Post

Micah Tillman

Dec 10, 2020 • 242 tweets • >60 min read • Read on X

You can now share the #fractals you find using L-System Visualizer. Here's one I found the other day:
micahtillman.com/lsviz/?FrBlFlB…

@unity3d

To hunt for fractals, go to micahtillman.com/lsviz/. The app (made with @unity3d) will generate a URL for each figure you find, which you can then send to others. They can then examine the L-system you created, and derive variants from it.

v 1.1 of the app also greatly improves memory usage, which means the complexity of the systems/fractals you create is now only limited by your patience and CPU speed.

(Well, we're still limited to 2GB of memory in browser, but C#'s StringBuilder class manages that memory much MUCH more efficiently than the standard string class.)

@unity3d

Here's another #fractal I found using my @unity3d L-system Visualizer app:
micahtillman.com/lsviz/?L&L=LOP…

You can click on the link above to see how the fractal develops, step by step (and then examine and tweak the L-system that generates it).

Another fractal (which appears to combine multiple candelabra):
micahtillman.com/lsviz/?a&a=aca…
(Click the link above to examine/edit the L-system)

@unity3d

A snowflake chain fractal I found with my @unity3d L-system visualizer app:
micahtillman.com/lsviz/?a&a=aoq…
(Click the link above to see the fractal develop, step-by-step.)

Enhance! (This is what the fractal above looks like up close.)

A Sierpinski Starflake, with quasi-Mandelbrot Set shapes in the negative space
micahtillman.com/lsviz/?I&I=rIr…
(Click link above to examine/edit the L-system, or to see the fractal develop)

What if you built a square of Xs and crosses . . . made of squares of Xs and crosses?
micahtillman.com/lsviz/?frfrfrf…
(Click the link above to examine the L-system behind this fractal.)

Or what if you take the Hilbert Curve's L-system, and tell it to stop ignoring the As and Bs?
micahtillman.com/lsviz/?A&A=pBF…
(Click the link above to examine and edit the L-system behind this fractal)

And if you tweak the Levy C-Curve a bit, you get a weird, Legend of Zelda-themed tiling of the plane:
micahtillman.com/lsviz/?F&F=pFm…

@0jenzee0

Inspired by @0jenzee0's design for Chaos (in Hades, by @SupergiantGames), I call this the "Chaos Kermit" fractal:

micahtillman.com/lsviz/?m&f=fml…
(Click the link above to view/edit the L-system)

@0jenzee0

If you want a better version of @0jenzee0's work (including other character designs), go to the @SupergiantGames site:
supergiantgames.com/games/hades/
Then buy the game and play it for hundreds of hours.

Not sure what's going on in this one:
micahtillman.com/lsviz/?F&F=Flr…

A weird Jack-Frost-looking fractal, akin to the Koch Snowflake:
micahtillman.com/lsviz/?f&f=flo…
(Click the link above to examine and edit the L-system)

Look at the negative space in this one. Like stars made of monster arms? Monstars?
micahtillman.com/lsviz/?f&f=flf…
(Click the link above to examine/edit the L-system)

Remember when everyone got into stamps for a few months? Not philately. The other kind. Well, this one reminds me of cool stamps:
micahtillman.com/lsviz/?f&f=for…

Chaos Kermit (see above) becomes Mutant Chaos Kermit. Or maybe this is Mutant Cookie Monster?
micahtillman.com/lsviz/?f&f=frf…
(Click the link above to examine/edit the L-system)

And these are the cookies that Mutant Cookie Monster wants to eat. (It's the same fractal, just with all the turns set to a different base angle):
micahtillman.com/lsviz/?f&f=frf…

A variant on the Dragon Curve, this one looks more like . . . shark fins?
micahtillman.com/lsviz/?fx&f=f&…

To create an L-system that generates a fractal, it's easiest to use palindrome replacement rules. But fractals in the Dragon Curve family instead use mirrored rules.

So, a normal fractal's L-system might have rules like...
F -> FrBrF
B -> BrrBBrrB
...where the string you're putting in place of each symbol reads the same forwards as backwards.

But Dragon Curve fractals have rules that mirror each other. One symbol's replacement string will be a reversed and inverted version of another symbol's replacement string
X -> XpYFp
Y -> mFXmY
(Where X is the "opposite" of Y and "p" is the opposite of "m")

So, in Dragon Curve fractals, the palindrome you would expect -- having both the forward and backward ordering of the same string -- is spread across pairs of rules. But also, the reversed version contains the opposites of the signs it is "repeating"

Another variant of the Dragon Curve. Let's call this one the "Pinwheel Dragon":
micahtillman.com/lsviz/?X&X=XrF…

A triangular variant of the Hilbert Curve:
micahtillman.com/lsviz/?A&A=pBF…

And another variant of the Hilbert Curve, that ends up looking much like the Koch Snowflake:
micahtillman.com/lsviz/?A&A=pBF…

And yet another derivation of the Hilbert Curve. Let's call it the "Woven Mat":
micahtillman.com/lsviz/?A&A=pBF…

Here's a variation on the Sierpinski Triangle. Let's call it the "80s Movie Poster R Curve":
micahtillman.com/lsviz/?F&F=FmG…

Fractals like this are self-similar "at all levels," but what I want to call the "true shape" of the fractal doesn't become clear till you're a few levels up. (And conversely, the low-level detail disappears at the higher levels):
micahtillman.com/lsviz/?FlFl&F=…

Sometimes the Dragon Curve turns into a vine maze:
micahtillman.com/lsviz/?FX&F=F&…
(Click the link above to examine/tweak the L-system)

Look at this weird variant of the Sierpinski Triangle (which also looks a bit like the Koch Snowflake if you use different angles). Let's call it the Sierpinski Shark:
micahtillman.com/lsviz/?F&r=r&F…
(Click link above to zoom in/out and edit)

Same curve as the above, just using a different angle. From Sierpinski to Koch/Cesàro, with a change of angle:
micahtillman.com/lsviz/?F&r=r&F…
(cf: en.wikipedia.org/wiki/Koch_snow…) $A line drawing of a fractal...$

Oh, and the Sierpinski Shark, this time with image description (my bad!)
(micahtillman.com/lsviz/?F&F=Frr…) $A line drawing of a fractal...$

Another fractal that looks interesting using different angles. Version 1 (45 degrees) looks like some pine tree branch:
micahtillman.com/lsviz/?F&F=FrF…
(click the link above to zoom in/edit the fractal) $A line drawing of a fractal...$

And here's the same fractal, but this time using right angles:
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

And once again, but this time using 120-degree angles. Perhaps this one should be called "Sierpinski Jack Frost".
micahtillman.com/lsviz/?F&F=FrF… $A line-drawing of a fractal...$

@CAPYGAMES

Check out this weird variant of the Levy C-Curve fractal (the "Levy Dragon"). Looks just like Jorj from the bizarrely delightful game Grindstone by @CAPYGAMES, raging at the sky:
micahtillman.com/lsviz/?F&F=FlF…
(Click the link above to edit the fractal/zoom in) $A line drawing of a fractal...$

If you're not familiar with Grindstone, see:

And if you're not familiar with the Levy C-Curve, see:
en.wikipedia.org/wiki/L%C3%A9vy…

Now, check this fractal out. It looks like a bunch of waves with fish-shaped holes:
micahtillman.com/lsviz/?F&F=FlB…
(Click the link above to zoom in/examine the rules that generates the fractal) $A line drawing of a fractal...$

It would appear the fractal above is a variant on the "Terdragon" fractal (en.wikipedia.org/wiki/Dragon_cu…). See: micahtillman.com/lsviz/?F&F=FlB…, which just tweaks the rules slightly

What if the Big Dipper (Ursa Major) were made of smaller dippers, all the way down? Well, I think it'd look a little something . . . like this:
micahtillman.com/lsviz/?F&F=Fll… $A line drawing of a fractal...$

Here's a fractal that can also be generated by the "3-adic valuations of n" (the powers of 3 in each integer's prime factorization). This is the first time I've found it using an L-system.
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

"Find me a fractal that looks like the silver lining on the underside of a bunch of clouds."
Okay:
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

"But make sure that fractal also looks like a fancy gust of wind from a cartoon when you change the angle."
Can do. Same fractal:
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

Not sure what's up with this one. I just know it's fun to watch it develop, fill in, and become clearer:
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

I'm confused about why this one is a fractal. The L-system that generates it doesn't appear to have the normal symmetry properties. Also, it uses two different angles
micahtillman.com/lsviz/?F&F=Frr… $A line drawing of a fractal...$

A closely related fractal to the above. This one is a tentacle made of tentacles.
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

This one looks to me like the leaves on a grape vine, for some reason. Not that I've ever seen an actual grape vine.
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

Ummmmmm . . . Something like the Dragon Curve, mixed with the Mandelbrot Set? Or, a horrible prickly pear.
micahtillman.com/lsviz/?F&F=FFB… $A line drawing of a fractal...$

That fractal is a combination of a rule set that generates a "slightly off" version of the Levy C-Curve:
micahtillman.com/lsviz/?F&F=FFB… $A line drawing of a fractal...$

. . . with a mirror version of that rule set that generates the same "slightly off" version of the Levy C-Curve
micahtillman.com/lsviz/?B&F=F&B…

I need help naming this one. It reminds me of the birds you will frequently see on traditional totem poles from the "American" Northwest, but I think there are other resonances as well.
micahtillman.com/lsviz/?F&F=FBr… $A line drawing of a fractal...$

If you let the app run long enough, you can get the fractal to this level of detail. (And I'm informed that at least one of the birds I was thinking of was the Thunderbird.) $A line drawing of a fractal...$

If you happen to recognize any of the various figures in the gaps of the fractal discussed above (and included again below), let me know. I like finding a connection between the Thunderbird and fractals at least (especially since we already have a Dragon Curve).

And at 12 iterations (which took my laptop forever to calculate), the fractal looks like this. micahtillman.com/lsviz/?F&F=FBr… $A line drawing of a fractal...$

At this level of detail, it becomes clear that the central "rectangle" is both horizontally & vertically symmetric. However, the rectangles making up its wings have their own wings, which overlap with the top part of each rectangle, masking the vertical symmetry.

Also at this level of detail, you can see that the "sides" of each rectangle consist of smaller copies of the overall figure. (So, the central "rectangle" is echoed to the left and right, and is made of copies of the entire fractal.)

@terrycavanagh

This fractal is kind of fun to watch develop. It's a V made of Vs, so let's call it the "VVVVVV Curve," after @terrycavanagh's classic indie game (store.steampowered.com/app/70300/VVVV…)
micahtillman.com/lsviz/?F&F=rFB…
(Click the link above to zoom in/edit the L-system) $A line drawing of a fractal...$

And as often happens with fractals, using a different angle for the various turns results in another cool shape. Less fun to watch develop, but still. Looks like a fox face made of fox faces.
micahtillman.com/lsviz/?F&F=rFB… $A line drawing of a fractal...$

And with a slight change to one of the substitution rules, you get this cool version of the Levy Dragon/C-Curve (made of Vs):
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

The Levy C-Curve is all over the place. Makes one think it's the Ur-fractal, how often it shows up in various forms.

(. . . and if THAT doesn't make you at least an Aristotelian -- if not a full-blown Platonist -- about forms, I don't know what will)

Speaking of that, here's a loopy, lacy version of the Levy C-Curve. Maybe?
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

And now, a creepy fractal. Here's the "Skull King Curve." It's a skull (no lower jaw) wearing a weird crown that contains an image of itself, wearing a crown that . . .
micahtillman.com/lsviz/?F&F=rlF… $A line drawing of a fractal...$

A strange variant on the Dragon Curve, it would appear. Looks to me like a string of dolls (with Sunbonnet Sue-style dresses and Princess Leia hairdos -- or headphones)
micahtillman.com/lsviz/?F&F=lFl… $A line-drawing of a fractal...$

Tweak the replacement rules slightly, and change the angle of rotation, and you get two Levy Diamond curves -- well, the diamonds are upside down -- holding hands:
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

The "fins/spikes" here look like Godzilla to me, or some other dragon-ish thing.
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

Here's a version of the Levy Diamond fractal, but it looks like fruit trees, to me:
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

As I've said before, Levy shows up over and over in different forms when you're exploring fractals. He's a variant of the Levy C-Curve that looks like a flower garden to me:
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

And if that one was a Levy flower garden, this one is a Levy butterfly garden:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

Consider this variant on the Koch Snowflake . . .
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

You may recognize it -- using different angles -- as a variant the Levy Diamond fractal:
micahtillman.com/lsviz/?F&F=lFr…

There are so many different versions of both Koch and Levy in this same fractal, just depending on what angles you use.
Koch: micahtillman.com/lsviz/?F&F=lFr…
Levy: micahtillman.com/lsviz/?F&F=lFr…
Koch: micahtillman.com/lsviz/?F&F=lFr…
Levy: micahtillman.com/lsviz/?F&F=lFr…
Etc.

DNA? Streamers? A confusing fractal.
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

tfw you're trying your best to be a Koch Snowflake but . . . You know, sometimes it's just hard to be neat and tidy and there's a pandemic on
micahtillman.com/lsviz/?F&F=Fl&… $A line drawing of a fractal...$

Not sure I've ever been so impressed by how *sloppy* a fractal can be.

Another one for the "Fractals don't have to be pretty to be fractals" list:
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

Here's the same fractal, but with different turn angles. Prettier. Looks like a mountain ridge with pine trees.
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

That version is one of the fractals it's fun to watch develop. You can see the pattern becoming clearer and clearer with each iteration. (Not sure why the algorithm I have set up for automatic camera control works so nicely with some fractals and not others.)

Another similar, "Pine Tree Mountain Ridge"-looking fractal:
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

I like the fractals where the gaps repeat each other at different "levels." (Though, this one may be most interesting when you're "zoomed in"):
micahtillman.com/lsviz/?FFrFF&F… $A line drawing of a fractal...$

Same fractal, but with different angles. This one is clearer, if that's a term you can apply to fractals
micahtillman.com/lsviz/?FFrFF&F… $A line drawing of a fractal...$

Another fractal where the gaps seem to be the most interesting part (especially as you get to higher and higher iterations):
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

The same fractal, but with different angles. Instead of looking like a mask with holes for eyes, it looks (to me) like the head of an alien fly with various eyes and appendages.
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

There's something unsettling about this version of the Dragon Curve. It's like someone used the "shear" tool on it, but sheared it in different directions.
micahtillman.com/lsviz/?X&X=FrY… $A line drawing of a fractal...$

Or maybe it's a pattern projected onto the floors and walls of a space?

The effect is more extreme in this version (same as the above, but with different angles). I've added a red line to the second copy below, to show where it looks like the "wall" and "floor" meet.
micahtillman.com/lsviz/?X&X=FrY…

A more . . . wiggly? . . . version of the Dragon Curve:
micahtillman.com/lsviz/?FXF&F=F… $A line drawing of a fractal...$

Note sure what to call this one. It's a bit like a box with a slash through it, made of boxes with slashes through them?
micahtillman.com/lsviz/?F&F=Fll… $A line drawing of a fractal...$

*Not

Let's call this one the Goldsworthy Leaf Chain Fractal (after Andy Goldsworthy's leaf chain piece, which you can see here: ):
micahtillman.com/lsviz/?F&F=lFF…
(Click the link above to zoom in/edit the fractal) $A line drawing of a fractal...$

Now for a small sub-series of fractals.

First, consider this version of the Terdragon.
micahtillman.com/lsviz/?F&F=FlF…
(For details on the Terdragon, see: larryriddle.agnesscott.org/ifs/heighway/t…)

The basic rule for the Terdragon is: Start with "F." Then, replace each "F" w/ "FlFllF"

Here, "F" = "move forward" and "l" = "turn left 120 degrees."

So: move forward; turn left once; move forward; turn left twice; move forward.

But what if we continue the pattern?

Instead of "F" -> "FlFllF," this fractal replaces each "F" with "FlFllFlllF." And it cuts the angle turned with each "l" in half:
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

That sort of shape shows up pretty frequently as you explore various L-systems. (See tweets earlier in this thread, for example.) Looks like a cartoon gust of wind to me.

But what if we continue the pattern?

We started with "FlFllF" and an angle of 120 degrees. Then "FlFllFlllF" and an angle of 60. What about "FlFllFlllFllllF" and an angle of 30?
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

Alas, if you try to continue on to "FlFllFlllFllllFlllllF," using 15 degree angles, you seem to just get a mess. But it was fun while it lasted.

AND it makes me want to build an app to explore these kinds of "meta" rules for iterating "first order" L-systems.

This fractal looks like snowflakes and angels:
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

tfw you try to draw a Levy C-Curve fractal while playing a bunch of stylized, isometric, voxel-art games:
micahtillman.com/lsviz/?F&F=lFF… $A line drawing of a fractal...$

Here, I found you another flower chain fractal for Friday:
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

Same fractal, but with one of the angles changed. Looks bit like how one is taught to draw seagulls in the distance. Except with more Tron.
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

Or, if you use a square for the seed, you get this fractal:
micahtillman.com/lsviz/?FrFrFrF… $A line drawing of a fractal...$

Turned this fractal upside down, but who's to say what the right orientation of a fractal is? You? Are you to say?
micahtillman.com/lsviz/?FBrFBr&… $A line drawing of a fractal...$

A chain of circles that are also squares. #fractal time continues:
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

Another in fractal in the "Levy C-Curve -- But Kind of Weird" family:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

A triquetra (the Celtic knot associated with the Trinity) fractal:
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

This fractal is a variant on the Koch Snowflake, I believe. The empty spaces between the "points" of the flake look like Shrek -- or some other large, tubby cartoon character -- asking for a hug.
micahtillman.com/lsviz/?FrFrFrF… $A line drawing of a fractal...$

A fractal based on Fibonacci:
micahtillman.com/lsviz/?F&F=FlF… $A drawing of a fractal. The...$

A relative of the Thunderbird Fractal, which we stumbled across earlier in this thread:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

https://twitter.com/MicahTillman/status/1362783538492407811

The original tweet with the Thunderbird Curve:

https://twitter.com/MicahTillman/status/1362783538492407811

On a happier note, here's a new fractal for Easter. Would be cleaner if it weren't for floating point errors (a computer can only store positions and angles to a particular level of precision, so the more details you add, the more things get wonky)
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

Here's something akin to the Dragon Curve, but built of triangles. Or guitar picks. I recognize it, but am not sure from where:
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

The same fractal, with different angles. Looks like of like some sort of sea creature? (Notice once again the places where the lines don't perfectly overlap, and thus look thicker, due to floating point errors, I assume.)
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

I enjoy these slightly-off fractals. Here's a not-quite-Sierpinski-Triangle:
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

A fractal wave, made of fractal waves. Or of cauliflower. Or something.
micahtillman.com/lsviz/?F&F=FpF… $A line drawing of a fractal...$

A related fractal, which also looks like waves and maybe cauliflowers:
micahtillman.com/lsviz/?F&F=FpF… $A line drawing of a fractal...$

Here's another one of the sloppy fractals. (I have a hard time believing these exist, even though I've found several.)
micahtillman.com/lsviz/?F&F=FFB… $A line drawing of a fractal...$

https://twitter.com/MicahTillman/status/1360294089443393541

The "clean" version of this fractal can be found in the thread above:

https://twitter.com/MicahTillman/status/1360294089443393541

https://twitter.com/MicahTillman/status/1359877340738834432

Previously, we've seen a "sloppy" version of the Levy C-Curve:

https://twitter.com/MicahTillman/status/1359877340738834432

Now for another in the "botanical fractals" series:
micahtillman.com/lsviz/?F&F=lFl… $A line drawing of a fractal...$

Lookit this weird Sierpinski Triangle. Sierpinski Diamond? Something.
micahtillman.com/lsviz/?FrF&F=F… $A line drawing of a fractal...$

Umm . . . Fractals continue to baffle me. Let's call this one "The Block Chain"
micahtillman.com/lsviz/?FBF&F=F… $A line drawing of a fractal...$

Looks like this fractal could "tile the plane" (if you put multiple copies of itself together like puzzle pieces). Looks like if the Ulam Spiral had a baby with the Dragon Curve
micahtillman.com/lsviz/?FrF&F=F… $A line drawing of a fractal...$

@numberphile

I'm always looking for connections between fractals and prime numbers. On the Ulam Spiral and primes, see @numberphile's video with @jamesgrime

@3blue1brown

Then, when you've finished that one, watch @3blue1brown's video to go deeper:

A circle with tassels made of circles (connected by triangles) with tassels made of circles (connected by triangles):
micahtillman.com/lsviz/?rrrr&r=… $A line drawing of a fractal...$

You can vary the size of the circles and triangles in that fractal relatively easily (by adjusting where/how many Gs you put in its Lindenmayer system). I messed around with it quite a bit before settling on the version above. But I don't feel like I've yet found its "true form."

Oh, also, if you zoom in on the final fractal, it looks like it's spinning. It's a fun optical illusion. Try it yourself:
micahtillman.com/lsviz/?rrrr&r=…
Or don't.

If the Dragon Curve were made of boxes, it might look a little something like this #fractal. Note how the two large "boxes" are connected by smaller copies of themselves, which are connected by smaller copies of themselves . . .
micahtillman.com/lsviz/?rr&r=rr… $A line drawing of a fractal...$

Here is the basic "unit" of the fractal above. (You get the fractal above by duplicating the seed for the fractal below.)
micahtillman.com/lsviz/?r&r=rrF… $A line drawing of a fractal...$

Fractals are often compared to coastlines. The most interesting thing to me about this one are the rivers stretching inland:
micahtillman.com/lsviz/?r&r=rrF… $A line drawing of a fractal...$

"Okay. Hear me out. What if we designed a circuit, but -- get this -- *made of trumpets*."
Or, what if we made a trumpet out of smaller trumpets? #fractal
micahtillman.com/lsviz/?r&r=rFr… $A line drawing of a fractal...$

Imagine a 3D grid of cubes, stretching to infinity in every direction. Or just look at this fractal:
micahtillman.com/lsviz/?rrrr&r=… $A line drawing of a fractal...$

https://twitter.com/MicahTillman/status/1367588076881547265

Another fractal that looks like animals made of DNA streamers (the first one I found is here

https://twitter.com/MicahTillman/status/1367588076881547265

):
micahtillman.com/lsviz/?Fr&F=rF… $A line drawing of a fractal...$

Another of the quasi-Koch Snowflake fractals. This one appears more organic to me. That, or it's based on a Sonic the Hedgehog pose.
micahtillman.com/lsviz/?r&r=rFF… $A line drawing of a fractal...$

A fractal that looks like a box made of levels from a platformer:
micahtillman.com/lsviz/?rrrr&F=… $A line drawing of a fractal...$

Um. A weird shuriken (or kunai?) with a bunch of smaller shuriken/kunai attached to it at random angles? Just when I thought I'd seen every sort of fractal. . . .
micahtillman.com/lsviz/?F&F=BrF… $A line drawing of a fractal...$

Here's the same fractal, but with different angles:
micahtillman.com/lsviz/?F&F=BrF… $A line drawing of a fractal...$

Slightly simplified versions of the fractal(s) above:
micahtillman.com/lsviz/?F&F=rFl…

micahtillman.com/lsviz/?F&F=rFl…

And another version of that fractal, with different angles. Looks like wheat to me, for some reason.
micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

A closely-related fractal
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

Same fractal, but different angles. Reveals itself to be a heavily-disguised Levy C-Curve. The combination of squares and squiggles makes this one look like the stones + vines you see in games like Fez (which I just recently played for the first time):
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

Another in the "wind gust"/"waves" series of fractals: micahtillman.com/lsviz/?F&F=rFl… $A line drawing of a fractal...$

A fractal moth? Perhaps?
micahtillman.com/lsviz/?r&r=FFr… $A line drawing of a fractal...$

A fractal chain made of ovals made of ovals.
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

As always, you can click the link (micahtillman.com/lsviz/?F&F=lFr…) to examine and edit the L-system that generates the fractal

The same L-system, with different angles, produces cactus-covered hillsides:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

If Tim Burton was asked to redesign the Levy C-Curve fractal:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

The same L-System will produce these three fractals. All you have to do is change the angles used for turns (and perhaps the number of iterations):
1) micahtillman.com/lsviz/?F&F=lFl…
2) micahtillman.com/lsviz/?F&F=lFl…
3) micahtillman.com/lsviz/?F&F=lFl… $Line drawings of three frac...$

A new fractal: a chain of triangles made of chains of triangles
micahtillman.com/lsviz/?F&F=FXY… $A line drawing of a fractal...$

A closely-related fractal to the one above:
micahtillman.com/lsviz/?F&F=FXY… $A line drawing of a fractal...$

Not as cool as the previous fractal, but here's a messy Dragon Curve. (And the two seem to belong to the same family at least)
micahtillman.com/lsviz/?X&X=XFr…

This fractal looks like a very serious cartoon guard dog posed for one of those "Me, but also me as a floating head" portraits.
micahtillman.com/lsviz/?X&X=XrF… $A line drawing of a fractal...$

This fractal's final form is a bit boring, but how it gets there is super interesting. It starts off looking like a rickety old building, but straightens itself out as it goes along.
micahtillman.com/lsviz/?X&X=XlF… $A line drawing of a fractal...$

A fractal star exploding into fractal stars
micahtillman.com/lsviz/?X&F=FF&… $A line drawing of a fractal...$

Another in the "messy fractals" series. This one belongs to the Levy Diamond family:
micahtillman.com/lsviz/?X&X=rXF… $A line drawing of a fractal...$

After watching this one develop (it seems to go through a two-step cycle), I'm pretty sure this one is a fractal. But it's hard to see at any particular stage
micahtillman.com/lsviz/?F&F=FrX… $A line drawing of a fractal...$

tfw you need a hand mirror, but it has to be the fanciest handmirror in human history
micahtillman.com/lsviz/?F&F=FrX… $A line drawing of a fractal...$

The same fractal, but with different angles. Now it looks like a bunch of fancy hand bells, trailing off into the distance
micahtillman.com/lsviz/?F&F=FrX… $A line drawing of a fractal...$

Your #FridayFractal (a thing I just made up but which is going to totally catch on) is a bunch of triangles attacking larger triangles like an infinite fractal Independence Day
micahtillman.com/lsviz/?X&X=rFr… $A line drawing of a fractal...$

You, a bad person who, like Jon Snow, knows nothing: "That's just a Sierpinski Triangle flipped inside out."

Me, a persecuted visionary: "No, it's a brand new fractal that no one has ever seen before which inaugurates a new epoch in Fractal Hunting."

As with all the fractals in this thread, you can click on the associated link and examine/edit the L-system that generates the fractal, and zoom in/out on the fractal

Same fractal as the above. To "flip" the triangles "inside," you just use a different angle (60 degrees instead of 120). But things don't quite meet up to form the Sierpinski Triangle
micahtillman.com/lsviz/?X&X=rFr…

"What if the Levy C-Curve, but Art Deco?"
micahtillman.com/lsviz/?X&X=rFr… $A line drawing of a fractal...$

Speaking of the Levy C-Curve fractal, lookit this double version:
micahtillman.com/lsviz/?X&X=lFr… $A line drawing of a fractal...$

Is this one a fractal?
micahtillman.com/lsviz/?X&X=lFr…

It *develops* like a fractal

Perhaps the reason why it looks so odd is that the "step forward, drawing a line" instruction is stripped out of the instruction list at each iteration. It's just that it's also added back into the instruction list at each step

Oh, and a slightly simpler version of the L-system behind the fractal(?) above makes the figure cleaner
micahtillman.com/lsviz/?X&X=lFr…

And if you remove just one more letter from the L-system, you get this fractal (which is just the Levy C Curve using 150 degree angles rather than 90):
micahtillman.com/lsviz/?X&X=lFr… $A line drawing of a fractal...$

Fractal skyscrapers
micahtillman.com/lsviz/?F&F=FXr… $A line drawing of a fractal...$

A related fractal, but now it looks like fractal antennae (though not like the actual fractal antennae: en.wikipedia.org/wiki/Fractal_a…)
micahtillman.com/lsviz/?F&F=FXr… $A line drawing of a fractal...$

What if every thorn on a thorny vine was itself a thorny vine? You ever consider that? No. You never did.
micahtillman.com/lsviz/?F&F=GlF… $A line drawing of a fractal...$

One of the challenges with these fractals is figuring out how to describe them in the alt text. It's a square with creepy trees growing from each side (looking at the empty space)?
micahtillman.com/lsviz/?FlFlFlF… $A line drawing of a fractal...$

The same fractal, but with different angles and a simpler seed (replacement rules are the same). It's a variant of the Koch Snowflake, just with two "arms" instead of one.
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

I named the image above the "Goro Flake" on my hard drive, but I think "Tardigrade Flake" may have been better

Another weird Levy Dragon/C-Curve fractal:
micahtillman.com/lsviz/?X&X=lFX… $A line drawing of a fractal...$

The same fractal, with different angles, looks like a bunch of fancy crosses made of fancy crosses
micahtillman.com/lsviz/?X&X=lFX… $A line drawing of a fractal...$

Remove one letter from the L-system that generates the fractals above, and you can get something that looks like the cross section of a cauliflower
micahtillman.com/lsviz/?X&X=lFX… $A line drawing of a fractal...$

Remove another letter, and you get a T made of Ts, rather than a + made of +s
micahtillman.com/lsviz/?X&X=lFX… $A line drawing of a fractal...$

Keep removing Fs from the L-system for the fractals above and the T will get thinner and thinner

This fractal looks a bit like a Koch Snowflake made of Xs instead of triangles. Like the Vin Diesel XXX logo (en.wikipedia.org/wiki/XXX_(film…). Or maybe it's a Koch Snowflake where the triangles point inward instead of outward? Maybe?
micahtillman.com/lsviz/?FrFr&F=… $A line drawing of a fractal...$

Another Koch Snowflake-ish fractal, but each outward-facing triangle is capped by an inward facing one?
micahtillman.com/lsviz/?FFFFFF&… $A line drawing of a fractal...$

I think I could tweak the L-system for this fractal a bit to make it cooler, but just for the record:
micahtillman.com/lsviz/?X&X=XrF… $A line drawing of a fractal...$

Was *not* expecting this fractal to turn into another Levy C-Curve variant, watching its first few iterations:
micahtillman.com/lsviz/?X&X=lXr… $A line drawing of a fractal...$

Another Levy C-Curve-ish fractal, but whose L-system is related to the Vicsek Fractal (en.wikipedia.org/wiki/Vicsek_fr…):
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

Same fractal, with different angles:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

This one reminds me of sheaves of wheat, so I feel like I should name it after Joseph:
micahtillman.com/lsviz/?F&F=BrF… $A line drawing of a fractal...$

Fractal Saturday continues. Imagine if Kermit the Frog was a fractal alien
micahtillman.com/lsviz/?F&F=rBl… $A line drawing of a fractal...$

Speaking of aliens, this is the same fractal, using different angles. Looks like a fleet of old-timey spaceships (reflected in water or casting shadows) lined up to take off
micahtillman.com/lsviz/?F&F=rBl… $A line drawing of a fractal...$

https://twitter.com/MicahTillman/status/1338940444315705345

The same fractal, using different angles, becomes a combination of the Chaos Kermit (

https://twitter.com/MicahTillman/status/1338940444315705345

) and Cookie Monster Mutant (

https://twitter.com/MicahTillman/status/1341068953473441793

) fractals, earlier in this thread:
micahtillman.com/lsviz/?F&F=rBl… $A line drawing of a fractal...$

Yes, okay. Fine, L-system. But . . . *why*? Why make two strings of hexagons? Why arrange them in pairs? Why connect the pairs on the right differently from the pairs on the left? Why have them grow larger, as if we were looking down a hallway?
micahtillman.com/lsviz/?F&F=lGr… $A line drawing of a fractal...$

"Hello! Welcome to the Fractal Shoppe. What can I get for you?"
"Whatcha got in the way of boomerang-shaped slugs?"
"How about this one?"
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

Same fractal, with different angles . . . but still a Sierpinski variant:
micahtillman.com/lsviz/?F&r=rr&… $A line drawing of a fractal...$

Again, a Sierpinski variant from the same L-system, but using different angles:
micahtillman.com/lsviz/?F&r=rr&… $A line drawing of a fractal...$

And, it turns out, this L-system can generate a perfect Sierpinski Triangle, in spite of not being the "correct" L-system for that fractal:
micahtillman.com/lsviz/?F&r=rr&… $A line drawing of a fractal...$

This one doesn't look like a fractal. But if you zoom in on this string of spiky rectangles (click the link below), you will see that they are themselves made of smaller spiky rectangles
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

A closely related L-system to the above produces this much more straightforward fractal:
micahtillman.com/lsviz/?F&F=lFr… $A line drawing of a fractal...$

The most boring fractal?
micahtillman.com/lsviz/?X&X=XFr… $A line drawing of a fractal...$

Same fractal, but using a different angle for the L-system
micahtillman.com/lsviz/?X&X=XFr… $A line drawing of a fractal...$

A weird combo of the Koch Snowflake and anti-flake. Click on the link below and try zooming in.
micahtillman.com/lsviz/?X&X=XFr… $A line drawing of a fractal...$

You: "Could you draw me a fractal that looks like it was done by a mathematically-precocious child?"
Me:
micahtillman.com/lsviz/?X&X=XrF… $A line drawing of a fractal...$

A fractal chain of 8s (pirates only ever get pieces of 8)
micahtillman.com/lsviz/?F&F=FXr… $A line drawing of a fractal...$

The most religious fractal (yet another Levy Variant) I've found to date
micahtillman.com/lsviz/?F&F=rFB… $A line drawing of a fractal...$

These two fractals are generated by the same L-system. It uses 120 angles on the left, and 30 degree angles on the right
micahtillman.com/lsviz/?FrrFrrF… $A line drawing of two fract...$

Also, it's weird how Twitter's image compression will take a color in one part of the image and apply it to another part. Red is only present in the central line of the image, but Twitter shows parts of the triangle as being red too.

The L-system for this fractal is so simple I'm surprised I hadn't stumbled across it till just now. Looks like a kind of toothy (or claw-y) Dragon Curve
micahtillman.com/lsviz/?F&F=Frr… $A line drawing of a fractal...$

The L-system Wikipedia (en.wikipedia.org/wiki/Dragon_cu…) and WolframMathWorld (mathworld.wolfram.com/DragonCurve.ht…) have for the Dragon Curve is much more complex than the one you'll find here:
cs.unm.edu/~joel/PaperFol…
As far as I can tell, the simpler version works:
micahtillman.com/lsviz/?F&F=FrG…

The factor by which the string of instructions increases in length, from iteration to iteration, is identical in both systems, however.
micahtillman.com/lsviz/?FX&F=F&…
It's just that the more complex L-system produces a string 2x as long.

tfw you're exploring fractals and suddenly find Frank from Donnie Darko emerging from a stunted Koch Snowflake:
micahtillman.com/lsviz/?F&F=FrG… $A line drawing of a fractal...$

("Why are you wearing that stupid *man* suit?" may be the most Platonic line in movie history, btw.)

Found another one of the swoosh/wind gust fractals, made of woven strands
micahtillman.com/lsviz/?F&F=FlG… $I line drawing of a fractal...$

A related L-system generates this fractal, which is more metal than I will ever be:
micahtillman.com/lsviz/?F&F=FlG… $A line drawing of a fractal...$

Another L-system in the same family gives us this fractal:
micahtillman.com/lsviz/?F&F=FlG… $A line drawing of a fractal...$

Still working in the same family of L-systems, we get:
micahtillman.com/lsviz/?F&F=rFr… $A line drawing of a fractal...$

Take a look at this fractal. Like a tree, with branches overlapping branches:
micahtillman.com/lsviz/?F&G=Frr… $A line drawing of a fractal...$

The same L-system, using different angles (60 instead of 30 degrees), generates this fractal:
micahtillman.com/lsviz/?F&G=Frr… $A line drawing of a fractal...$

Instead of tweaking an L-system's sign-replacement rules, or its interpretation rules, you can also change the seed the system grows from. So, take this fractal, for example:
micahtillman.com/lsviz/?F&F=FlF… $A line drawing of a fractal...$

But, if instead of having the L-system start with a single line, we have it start with a two-part line that doubles back on itself, we get this:
micahtillman.com/lsviz/?FmF&F=F… $A line drawing of a fractal...$

Feed the L-system and equilateral triangle to start, and you get this:
micahtillman.com/lsviz/?FmFmF&F… $A line drawing of a fractal...$

Feed the L-system a pentagon, and you get this:
micahtillman.com/lsviz/?FmFmFmF… $A line drawing of a fractal...$

Given that the fractals above all come from the same L-system -- but end up looking so different based on the shape they start with -- means there's some interesting but difficult work to be done on the identity conditions of a fractal, and of an L-system.

That work will, it seems to me, depend on finding a good algorithm for determining how "close" two L-systems are to each other in the "possibility space" of all possible L-systems. 🤔

Another version of the tree branches fractal, above:
micahtillman.com/lsviz/?F&F=Frl… $A line drawing of a fractal...$

This is a fractal, maybe? Like a map of a mountain range? Two islands made of shuriken/throwing stars?
micahtillman.com/lsviz/?F&F=lGF… $A line drawing of a fractal...$

It's the Dragon Curve, but filled with lots of happy axolotl and angry frog faces? Or maybe the happy faces are puffer fish? Fractals are weird.
micahtillman.com/lsviz/?F&F=rBr… $A line drawing of a fractal...$

This fractal goes through a cycle of three different shapes as the number of iterations increases.
micahtillman.com/lsviz/?F&F=BrF… $A line drawing of a fractal...$

This fractal is a variant of a rather normal, Koch Snowflake-ish figure. But if you change the angle from 90 to 135, you get a pillar of angry triangles (reminiscent of the Sierpinski Triangle)
micahtillman.com/lsviz/?F&F=rlF… $A line drawing of a fractal...$

Set the angles to 120, and you get a Sierpinski Triangle (lacking an outside border). Set the angles to 120 and 60, however, and you get this. Looks like trees on the savanna of another world
micahtillman.com/lsviz/?F&F=rlF… $A line drawing of a fractal...$

That feeling when you're trying to be a triangle, and are surrounded by other shapes that are trying to be triangles, but none of you are quite living up
micahtillman.com/lsviz/?F&F=rlF… $A line drawing of a fractal...$

A fractal that poses the question of whether sailboats could be made of hourglasses, but also other hourglass sailboats
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

The same L-system, with one of the angles changed, produces this fractal:
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

Change both angles to 90 degrees, and you get this. And when you zoom in, you find that the blob is a version of the Levy C-Curve
micahtillman.com/lsviz/?F&F=FrF… $A line drawing of a fractal...$

I was surprised to find the Python logo, going very fast, in this fractal:
micahtillman.com/lsviz/?B&F=FFr… $A line drawing of a fractal...$

When fractal hunting, it's always interesting to find something that looks so botanical being generated by a an algorithm using only 90-degree angles. (I believe it's related to the Dragon Curve.)
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

Cut the main replacement rule in the above L-system in half, add a turn to the end of it, then change the angle to 135 degrees, and you get this fractal. Looks very much like half a Dragon Curve.
micahtillman.com/lsviz/?F&F=FFr… $A line drawing of a fractal...$

Triangles supported by buttresses made of triangles. That's just good architecture. #fractal
micahtillman.com/lsviz/?F&F=FFl… $A line drawing of a fractal...$

For v.2 of the app, I have moved over to a new thread:

https://twitter.com/MicahTillman/status/1426180031794003971

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