## Dougstooth

05.01.2015 § 2 Comments

Dougstooth is the three dimensional equivalent of the two dimensional houndstooth pattern. “A” dougstooth, then, is a single tile of this tessellation, just as one could refer to a single tile of houndstooth as “a houndstooth”. Here is what a dougstooth looks like:

Or you can play around with one here: douglasblumeyer.com/dougstooth

The .obj file is available for download here: https://www.dropbox.com/sh/tormoqk0ut7ejmz/AADUG_-88sa2mKCi_x1aR4kia?dl=0. I hope to have a description of the triangles geometry up soon.

I do not use the definite article “the” lightly. I am putting my name on this thing, after all! I will now make the case that this form is the most logical 3d analog of the 2d original. I did not arrive at it straight away, and in fact went through several failed attempts at designing it. Along the way I learned a lot about what makes houndstooth essentially houndstooth, which was honestly a little surprising given how closely I’ve worked with the pattern in the past (see my musical project Othoundsto, press any key to start and turn your volume way up).

More Than Extrusion

Now, I did at least know from the start that I was interested in something more than simply extruding the traditional houndstooth out of the cloth, adding a z-axis in an only superficial way. This is the easiest to imagine three-dimensionalization of houndstooth, but it is not really interesting. For reference, this is what that looks like:

this is stupid.

All we’ve done here is converted a flat puzzle into a tall puzzle. The puzzle is no more interesting. The third dimension in this version brings nothing to the table. Every cross-section parallel to the surface is exactly the same: an original houndstooth tiling. I knew that I wouldn’t be satisfied until I came up with a solution that embraced all three dimensions equally, preferring nor spurning none.

And more importantly, I knew that however I three-dimensionalized houndstooth, I wanted it to derive from what defined 2d houndstooth in essence already. My intention was to abstract the design rules which give rise to houndstooth, and generalize those to a third dimension, creating 3d houndstooth by induction. Clearly, this extrusion method was not what I was looking for.

The Right Kind of Diagonal

So I returned to the original houndstooth pattern and contemplated it. One way to think of it is like a checkerboard, but more complex. In place of black tiles and white tiles it instead has solid tiles and diagonally striped tiles. Each striped tile has a total of four diagonal stripes, either white-black-white-black or black-white-black-white.

Checkers superimposed on houndstooth. Red checkers overlay with the solid squares, and yellow checkers overlay with the striped squares.

I realized that to move from a 2d houndstooth pattern into a 3d houndstooth pattern, squares would become cubes. Solidly colored cubes I had no trouble imagining; it was going to be these diagonally striped cubes that would define the challenge and the beauty of this mission.

I knew that if I wanted diagonal stripes through a cube, and I wanted them to embrace all three dimensions equally, I would need to stripe perpendicular to a line connecting one of the cube’s vertices to its opposite vertex, thereby traveling across the cube in each of its three dimensions.

To be clear, what I would have ended up with earlier had I pursued the superficial extrusion method – extruding off the page 1 grid unit to produce a single layer of houndstooth building block puzzle pieces – would be stripes perpendicular not to a line connecting opposite vertices, but a line connecting opposite edges (specifically connecting their midpoints, or any pair of points really producing a line perpendicular to the edges). Viewed looking down on my extruded houndstooth I would see diagonal lines just as I would in normal houndstooth, but if I looked at the top, bottom, right, or left sides of my tabletop puzzle I would see only straight lines down.

I need those lines to be diagonal too if I am to consider myself to have to truly achieved 3d houndstooth! And I would get that if I striped from vertex to opposite vertex.

Thought about another way: in traditional houndstooth, stripes go from square vertex to opposite vertex; were they to go from one edge to the opposite edge, I wouldn’t have diagonal lines at all, but horizontal or vertical stripes. Diagonality is a relationship between two dimensions; in the case of traditional houndstooth, x and y. For 3d houndstooth, I would need to achieve diagonality between every pair of dimensions, not just x and y (as in the extrusion method) but also x and z, and also y and z.

A First Attempt at the Striped Cube

The first method I tried for dividing the cube up into three-dimensionally diagonal stripes was to draw tick marks at the midpoints and endpoints of the cube’s edges, and slice along planes intersecting those points. I chose this method by observing that the diagonal stripes of the original houndstooth pattern intersect the edges of squares at their midpoints and endpoints.

However I ran into an issue at this point. Dividing a cube up into diagonal stripes this way yields not four stripes, but six stripes! What would I do with six stripes?

I could alternate black-white-black-white-black-white, but I’d be diverging significantly from a recognizably houndstoothy look. Of course it made sense on some level that in moving up dimensions I could end up with more of some resource than I did at a lower dimension, just as a square has 4 edges and 4 vertices while a cube has 6 and 8, respectively. But there were more options to explore, so I set this method aside for a time.

A Second Attempt at the Striped Cube

An alternate angle to stripe the cube would result in only four stripes. This involved setting the halfway point as the plane slicing the cube along the two other vertices co-planar with the chosen opposites.

And the other two slices each being the other pairs of vertices. Not all midpoints being stopped at, only those incidental to these endpoints/vertices being chosen.

This wasn’t as aesthetically pleasing to me, since looking at the square faces of this cube, some diagonal lines were now not at 45 degree angles to the sides (in the 6-slice version, that was still the case, as it is in the original pattern). But it was superficially more similar to the original pattern, so I decided to proceed.

Cube Tiling Struggles

Unfortunately I ran into problems when I began to try to tile 3d space with this pattern. I could not come up with a repeating 3d layout of cubes that corresponded to the pattern in the 2d original.

Reviewing the original, one notes that within the solid tiles, half are white and half are black, as two interlocking lattices of alternating rows and columns. Within striped tiles, similarly half are type A and half are type B as interlocking lattices, where type A begin in the top left corner with a white stripe, and type B begin in the top left corner with a black stripe.

I realized that thinking about it carefully enough, I had made the assumption that it was possible to create two interlocking 3d lattices of alternating rows and columns. This does not actually make sense though. Consider the solid black tiles in the original. Vertically, they are connected to other black tiles through type A striped tiles, and horizontally, they are connected to other black tiles through type B striped tiles. Now, it is certainly easy enough to imagine that in 3d houndstooth I could have a type C striped tile, and that this is what I would put on the third axis both immediately above and below a black tile (replace tile with square here) between it and the next black tile in the z dimension. And I could analogously do this with the white tiles. But, critically, what then would go immediately on top of and below the tiles in the original houndstooth that were already striped? Nothing makes sense to put there that is motivated by the original houndstooth design. And leaving nothing there means that my pattern doesn’t really tile 3d space completely; it leaves every other layer halfway full of air, plus these layers are only with respect to 2 dimensions, thus failing my non-preferential test; this pattern is not symmetrical in any orientation.

The main issue with interlocking lattice strategy is that it tries to be agnostic to the type of striped tile, which it cannot remain.

Breakthrough: Weave

Except then I realized that this was rooted in the assumption that a cross-section of 3D houndstooth would be 2d houndstooth. This was false, as you shall soon see. Ironically that getting away from the geometry and back to the original how it would have been threaded saved the day!

Another way to describe houndstooth is with every square consisting of diagonal stripes. That is, don’t think of the solid black square as “solid black” so much as a square where all diagonal stripes are black (and vice versa for white). Then it quickly becomes apparent that diagonal stripes come in two different sets:

• Set A: The first and third, i.e. the top left triangle and the bottom-right-er of the two trapezoids;
• Set B: The second and fourth, i.e., the top-left-er of the two trapezoids and the bottom right triangle.

And the rows of houndstooth can be described as alternating Set A between being white and being black, and the columns of houndstooth can be described as alternating Set B between being white and being black. When both Set A and Set B are white, you have a white square. When one of Set A or Set B are black, you have one or the other of the two types of striped squares. When both Set A and Set B are black, you have a black square.

As compared with the previous “interlocking lattices” example, here no column’s color matches a row’s color. They are compared only within themselves. The blue rows alternate dark and light depending on whether the top left triangle is black or white. The green columns alternate dark and light depending on whether the bottom right triangle is black or white.

This is probably exactly how houndstooth is sewn: with diagonal stretches of thread across a grid of holes! (It’s like gingham, but with way fewer stripes)

Now this is much more conceptually easy to generalize into the third dimension. I would have alternating rows, alternating columns, and alternating stacks (stack being the best term I can muster for the equivalent of a row or column for the third dimension – honestly surprised consensus hasn’t been achieved on this already, at least not as far as I can tell through Google searching) such that every 2x2x2 section of 3D houndstooth would have one cube where all three sets were white, one cube where all three sets were black, and the other 6 cubes would cover all the other combinations of sets.

Conflicts

So the first thing I realized after this breakthrough was that my original and more aesthetically pleasing realization of the diagonally striped cube, the one which yielded 6 stripes, was correct after all. If I was going to get three sets of stripes out of each cube, I sure couldn’t do that with only 4 total, at least not evenly! (Could consider grouping the two end triangles together?) I was growing comfortable with the idea that the 3D analog of a houndstooth might not just look like a tooth, but might be somewhat surprisingly shaped.

The first way I tried to divide up the 6 stripes was like how houndstooth does it: alternating. My three Sets were:

• Set A: The first and fourth
• Set B: The second and fifth
• Set C: The third and sixth

Unfortunately, once I built this, I realized that the resulting houndsteeth of the same color shared faces. Meaning that they weren’t separate. Which is definitely a radical departure from original houndstooth. At least, I consider it a pretty important property of houndstooth that individual houndsteeth of the same color touch only at vertices.

[I can’t seem to find my documentation of these incidents where houndsteeth of the same color touched each other, but screenshots of Unity game engine will go here when I do]

So I experimented with grouping like this:

• Set A: The first and second
• Set B: The third and fourth
• Set C: The fifth and sixth

This is worse because it’s missing the alternating property of houndstooth stripes, and besides, it doesn’t solve the problem of houndsteeth not touching those of the same color.

Next Breakthrough: Trois Coleurs

I eventually realized that this was not possible to do with only two colors. The major realization came when I remembered a shirt a guy had worn in front of me at a fluxus performance: he had a shirt with vertical orange, green, and blue bands, and horizontal orange, green, and blue bands. When the two oranges intersected, a square was solid orange.

So there you have it. Not what I originally imagined – something that would look like a houndstooth / project a houndstooth shadow from multiple angles.

[I do have a screenshot out of Unity somewhere where an earlier 4-stripe diagonal experiment could look like a 2d houndstooth from certain directions… I’ll see if I can dig that up too.]

The final Dougstooth, unlike a houndstooth, is not quite contiguous. There are pieces that hang by edges, so if I wanted to, say, 3D print one then I’d have to add little connectors or something. But at least they only touch other houndsteeth of the same color at edges too. So I’m not sure if that qualifies as satisfying the essence of houndstooth, but it’s certainly better.

I tried to get any rearrangements stacking out of the 3×3 core to be symmetrical, just as they are in the original “a houndstooth” which spills off a 2×2 where the cusps (the smaller triangular spikes, as opposed to the roots which are the longer trapezoidal spikes; this is just my terminology for the houndstooth, not established by any means, but logically derived from orthodontial terms) would be danglies.

Ultimately you end up with a 3×3 supertile. I left the third color as alpha so you can see in, and black as reddish to better distinguish it given all the shadows and lighting necessary to understand geometry.

I’ll leave you with my 3D printed version of it: