the torus can be split into three mutually interlocked parts.

Since the situation is somewhat on the congested side, when the central hole is equal to the worm diameter,

I made another toroid with a hole twice as large, relatively;

I also changed the swept profile to a hexagon, (the individaul strands now have a rhombic profile),

so that inner and outer surfaces have the same geometry and can thus be matched up easily.

--

And these are the kind of configurations that you can make with these three rings ...

Here is an example of a torus split into 4 mutually interlocking rings:

it does not cut the torus into two parts, but splits it into a semi-circular beam traveling around the toroidal loop twice

before it reconnects to itself. This is because the enclosed splitting space is "single-sided", i.e., the same material is

on both of it side.

Nat Friedman suggested, that this empty Moebius space might itself be made the key feature of some sculpture.

To that effect, the gap should be made wide enough so that one can easily see inside. Also, the center hole

of the torus should be made rather small. So, Nat, these pictures are for you ...

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to make serrated surfaces that would tightly interdigitate like the teeth in a well-designed gear.

One way to do this, is to make longitudinal serrations on one torus, and latitudinal ridges on the other.

But can the two parts be made identical ?

Below is a seven-tooth version. The two tori can actually move with respect to one another

-- forming some kind of a pathological worm-gear.

Split Trefoils

By choosing suitable geometry of the generating path and the right amount of twist,

the band can be made into a single-sided surface.

This Moebius trefoil band can now be split down its middle,

resulting in a band of half the width and twice the length,

that travels around the trefoil twice.

Alternatively, we can split the band in its thickness direction into "two layers".

--

With different parameter settings, the geometry can be modified

so that the overall twist is now an odd multiple of 90 degrees.

This allows us to split the knotted prism "twice" lengthwise

(it is really just a flat cut going around the knot twice,

or alternatively a cross-shaped cut going around once).

This results in a single prismatic beam, traveling around the knot
four times.

Here is a bigger picture, including my user interface.

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