#tiling

[xanaxfarts]

Tile flooring at the Oceanário de Lisboa, an aquarium in Lisbon, Portugal

[xponentialdesign]

Spectral Cohesions

[lcatala]

Inspired by this post by @thanergetic-hyperlinks, I present to you

Tessellations of the Nine Houses

(Or "I can't really draw figurative art so my Locked Tomb fanarts are geometrical vector drawings")

"A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps." — Wikipedia.

Making tilings themed after each necromantic House seems obvious: for each House you pick a tile with the same number of sides as the number of the House; but this does present some challenges for some of the Houses.

note 1: this might give the impression that I first decided on the symbols and then found patterns to match them in a very organized and motivated manner; in practice it was much more chaotic and multidirectional, the patterns informing the symbols as much as the symbols informed the patterns; this is fine since symbolism is entirely associative and arbitrary anyway

note 2: I added alt-texts for all the images, but I have no idea of how to properly describe abstract geometric art; if you feel you can do a better job than I did, feel free to put your fingers where your mouth is--wait, hang on-- I mean feel free to provide better descriptions if you can

note 3: looking forward to the geometry nerds explaining to me how I got basic geometric details wrong, friggin nerds

The First House

The First House seems obvious, as a shape with one side is an ellipse (of which the circle is a special case). There's just one problem: ellipses do not tile the plane. No matter how much you stretch them and deform them, the very nature of ellipses means you'll always have gaps or overlaps.

So we cheat and we work with overlaps: turns out there is a history of tilings that use circles as a construction pattern, then turn the overlapping sections into the actual tiles. Such patterns have been used extensively in European and Middle Eastern art, and have also been associated with the New Age movement, so it fits Jod's style perfectly. And so we get this:

The different cells correspond to different House colors, with the resulting gothic stained-glass appearance quite in line with the Roman Catholic Empire vibe Jod is going for. The overlapping circles convey the intricacy of the relation between the First House and the eight other, both autonomous from it yet intrinsically part of it.

The Second House

There's a variety of geometrical shapes that have two sides, but most of them don't tile the plane, altho there is one that does — if we take a crescent shape and slightly thicken it so that the inner and outer curves are identical, we can do this:

The waving pattern is of course evocative of the flag of conquest which the Cohorts of the Second House have planted on many worlds.

The Third House

With the Third House things get a lot easier, because equilateral triangles are one of the three regular polygons (where all sides are the same length and all angles are identical) that tile the plane all by themselves without needing any other shape! Which however doesn't mean we have to be boring; we can have a little bit of fun:

Flowers for the beauty and ionizing radiation warning signs for the rancid vibes.

The Fourth House

Squares are the second regular polygons that tile the plane by themselves, so again our job is easy here, altho we still want to not go for the easiest option in order to be able to work in some symbolism:

The four big navy squares with a small white square at the center of course evoke the number five and the shadow of the Fifth House's regency over the Fourth.

The Fifth House

Regular pentagons do not tile the plane, so we have to use a more unusual shape — there are many options, but obviously we want to again pick one that offers some interesting numerical symbolism:

The cross-like patterns of course bring up the number four and the hold of the Fifth House over the Fourth. As for the crosses themselves and the fact that they appear to be made of wooden stakes, well uh… Abigail Pent, Vampire Hunter??? She does have Van Helsing vibes.

The Sixth House

Hexagons are the third and last regular polygons that tile the plane on their own. But this is the Sixth House we're talking about, things need to look orderly but in a convoluted way. So how about multiple levels of recursion:

The apparent complexity of the pattern is created by different orientations of a small number of elements, either 3 irregular hexagons, or 1 patterned regular hexagonal tile, depending on how you look at it, in line with the kind of hermetic scientism one imagines the Sixth House indulges in. The result is those apparent three-dimensional elements and emerging higher-order patterns, including that of ꙮ, the Multiocular O found in exactly one word of one 15th century Old Church Slavonic translation of the Book of Psalms ("серафими многоꙮчитїй" many-eyed seraphim).

The Seventh House

Regular heptagons do not tile the plane, but they don't need much tweaking to work, which is fine since for the Seventh House we want something deceptive yet simple (deceptively simple? deceptive in its simplicity?):

Hearts for the beauty, snake scales for the poison [the Seventh House is on Venus, the planet named after the Roman Goddess of love, but etymologically "Venus" is actually the same root as "venom", and of course "Septimus" resembles "septic" — tho in that case there's no etymological connection, it's just a happy coincidence].

The Eighth House

Octagons do not tile the plane, but they come pretty close, so we can give the Eighth House a simple, stern, but slightly threatening pattern:

Boring sterile bleached temple mosaic, with just a little bit of passive-agression, a perfect fit for Evangelical Christians Tumblr puritans the Eighth House.

The Ninth House

And so we reach the Ninth House. Now the thing about the Ninth House is that, even by imperial standards, they're huge freaks, like they're completely unhinged heretical weirdoes. So, when it comes to their tiling, we need to get weird, like, a lot weirder than we've been so far, and this will require some context, so get ready because now we're officially going on a wild tangent.

So far all the tilings we've seen were periodic. That is, they were drawing a pattern that repeats itself indefinitely in all directions.

But starting in the 1960s, mathematicians began to study aperiodic tilings, tilings that don't repeat; you can keep expanding them forever and never exactly find back the original pattern you started with. The first mathematical proof of such a pattern was made in 1964 and theoretically required 20,426 distinct tile prototypes… This was soon refined to just 104 tile prototypes, then a mere 40. By 1971, it was mathematically demonstrated that you could make such a pattern with just 6 tile prototypes.

Except that was a lie.

Note that I said mathematically demonstrated. As it turns out there was an aperiodic pattern with just 5 tile prototypes, known as Girih, that had been used in Islamic art… since at least the 13th century — but it had historically been treated merely as an element of architectural design, and its mathematical properties weren't studied until 2007.

Then in 1973 this guy Penrose came along and demonstrated you could make an aperiodic tiling with just 2 tile prototypes. So now the goal was to find the ultimate aperiodic tiling, the one that would use only one tile prototype. Given how fast the field had progressed so far, it seemed that this discovery was imminent.

It took 50 years.

Not only that, but it was the work of amateur mathematician David Smith who accidentally discovered a 13-sided polygon that could make an aperiodic tiling all by itself (he then had his discovery checked by and co-authored a paper with a number of professional mathematicians).

EXCEPT THAT WAS A LIE AGAIN.

In turns out an aperiodic tiling using only one tile prototype had already been found… in 1936. But since the study of aperiodic tilings only started in the 60s, its significance in that domain wasn't understood at the time. It was seen as significant, but for an entirely unrelated reason: it was the first demonstration of a polygonal shape that needed only two copies of itself to completely enclose the original one — many mathematicians before that point thought the minimum possible was 3 (think of the Triforce from Zelda, with one equilateral triangle completely enclosed between three other identical triangles).

And coincidently, that shape happens to be a highly-irregular nonagon [yes "enneagon" is """technically""" more correct but "nonagon" has been used since the 17th century and is more common and it has Nona in it and Nona loves you]. So here it is, the Voderberg tiling, the freakish freakish tessellation of the Ninth House:

Like you see this and you're like "what is this, what is that thing, that's not a tiling, what the fuck is that" — but it is, it is a tiling, you can keep adding the freaky polygon and it keeps expanding outward forever, with no gap, no overlap, and with an ever-changing pattern. A double-spiral radiating outward, for Anastasia and Samael, Anastasia and Alecto, Alecto and Harrowhark, Harrowhark and Gideon.

And if you were thinking that this last one must have been significantly harder to draw than the other ones, you would be correct.

[szimmetria-airtemmizs]

Does the inner pentagon have the same size as the others?

[medeaft]

Exit from the rat tunnel (originally: Kijárat a patkányalagútból) 2025 Digital

This is my 6th day of Rattober. I made this for the prompts 2. accu-rat and 29. puzzle rat. I knew from the beginning that what I was desiring to make for these two (assuming I connect the prompts) is a rat-shaped tiling. But I wanted to put the tiling on something, preferably a curved surface, so that it could better be seen. And this is what I came up with. I built the scene in blender. HDRI and concrete texture are from polyhaven.

And here you may see the tiling itself without being put on anything. I drew it with vectors.

And the sketch for the tiling, when I designed it yesterday on grid paper:

[2minutetabletop]

The DUNGEON MAP TILES

Building a 2MT-style dungeon has never been easier! Please try our new map tiles and let us know what you think. :)

→ Find them here!

[mathhombre]

Nakhchyvan Tessellation

Chris Cambré has a lovely analysis of a hexagonal tiling from the Yusif ibn Kuseyir Mausoleum in Azerbaijan. I wondered about generalizing it as it's completely determined by the 2nd hexagon relative to the first.

At first I didn't realize that some of the green hexagons were missing...

Play in GeoGebra.