3DOF Wheels

In this post I described a type of void-sponge based on the 6 regular polychora (5-cell, 8-cell, 16-cell, 24-cell, 120-cell and 600-cell). A nice property of these is their symmetry to 4D rotations projected stereographically into 3D, in other words transformations that rotate around a circular ring. 

If we apply this rotation with translation you get a nice 'swimming' motion:

You could imagine that if there was such a flexible object then it could traverse through water because the outer region pushing downwards is larger than the inner region pushing upwards. 

And due to its symmetry it can move in any direction in 3D. The principle axes require the least expansion and contraction, and are shown here with it moving in each axis direction in turn:

You can think of it as a 3 degree-of-freedom version of a wheel. It can move a payload in 3DOFs where a ball-robot can move it in 2D, a wheel moves a payload in 1D and something like a table leg moves a payload in 0D i.e. nowhere, it just supports the payload.

But there are 6 DOFs of Mobius transformations, so the structure can do more than just translations. If we offset the circular rotation then it should turn as it pushes through the water, since the outer edge is pushing down faster on the right side than the left in this animation:

Even though there are 6 DOFs in Mobius transformations, the three rotation degrees of freedom are rigid so uncontrollable by the structure itself. The remaining degrees of freedom allow it to get around, underwater in the above case, but also on land.

For example, something like this transformation could allow the structure to move forwards, by pushing more of its mass forward:

Notice that this is different from a rigid roll, notice the change in size of the large circular edge.

It could also transform into more of a 1DOF wheel shape:

and there is still freedom to distort the wheel while keeping its rim circular, in order to push in a particular direction by offsetting the centre of mass:

We can do all of these things with the other regular polychora-based void-sponges. Here is the swimming motion for the 24-cell sponge:

The largest holes are the best locations for new directions as they minimise the expansion and contraction required. The 24-cell has more symmetry than the 8-cell, so change its movement with a choice of six different directions naturally, compared to four for the 8-cell (translate left, right, fwd, bwd).

Incidentally, the 'rotation around a circular ring axis' transformation is called an elliptic Mobius transformation. But since this is somewhat misleading terminology I prefer the term poloidal rotation, which couples nicely with its counterpart toroidal rotation (along the ring): https://en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates.

You could also call it vortex rotation or vortex circulation, but that is more suggestive of a vector field.

There is another form of locomotion that seems better suited to burrowing underground as the poloidal motion above would require displacing a lot of earth. The motion is a hyperbolic Mobius transformation:

Unlike the elliptic transformation, this contains unbounded contraction (at the top) and expansion at the bottom. It would be harder to generate biophysically, but not impossible, it would need the physical cells to redistribute as it moves. Or in other words, the cells would need to coordinate the holes to move locally upwards without moving the cells upwards.

As with the swimming motion, this method can burrow along any of the three axes, and turn its heading. It is also possible with the other void-sponges too, such as the 8-cell version.

There is one Mobius transformation not mentioned yet, the parabolic transformation. This is equivalent to addition on the Riemann sphere, it leaves one point fixed on the structure's surface. This could be useful for turning on the spot when burrowing.