## 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.

- First, I felt joy at the prospect of connecting with a fellow mathematical houndstoothnaut.
- 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.
- 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.

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 Mondrian, used 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 often^{1} 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.

- Here’s a quick gallery of those fractal houndsteeth of mine I mentioned (thousoondth is first).

- 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.

- 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.

- 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:
- 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.
- Besides thousoondth, I did write full-fledged blog posts for a few of my houndsteeth. You can find those here:
- dougstooth: of all the 3D houndsteeth I’ve invented, this is the first, and it is what I believe to be the truest, most fundamental 3D analog to the 2D houndstooth shape
- ginghoundstootham
- gingham-herringbone continuum (later re-named to gingham-chevron continuum)
- derasterized

# Further Ado

# The Dump

## VARIATIONS I’VE HAD DIFFICULTY CATEGORIZING

### ht2d_023: hextooth

### “chevrons on a triangular grid”

### “a different type of pinwheel”

### “laterally split houndstooth cards”

### ht2d_022: (randtooth variations)

(randomizing various properties)

## VARIATIONS ON STRIPE (WIDTH, ANGLE, COUNT, COLOR ASSIGNMENT, ETC.)

### 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

### horiztooth

### 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”

**VARIATIONS ON STRIPE SHAPE**

### ht2d_011: radiate houndstooth

### ht2d_028: squared-off houndstooth

### ht2d_031: foldingtooth

### ht2d_033: yagasuri-checkers continuum

### counterweight houndstooth

### amime / chaintooth

### crinckle-tooth

### “censored midroot ridge / made all of trapezoids”

### “something to do with a middle tile”

### “shower tiles from Kyoto Airbnb”

### “shave-merge-corners”

### “variation on shave-merge-corners”

### substitution tiling houndstooth

### “invert trapezoid”

### blue sea wave houndstooth

### woundstooth

### houndstooth of lakshmi

### seven treasures houndstooth

### chevrontooth

### “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

### barkstooth

### dissection tiling of pentagon houndstooth

## VARIATIONS ON GRID SHAPE

### ht2d_009: houndsmorphosis

### ht2d_017: staggertooth

### ht2d_018: houndspiral

### houndseye (bullseye houndstooth)

### ring of overlapping circles

### ht2d_020: gridstooth

### 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”

### Pythagorean houndstooth

### Cairo houndstooth

## VARIATIONS BASED ON INVERTING LAYERS

### invertooth

### “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”

### inversion fractal

### perpendicular grain inversion

### invert circles houndstooth

## VARIATIONS INVOLVING TRANSPARENT LAYERING

### ht2d_007: cmyktooth

### slacktooth

### “overlap/overlay very slight offset translucent blend type”

### “layer see through reverse grain”

### “overlapping semitransparent teeth”

### “overlap/overlay polyrhythm”

### tattersall fractal

## VARIATIONS THAT ARE JUST RE-TEXTURINGS OF HOUNDSTOOTH

### ht2d_004: gongram

### ht2d_003: ginghoundstootham

ht2d_010: houndazzle

### “‘dazzle-‘strude hybrid”

### ice ray houndstooth

### “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”

### argyle variation

### 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”

### houndscomb

### “rotatey substripes”

### noisetooth

### Kusama tentacles

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

### 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”

### thundstooth

### “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”

## VARIATIONS THAT ARE ASSEMBLED FROM PIECES

### ht2d_015: hndstth

### ht2d_027: which / outline houndstooth

### higher order houndstooth

### “made out of herringbone”

### quadrisoscelestooth

### 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”

## VARIATIONS BY ARRANGING HOUNDSTEETH IN SPACE

### ht2d_024: cogstooth

### ht2d_032: houndsring

### houndsverge

### houndstone (halftone)

### modulation houndstooth

### “spaced out”

### “tips-to-cusps”

### “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”

### sprouttooth

### “cast shadow houndstooth”

### propellertooth

## VARIATIONS WITH NON-FLUSH TILES AND TEETH

### twist houndstooth

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

beveltooth

## VARIATIONS THAT ARE OUTLINES, NO FILL

### “outline houndstooth”

### “overlapping outlines”

### “interlocking outline houndstooth”

### moundstooth

### “inline”

### “criss cross outline”

### “connect all vertices”

### peace psign

## VARIATIONS BASED ON DISTORTING OR STRETCHING THE WHOLE

### “new patterns from shrinking half the tiles into oblivion”

### hypertooth

### pointstooth

### anamorphic houndstooth

### kaleidoundstooth

### 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”

### vorondstoith

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

## VARIATIONS THAT ARE NON-TILINGS OR STANDALONE SHAPES

### ht2d_001: thousoondth

### ht2d_026: hatchtooth

### holestooth

### 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)

### hohoundstoothoth

### houndsundstotooth

### “gingham fractal A”

### “super gingham, sub houndstooth”

### “fractal based on all trapezoids”

## VARIATIONS BASED ON CHECKERING HOUNDSTOOTH LOOK-A-LIKES

### “mutant merged”

### “giraffe mutant merged”

### “infinitely 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”

### houndsmooth

### shadowtooth

“repeatedly connect midpoints of a houndstooth”

## 3D HOUNDSTOOTH VARIATIONS – FUNDAMENTAL

### ht3d_001: dougstooth

## 3D HOUNDSTOOTH VARIATIONS – PRISMATOID

### 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)

## 3D HOUNDSTOOTH VARIATIONS – PROJECTIONS, FACES, AND COMPOUNDS

### ht3d_018_houndstampGreaterExtruded

### 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)

## 3D HOUNDSTOOTH VARIATIONS – MISCELLANEOUS

### ht3d_026: parallelopiped striped houndstooth

### houndsdome

## 3D HOUNDSTOOTH VARIATIONS – LATHED

### 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

## 3D HOUNDSTOOTH VARIATIONS – TOPOLOGICAL

### 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

# Footnotes

^{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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