the origin and identify other points on the circle by their distance from the origin.
(Similarly, mathematicians use "sphere" to refer to a two-dimensional surface and "ball" to refer to a three-dimensional solid object; see https://en.wikipedia.org/wiki/Sphere for a more detailed discussion.)
I can't actually work out a useful way of displaying Pythagorean fifths and the theoretical infinity of double, triple, quadruple, etc., flats and sharps as a helix. I find a straight line works better:
...-B♭♭♭-F♭♭-C♭♭-G♭♭-D♭♭-A♭♭-E♭♭-B♭♭-F♭-C♭-G♭-D♭-A♭-E♭-B♭-F-C-G-D-A-E-B-F♯-C♯-G♯-D♯-A♯-E♯-B♯-F♯♯-C♯♯-G♯♯-D♯♯-A♯♯-E♯♯-B♯♯-F♯♯♯-...
Because of enharmonic equivalence, we can wrap this line around a circle whose circumference is 12 times the distance between two adjacent elements. It doesn't matter whether that distance represents a ratio of 1.5 or the 7/12 root of 2 or anything else; the only thing that matters is that we can consider (e.g.) A♭ as being enharmonically equivalent to G♯.
Some representations of the circle of fifths (or circle of semitones) use variable distance around the circle to represent different temperaments: 30 degrees represents 700 cents (or 100 cents) and longer and shorter arcs show which fifths (or semitones) are wider or narrower. But this is still one dimensional, and if one extends into multiple sharps or flats one doesn't get a helix but rather (for example) a B♯ that is a bit to the left or right of C♮
To represent these relationships in two or three dimensions, you would need to define some aspect of the relationship that you could represent using a second or third dimension. E major is just twice as distant from C major as D major is: a larger magnitude in the same dimension.
There is a 2-dimensional representation of pitch relations called the Tonnetz, but I suspect that even if it is possible to bend it into a curved surface using enharmonic equivalence, doing so would not yield a sphere.
Assuming enharmonic equivalence, any mapping of (perfect) fifths will eventually wrap around onto itself.
As such, in a two-dimensional representation, distance isn't always as clear as one might hope since the mappings of a key onto itself are, by definition, in another location. Starting at x, the distance to y is clear enough; but the distance to z, which is equivalent to x, may not initially be obvious if one isn't primed to look for multiple x equivalents.
In a three-dimensional representation, however, distance could be made more clear in the right circumstances. If the representation were a sphere with only one instance of each key, for instance, showing maximum distance (i.e., the pole) is immediately obvious and musically very meaningful.
A 3D model of fifths would not be a sphere.
A friend of mine says that this is what bothers him about the "circle of fifths": That it isn't in fact a circle; it's a helix. He'd like that fact acknowledged.
Pretending it actually is a circle is a lie-to-children, and like all lies-to-children, it can mislead or get confusing.
I'm not sure if he'd actually want to see the helix of fifths represented with a 3D model rather than a flat image, though. That seems kind of unwieldy for practical purposes (reproduction, transporting, easy reading).
But it does seem to me that a more representative visual than a circle would be a spiral in which for instance Gb and F# would be in the same direction from the centre, but drawn distinctly, one on the inside and the other on the edge, each connected to their own (and not one another's) mates.
Note that the "circle" of fifths is schematically a polygon (dodecagon), in which we can model the keys (or pitches) as vertices, and the relationships between them as edges. (Whether the edges are straight or curved doesn't matter.) I'm assuming enharmonic equivalence, without which no cyclical model makes sense.
So to extend this to another dimension, mapping key relationships onto a sphere would schematically mean mapping onto a polyhedron, in which vertices represent keys and edges represent relationships. The OP mentioned a tetrahedron, which would be an example of this.
In order to represent all 12 enharmonic pitches, you would want a polyhedron with 12 vertices. And you would want the vertices to be symmetric to each other; e.g. every vertex should have the same number of edges. (It wouldn't make sense for some keys to have more relationships than others.)
This basically gives you Platonic solids, Archimedean solids, prisms, and antiprisms.
Of the first two, only the icosahedron, truncated tetrahedron, and cuboctahedron have 12 vertices. The vertices of these polyhedra each have 5, 3, or 4 edges respectively. So if you want each key to have a relationship to 4 other keys, you might try a cuboctahedron.
In a prism, each vertex has 3 edges, and in an antiprism, 4. Imagine an antiprism in which the top is a ring of 6 vertices: A - B - C# - D# - F - G - A. The bottom is a ring of the other six keys: D - E - F# - G# - A# - C - D. Each top note is connected to two bottom notes that are a 5th away from it, sort of like this:
|--A---B---C#--D#---F---G-|--A | / \ / \ / \ / \ / \ / \| / |D---E---F#---G#--A#--C---|D--
(showing A and D twice to illustrate where they connect on both sides).
Does this give any new insights? Well, it illustrates the "closeness" of whole steps, in addition to the relationship of 5ths. You could choose other relationships instead.
For example, if each horizontal ring is a cycle of minor 3rds (A--C--Eb--Gb--), then you need 3 such rings:
|--A--C--Eb--Gb--|
|--A#--C#--E--G--|
|--B--D--F--Ab--|
But then if you still want to connect the rings to each other in a cyclical way, you can no longer use a prism or antiprism, which only allow 2 rings. To connect a grid cyclically in two different dimensions, you need a torus.
So, as a couple of people have said, you could wrap your grid around a torus. If you do that, your grid could be basically like the Tonnetz. Or you could emphasize different relationships/intervals. But it's hard to imagine how you would improve on the Tonnetz.
