Houndstooth Taxonomy

08.13.2018 § Leave a comment

Inventing variations on the houndstooth pattern is one of the things I do. Of the over two hundred of them I’ve thought up, though, before today I’d only gotten around to sharing a handful. And while I’d always planned to get them all out into the world eventually, a recent event has inspired me to bypass my usual perfectionism and just dump all of them here, now, however much works in progress they may be.

Why today

This recent event is a comment left on my very first houndstooth post. It was back in 2015 when I posted “thousoondth“, a fractal version of houndstooth, but not until now, three years later, that someone responded, claiming to have invented houndstooth fractals of their own.

My reaction to the email notification I received from WordPress went through several stages in rapid succession:
  1. First, I felt joy at the prospect of connecting with a fellow mathematical houndstoothnaut.
  2. Second, I felt regret that it had been thousoondth serving as the representation of my houndstooth fractal abilities… because since 2015 I had invented several additional houndstooth fractals, all of which I considered to be more compelling, but I hadn’t gotten around to shipping them yet.
  3. Third, I felt anxiety that this person might have in parallel come up with one of the same fractals that I had… and beaten me to the market.

This series of feelings all occurred just in the few seconds it took to read this mystery man’s message, before I clicked through to his link and saw his work.

Once there, my anxiety fell away, and my joy intensified. He hadn’t stolen any of my fractal houndsteeth ideas out from under me. He’d come up with two I hadn’t imagined yet: one inspired by Cantor dust, and one line fractal.

This man is a fascinating fellow named Loe Feijs, a PhD in Computer Science and a professor at the Eindhoven University of Technology in the Netherlands, and he is quite a bit more advanced in his houndstooth game than I am. And that’s not just because of his houndstooth bowtie.

Prof. Loe Feijs

Both of Feijs’s houndstooth fractal fashion items were collaborations with Mariana Toeters, a designer and researcher in fashion technology. Both were showcased in different years at Bridges, an annual international conference of technology and mathematical art (the 2018 edition was just a couple weeks ago). According to Professor Feijs, quoted prominently on his profile at the link above, “Algorithms are our new materials.” What a cool sentiment. Here’s a guy who has also written code to generate pieces inspired by Mondrianused cellular automata to weave emergent patterns into textiles, and to top it off produced an artwork bridging Duchamp’s Nude Descending a Staircase to Conway’s Game of Life using klein bottles and houndstooth. He and Toeters have inspired me to keep up my crafting, and keep reaching out to the world with it. I’ve messaged Feijs back directly too, of course, and hope that one day we can talk, meet up, or even collaborate.

Sometimes I feel a bit lonely, like I’m just hurtling my musings into the abyss on this blog, but every so often1 I discover other people out there with the same interests and it makes it all feel worth it.

Points of order

Before we dive into my houndstooth brain dump, I want to provide links to few other houndstooth creations of mine.

  1. Here’s a quick gallery of those fractal houndsteeth of mine I mentioned (thousoondth is first).
  2. houndstoothcraft: “a place to combine houndstooth effects”. My vision for this tool is that it should support all possible houndstooth variations (fairly ambitious; as Professor Feijs puts it, houndstooth is a “non-exhaustible source of inspiration”). Thus I’ve designed the tool such that casual users can just use the available interface to customize simple things like stripe count, coloration, and orientation, while advanced users can build their own effects from scratch to submit for inclusion. I’ve only guinea pigged six of my 2D houndsteeth there so far, though I’m not terribly concerned about the speed of progress yet. I did incur significant overhead for the tool itself: testing infrastructure, animation controls, the customization interface, etc. But perhaps the greatest cost has been, for each ‘tooth effect, finding the proper abstractions such that it can be best integrated and cross-pollinated with all the other effects. My desire for this programmatic perfection is perhaps the single greatest culprit in my houndstooth shipping procrastination.
  3. houndstoothtopia: a web VR experience housing all of my 3D houndsteeth. Without a VR headset, you can still zoom, pan, and rotate the view to get a look at them all in the courtyard. If you have a VR headset, though, you get to walk around inside the hedges. Plus, if you also have VR controllers, you can turn your hands into all the different houndsteeth too and get your eyes all up in ’em.
  4. The music that plays inside houndstoothtopia is shaped like a houndstooth. If you can’t make it inside to experience it spatially, here is a rendition of it:
  5. Also back in 2015 I used houndstooth as a substrate for demonstrating one of my Fun Musical Ideas. That idea is called “yaw“, and the demonstration is called “othoundsto“. You can find that one here.
  6. Besides thousoondth, I did write full-fledged blog posts for a few of my houndsteeth. You can find those here:

Further Ado

In my personal files I keep my houndsteeth in what I call a “Houndstooth Taxonomy” (hence the title of this post). Unlike the version of the taxonomy in my personal files, though, I do not plan to maintain this post as my ideas develop. This is merely meant as a snapshot of the current moment. I do still plan to get these out there in refined forms one day. I just fear that if for each one I waited to implement it in houndstoothcraft or prepare an in-depth post for it, that could take me several lifetimes.
As you will notice below, I’ve graduated some of the more finished of my houndsteeth to the level of having IDs, as my names for them shift occasionally and that gets really frustrating when I’m looking back through old notes and can’t figure out what I’m referring to (digitally mind-mapping thought with hyperlinks only helps so much).
Some of them not only don’t have IDs or names at all, or digital realizations, but not even sketches yet. Only descriptions in words and/or pictures of inspiration sources.

The Dump


ht2d_023: hextooth

“chevrons on a triangular grid”

“a different type of pinwheel”

“laterally split houndstooth cards”

ht2d_022: (randtooth variations)

(randomizing various properties)


ht2d_002: gingham-chevron continuum

ht2d_005: derasterized

ht2d_006: flippy

ht2d_008: harmonitooth

ht2d_012: all-cusps & all-roots

ht2d_016: wavestooth

ht2d_021: houndstrade

ht2d_030: crowntooth


hairline houndstooth

“stripe count changes within each square but perpendicularly to the stripes”

“in my picture i’ve got it kind of blocky, but that’s not my goal, my goal would be for it to be smooth”

“different stripe counts per column”


“each striped square is spinning and scaling”

“grid is rotating but stripe orientation is rotating to cancel it out, but stripe count stays the same, so the stripes are growing and shrinking within the range of the square root of 2 (i.e. it’s not just like a stripe layer underneath a checkers layer where white is alpha, because each striped square is a window into its own world)”


ht2d_011: radiate houndstooth

ht2d_028: squared-off houndstooth

ht2d_029: (roundtooth variations)


ticked stripes

“kind of evokes the zollner illusion – could do a variation with its askewness”


pinwheel fractal


“where it doesn’t matter what happens inside the tile as long as the same pattern of black and white on the edges”

parallelogram houndstooth

alternate root and cusp shapes, A

alternate root and cusp shapes, B

“old logo inspired”

arrow pointers

herringbone inside the striped tiles

spiky subsquares



ht2d_013: halftooth

ht2d_014: houndsbone

ht2d_031: foldingtooth

ht2d_033: yagasuri-checkers continuum

counterweight houndstooth

amime / chaintooth


“censored midroot ridge / made all of trapezoids”

“something to do with a middle tile”

“shower tiles from Kyoto Airbnb”


“variation on shave-merge-corners”

substitution tiling houndstooth

“invert trapezoid”

blue sea wave houndstooth


houndstooth of lakshmi

seven treasures houndstooth


“based on a uniform tiling based circle packing plus the next largest possible circle – uniform tiling being square”

“octagons and stars merged”

“half of tiles are octagons”

xn by yn

manji houndstooth

tuple houndstooth variations

“trapezoid stripes play”

pinwheel star houndstooth


dissection tiling of pentagon houndstooth


ht2d_009: houndsmorphosis

ht2d_017: staggertooth

ht2d_018: houndspiral

houndseye (bullseye houndstooth)

ring of overlapping circles

ht2d_019: herringstooth-staggertooth hybrid

“use rectangle style herringbone instead of parallelogram but then have blocks of it before it changes grains – inspiration attached”

ht2d_020: gridstooth

ht2d_025: pentooth

This one has an ID but I can confirm is a stretch goal: applying the essence of houndstooth to the Penrose tiling in some way.

zenigata houndstooth

italicized houndstooth

octagons and squares

“based on a uniform tiling based circle packing plus the next largest possible circle – uniform tiling being triangular”

snub houndstooth

“columns shrink/grow until replaced; do this over and over to evolve; instead of squishing diagonal stripes try shaving them off”

(not actually sketched yet – this is just the basic inspiration)

“cafe wall”

(also not sketched yet – just transfer houndstooth onto this)

“concave/convex grid”

(also not sketched yet; just apply houndstooth onto this grid)

Pythagorean houndstooth

Cairo houndstooth



“invert along perpendicular diagonal”

“the inverting layer on top is just checkers, half the size of the underlying houndstooth tiles, centered on them (not lined up edges)”

“tooth growing from inside, overtaking, inverting color”

“you could have just a single tiny houndstooth of opposite color gradually grow from center of each houndstooth until all of them overtake their parent at the same time thus flipping colors of the tiling, and if it wasn’t with respect to the perimeter then as it grew, parts of it would be outside the parent (in the mid-root void) so it could invert the neighbors!”

inversion fractal

standard houndstooth, then 1/2 size overlaid set so that in the center of each big square is a smaller square, and the overlay instead of being black and white either inverts or does not invert the colors of the previous layer, then repeat this forever and maybe it will make a cool fractal

perpendicular grain inversion

standard houndstooth, then an overlay of houndstooth which instead of being black and white either inverts or does not invert the colors of the previous layer, except that the overlaid houndstooth is in perpendicular grain

invert circles houndstooth

invert houndstooth with circles either centered on each square or split into four quadrants and positioned in each corner


ht2d_007: cmyktooth


“overlap/overlay very slight offset translucent blend type”

“offset two 50% blended houndsteeth by just a tiny amount, obviously related somehow proportionally, and maybe in the diagonal direction perpendicular to the stripes or maybe parallel”

“layer see through reverse grain”

“layer a houndstooth of transparent and black over a houndstooth in the opposite grain, offset to be centered on the corners of the other, and which uses another color besides black”

“overlapping semitransparent teeth”

“overlap/overlay polyrhythm”

“a houndstooth that is 50%50 blend but it’s like a 4×4 against 5×5 polyrhythm, stretched to take up the same space; or even a 4×5 against a 5×4 one squished one stretched to take up the same macrosquare”

tattersall fractal


ht2d_004: gongram

ht2d_003: ginghoundstootham

ht2d_010: houndazzle

“‘dazzle-‘strude hybrid”

“where the lines are always concentric but which color does it start at just inside the border”

ice ray houndstooth

“use the borders of houndstooth as the initial lattice – to spawn an ice ray pattern from”

“solid tiles have non-diagonal stripes lining up with main diagonal stripes of striped tiles”

“half of substripes end, implying boundaries”

“comprised of traced lines”

“four color rainbow”

houndstooth packing

“a weave but the threads are cycling through a gradient”

(not sketched yet)

“what if you did coloration not by tile, but by a grid that connected centers of squares”

“meaning that each houndstooth would be split into four unequal quadrants, each a different color and it would share that color with the nearest quadrant of each of the three nearest teeth that would have been the same color as it in a standard tiling”

argyle variation

“argyle red and green thin stripes one v one horizontal – with the thin stripes weaving above and below the houndstooth black and white respectively (r above b below w, g above w below b e.g.)”

circle-packing houndstooth

“houndazzle, but the two contrasting patterns are black and white herringbones (like the Black Lodge floor) oriented vertically vs horizontally”

(not sketched yet)

“the two textures are both houndsteeth”


“rotatey substripes”


“chrominance vs luminance, i.e. greyscale vs. colored”

Kusama tentacles

(apply this to houndstooth)

“blend a color of a vert weave from one color to another over the course of x columns”

(not sketched)

blendstooth / ombretooth

“some solids are stripes but houndstooth is still discernible”

“in the white area, only one of columns or rows (of half black, you know, parallelogram alternation there vs non-there), but in the black area, the rows are visible too”


“stumbling stripe houndstooth”

“through checkers” houndstooth

“continuous threads”

“every stripe segment is part black and part white; all that changes is whether black is thicker or white is thicker”


ht2d_015: hndstth

ht2d_027: which / outline houndstooth

higher order houndstooth

“made out of herringbone”


log cabin houndstooth

“a 2×2 of log cabin tiles could become a houndstooth tile”

“serrated fractal”

“out of squares with arc dividing them in half which is from a circle centered on one corner”


ht2d_024: cogstooth

ht2d_032: houndsring


houndstone (halftone)

modulation houndstooth

“spaced out”


“noise of houndsteeth”

“drop black houndsteeth onto a white background randomly, at random orientations and positions, with the caveat that it still comes out 50% white 50% black”

“free rotate and scale distort”

“only free scale allowed (not free rotate)”

“changing grains in different zones”

“pinwheel clusters”


“cast shadow houndstooth”



twist houndstooth

each square shrinks within itself gradually rotating and pasting itself over so you get a kind of twist or swirl

“checkers with standalone teeth of opposite color inscribed in each square”



“outline houndstooth”

“overlapping outlines”

“interlocking outline houndstooth”



“criss cross outline”

“connect all vertices”

peace psign


“new patterns from shrinking half the tiles into oblivion”



anamorphic houndstooth


camouflage houndstooth

“change shape to be like splotches except vaguely tooth-shaped”

“take color from houndstooth”

“Voronoi or Penrose or ice ray or anything cool – just take the center points of all the tiles and whatever color those points would be on an underlying standard houndstooth, make the tiles that. it won’t be a new pattern, just a kind of distorted standard houndstooth”


Voronoi tiling derived from a set of points based on houndstooth – whichever looks better – the centerpoitns of all the constituent trapezoids / triangles, or the midpoints of all the edges


ht2d_001: thousoondth

ht2d_026: hatchtooth


holestooth (alternate)

“yin yang out of houndstooth”

“overlapping lenses houndstooth”

houndstooth kanji

“another fractal attempt… tooth in cusps”

“star spangled tooth”

“built out of circles or arcs”

“houndstooth mandala”

(no sketch)


“fractal: one houndstooth with another inside its square part of the opposite color; repeat”


“like hohoundstoothoth, but you could rotate each next inner tooth 180 degrees, with roots in cusps”

“gingham fractal A”

“the grey squares are always a smaller gingham, let’s say 3×3, and all corners are grey, so the two striped tiles are different: one has black on left and right subtiles, the other has black on the top and bottom subtiles”

“super gingham, sub houndstooth”

“gingham where the vertical dark stripe and the horizontal dark stripe are two opposite only-blacks-are-real houndsteeth patterns, so when they intersect it’s solid black”

“fractal based on all trapezoids”


“mutant merged”

“giraffe mutant merged”

“infinitely mutant merged”

“staggered infinite row and column mutant merged”

“hexagons with a pattern of roots and cusps protruding from their sides, extrapolating from thinking about standard houndstooth in that way for a square”

“variation on hexagons with a pattern of roots and cusps protruding from their sides, heterogenous clockwise/counter”

“octagons with a pattern of roots and cusps protruding from their sides, extrapolating from thinking about standard houndstooth in that way”

“pied-de-poule / fleur-de-lis”

“serif houndstooth”



“repeatedly connect midpoints of a houndstooth”


ht3d_001: dougstooth


ht3d_003: houndstrude, convex

ht3d_004: houndstrude, compound

ht3d_005: houndstrude, concave

ht3d_006: houndstrude, freshmans

ht3d_027: houndstooth prism

ht3d_028: houndstooth prism, composite

ht3d_029: houndstooth prism, composite with swirl

ht3d_030: houndstooth prism, composite, aligned halves

ht3d_031: houndstooth prism, composite, swirled halves

ht3d_033: houndstrude convex, mitered by edges

ht3d_034: houndstrude compound, mitered by edges

ht3d_035: houndstrude concave, mitered by edges

ht3d_036: houndstrude, freshmans, mitered by edges

ht3d_055: houndstrude, outward convex

(no image available)



ht3d_019: octahedstooth with its homogeneous faces roots

ht3d_020: houndstamp, lesser

ht3d_021: houndsject, greater

ht3d_022: houndsject, lesser

ht3d_023: houndsject, lesser with inversion

ht3d_032: rhombic dodecahedstooth

ht3d_037: octahedstooth with its homogeneous faces cusps

ht3d_041: vertooth, cusps

ht3d_042: vertooth, roots

ht3d_043: vertooth, compound cusps

ht3d_044: vertooth, compound roots

ht3d_045: vertooth, compound cusps and roots

ht3d_046: vertooth, cusps spun

ht3d_047: vertooth roots spun

ht3d_048: vertooth compound cusps spun

ht3d_049: vertooth compound roots spun

ht3d_050: vertooth compound cusps and roots spun

ht3d_051: vertooth swirl

ht3d_052: vertooth swirl compound

ht3d_053: vertooth swirl compound, counter and clockwise

ht3d_054: houndstamp, greater extruded with swirl

“connecting vertices of diagonally combined houndsteeth in 3d”

“perimeter tooth bc you can always trace an outline of the tooth in each dimension”

“connecting vertices of non-diagonally combined houndsteeth in 3d”

(no image)

“composite compound pyramids”

(no image)


ht3d_026: parallelopiped striped houndstooth



ht3d_007: houndslathe, interior

ht3d_008: houndstooth horn toroid, root to root

ht3d_009: houndstooth horn toroid, cusp to cusp

ht3d_010: houndslathe exterior

ht3d_011: houndstwist

ht3d_012: houndstooth arch

ht3d_013: houndstooth horn toroidal polyhedron, root to root

ht3d_014: houndstooth horn toroidal polyhedron, cusp to cusp

ht3d_015: houndslathe exterior, toroidal polyhedron version

ht3d_017: houndstooth arch, toroidal polyhedron version

ht3d_024: helixtooth

ht3d_025: helixtooth, squared off

ht3d_038: houndslathe interior, toroidal polyhedron version


ht3d_002: hosotooth

ht3d_016: hosotooth, octahedrified

ht3d_039a: houndstorus, toroidal grain, solids top

ht3d_039b: houndstorus, toroidal grain, solids out

ht3d_039c: houndstorus, toroidal grain, solids bottom

ht3d_039d: houndstorus, toroidal grain, solids in

ht3d_040a: houndstorus, poloidal grain, solids top

ht3d_040b: houndstorus, poloidal grain, solids out

ht3d_040c: houndstorus, poloidal grain, solids bottom

ht3d_040d: houndstorus, poloidal grain, solids in


1 Actually it was only a few months ago when a fellow named Pandu Rendradjaja commented on my post Relative Frequencies of English Phonemes, having spent some effort “mostly-replicating and extending [my] work”. That gave me a similar rush of happiness that I wasn’t the only one interested in such specific and bizarre things. You can see his work on that here.

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