The above, like most of the "3D truchet tiling" I have found on the web, uses tubes: each cubic tile contains three tubes, and each tube (rendered as a quarter-torus or something similar) connects the centers of two adjacent cube faces.
(Or else it has just been a 3D rendering of a 2D truchet tiling.)

My question has to do with another 3D analog of truchet tiling.

The 3D analog I've been thinking of uses surfaces instead of pipes.
It'starts by making each face of the cubic tile a copy of the 2D truchet tile, i.e. it has two quarter-circles in the plane of the face:

.

Then we connect the quarter-circles in the cube faces to each other using surfaces.
For example, using a minimal surface. Sketching an example by hand, the minimal surface stretched between these curves looks pretty interesting.

But this is where I get stuck: I don't have near the math knowledge to model such a surface. I've read some things about minimal surfaces, but there are many different kinds, and I don't know to apply them to this situation.

The good news is, I don't think there are very many different possibilities for this family of 3D tiles, when you eliminate duplicates under reflection and rotation. (I've counted 5, and there may be a few more.) The trivial case has four sphere-octants, one in each of four corners of the cube:

That's the only case where I know how to model all the surfaces.

Actually, I'm not sure that minimal surfaces are the best answer. In order for tiling to work, the surfaces need to follow two constraints:
(1) They need to intersect the cube faces (only) along the quarter-circles.
(2) They need to intersect each cube faces at a right angle to the face, so that they fit'smoothly with the surfaces in neighboring cubes.
A minimal surface may not always meet constraint (2). In fact I suspect in the trivial case in image 2, a minimal surface (instead of a sphere-octant) wouldn't intersect the faces at right angles.

On the other hand, constraint (2) is somewhat arbitrary. The surfaces don't necessarily have to join smoothly. I just think it would just look nicer.

My goal was to write an OpenGL animation, for viewing pleasure (e.g. as a screensaver). So ultimately I’ll need to make it fast enough to animate.

Any ideas for how to mathematically model smooth surfaces that would join the curves? Some sort of quadric surface?

Thanks,
Lars

I looked for a relatively simple surface to use as a starting point to figure out how to model the surfaces. Hopefully if I can succeed with this one, it will be a step toward solving the harder ones. One simple example is:

As you can see, the Truchet curves delimit two sphere-sections and one "other" surface which lies along the "floor". (The vertex nearest the viewer is <1,1,0>; the opposite vertex is <0,0,1>.)

In the enumeration of the 64 possibilities, this is cube 111001, classified as 2:1b (2 spherical sections, 1 other surface, unique variant b).

The cube is the unit cube (size s = 1).

Here are the equations I'm able to come up with for the surface.
(A) The surface intersects the cube faces along quarter-circles as shown. This is expressed as:
When z=0, x^2 + y^2 = r^2.
When x=0, y^2 + z^2 = r^2. (So far, x^2 + y^2 + z^2 = r^2, but not for long.)
When y=0, x^2 + (z-s)^2 = r^2 or (x-s)^2 + z^2 = r^2. (The latter if x>z; the former otherwise.)
When x=1, y^2 + (z-s)^2 = r^2.
When z=1, (x-s)^2 + y^2 = r^2.

Can we unify that all into one equation?

We want the surface to be smooth, so that means there is a first derivative and it is continuous everywhere in the domain (0 to 1 for each axis). (I don't think the 2nd derivative needs to be continuous... what do you think?) So I suppose what that means is I have to restrict my equations to those that use functions that are differentiable and whose first derivative is continuous.
Differentiable with respect to what? I'm not sure. In the equation for a sphere, x^2 + y^2 + z^2 = r^2, I guess the functions for x, y, and z each need to be (and are) differentiable with respect to the other two variables...?

Also, we would like the surface to intersect the faces at right angles; that's expressed as:
When z=0 or 1, dy/dz = dx/dz = 0.
When x=0 or 1, dy/dx = dz/dx = 0.
When y=0, dx/dy = dz/dy = 0.

The shape of the surface is saddle-like in the middle, so maybe I need something like a hyperbolic paraboloid. That doesn't work at the faces though.

Any ideas? Maybe I could use whatever they use for "blobs" in raytracing?

Thanks,
Lars

>On the other hand, constraint (2) is somewhat arbitrary.

They may join smoothly even if not orthogonal to the face of a square. But if you allow that, you'll have to exercise more care while tiling.