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 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.
- 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)
maybe… 1D? absval houndstooth outline
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”
x count of underlying stripes
xn by yn, still squares
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
incursed
compound houndstooth / diagonal checkers continuum
different compound houndstooth and their tilings
animating (whether by time, or across page) into checkers
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
at least when you do it so that the crotches of the 1/2 sized nested teeth are the same point
it works out so that an order-2 one (just 2 layers of differently sized teeth interfering)
there are 4 different shapes that come out, and none of them are a basic houndtooth
but when you do an order-3 one
9 different shapes come out, and one of them is a basic houndstooth
so I wonder if you did order-4 (at this point you’ll really want to automate it, though)
if there’d be 16 different shapes, and no basic houndstooth again
i.e. that the number of shapes is always the square of the order, and whether or not there’s a basic houndstooth among them alternates (always there for odd orders)
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
swastika kite houndstooth
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”
made of little teeth
connecting points
all possible right triangles
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
roots and cusps grow outward by rotating gradually
make blurry houndstooth out of placing on void with varying density although any given row and column adds to same total overall density i guess
(no image)
jagged straight stripes made out of teeth
houndsring but on grid
bizarro cursive from pythagorean
shrinking bands
transforming into 14-gons
Houndstooth Moiré
of black and alpha
like just overlay two of them at an angle
the end result will be more black than alpha but that’s okay
a kind of tatewaku i guess
sierpinski houndstooth
placing root to conid, but shrinking each time
yabanesashi
kind of arabic barely overlapping
shrinking houndstooth packing
chained houndstooth spiral
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
falling houndstooth
circle inversion
incomplete supertile rotated randomly
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”
surely there is some way to pick a subset of these that makes a cool kind of fractal yeah?
recursive quarter tile – squares all the way down – so it’s like ultimate rasterized fractal
bendy spiral
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”
mutant merged with flipping
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)
rotato-prism
a new kind of prism
trying to link a houndstooth to a regular heptagon – maybe i should try doing an irregular heptagon that is average of each pair of sides which would be interesting because they would be 7 diff kinds
houndstooth table
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)
houndsrhombicuboctahedrontooth
lame houndstamp
houndstamp – dumb with inversion
houndstamp – dumb with inversion – similar
3D HOUNDSTOOTH VARIATIONS – MISCELLANEOUS
ht3d_026: parallelopiped striped houndstooth
houndsdome
swirl around a cube
trying to have inversion perpendicular
houndsring pyramid/cone
What x houndsring can you make id you bend it down into cone/Pyramid
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
houndslathe but it does checkers-yagasuri continuum as it rotates and goes past checkers into facing the other direction each 90 degrees so it’s kind of like the other “with inversion”
(no image)
double houndslathe
2D but mapped onto 3D curved surface for different effects
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.
Leave a comment