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Micah Tillman

@MicahTillman

Aug 13, 2021 • 94 tweets • 45 min read • Read on X

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@madewithunity

Version 2.0 of my (free, online) L-System Visualizer App is out. (#fractals @madewithunity)
micahtillman.com/lsviz/
What's the big deal? Well, color, for one: $A line drawing of a fractal...$

@madewithunity

@madewithunity The colors show the iteration of the L-system's development that introduced/added/inserted that portion of the fractal. The most obvious way to see that as happening is via interpolation. That is, you can see the fractal as growing from within, with new "nodes' being inserted:

Some fractals, however, start with the same set of moves in each iteration. Thus, you can see new nodes as being appended to the previously-placed nodes. Instead of scattering colors throughout, you end up with regions of new color:

The app now comes pre-programmed with over 200 fractals and fractal variants. You can browse the available fractals, have the app show you one at random, or create new fractals yourself.
micahtillman.com/lsviz/

The app is now animated as well, so you can watch the app trace the path of each fractal in real time. Doing so reveals structure you couldn't see just by looking at the completed drawings.

There is also a background info page: micahtillman.com/lsvizinfo/
An FAQ page: micahtillman.com/lsvizfaq/
And the v2.0 release notes: micahtillman.com/lsvizv2notes/

And if you miss v. 1 of the app, just add a "1": micahtillman.com/lsviz1

https://twitter.com/MicahTillman/status/1421549789838184455

Most of the fractals that are now built into v.2's library were found using v.1, and can be seen in this thread:

https://twitter.com/MicahTillman/status/1421549789838184455

For the techy among you:
Each figure the app draws consists of a single line renderer and a single particle system. The user can show both, or just show one, to get different effects. You can also change the width of the line and the sizes of the particles.

I found this fractal using v1 of the app. But I think it's my favorite so far when you turn the colored dots on (and the lines off) in v2:
micahtillman.com/lsviz/?F&F=FrX…

But in v2 of the app you can have lines on and dots off, dots on and lines off, or both on. Here's a zoomed-in view:

If two dots are the same color, they were introduced during the same iteration. The latest introduced the magenta dots. The iteration before: the dark blue dots. The iteration before: lighter blue dots. The iteration before: green dots. Before that: yellow, etc.

So, this is a kind of archeological view of the fractal. dots that are the same color used to be much closer together, but dots with new colors have grown up between them and pushed them apart.

One advantage of v2 of my L-System Visualizer app is that figures that used to just look like a mess -- with just white lines to see -- now can show their hidden structure:
micahtillman.com/lsviz/?F&F=FrF… $Two copies of a fractal. Th...$

In this one, setting app to treat new dots as being appended to the end of the fractal leads to results that are more interesting at large scale. But the zoomed-in view of the "new dots are interpolated" view is cool too.
micahtillman.com/lsviz/?B&B=Brl…

@celeste_game

This fractal looks like a zoomed-out view of Chapter 1 (Forsaken City) of @celeste_game to me (e.g., celestegame.fandom.com/wiki/Chapter_1… &

https://twitter.com/OwenGoss/status/1012681586045747201

):
micahtillman.com/lsviz/?F&F=FrH… $A fractal with a roughly-re...$

This fractal looks to me like a stylized wisp of smoke -- seen from the side -- if you leave it in the "original" rotation. Rotate it to make it fit the frame better, however, and you get stylized whirlpools seen from above.
micahtillman.com/lsviz/?F&F=Fll… $Two copies of a fractal. Th...$

Was exploring L-systems (as you do) and stumbled across another version of the Puffer Fish Dragon (it looks like a bunch of puffer fish, but also looks like it's in the Dragon Curve family). Top is original; bottom is new:
micahtillman.com/lsviz/?F&F=FrG…

Differences between the two L-systems (which seem to be generating identical figures) include: The instruction string generated by the original scales by a factor of 2 with each iteration, while the new one scales by a factor of 4.

Though talk of the "fractal dimension" of various figures is often discussed, and often involves scaling factors, I haven't seen anyone talk about the scaling factors of the strings (of which the fractals are a visual interpretation). Then again, I haven't really looked. . . .

Interestingly, the fractal dimension of the Dragon Curve is supposed to be 2, and the scale factor of its L-system's string is also 2.
micahtillman.com/lsviz/?F&F=FlG…

Haven't posted one of these in a while, but . . . new fractal dropped:
micahtillman.com/lsviz/?f&f=frf… $An image of a fractal. The ...$

I feel like I've seen this fractal before, but just in case. Stumbled across this one yesterday:
micahtillman.com/lsviz/?f&f=frf… $A picture of a fractal. The...$

And here's a closely-related fractal that looks like a fox head made of fox heads, each of which is wearing a donut for a crown. The main fox head, though, is missing its donut.
micahtillman.com/lsviz/?f&f=frf… $A picture of a fractal. The...$

A fractal with the general structure of the Koch Snowflake (or the p-adic valuations of 2), but it looks like a series of abstract torii gateways:
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal. The ...$

A closely-related fractal to the above seems to be a space-filling curve, akin to the Hilbert or Peano curve:
micahtillman.com/lsviz/?f&f=f1g…

The same fractal, but with one of the angles changed:
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal. The ...$

The same fractal again, but with that same angle changed to a different value:
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal, cons...$

And again, the same fractal with multiple angles changed. This version looks like the "Mass of Souls" enemy from Dark Souls 1:
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal. The ...$

The same fractal yet again, with a changed angle. Spider skull?
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal curve...$

Same fractal again.
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal curve...$

I love fractals that occasionally pause and pretend they're doing something completely different
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal. The ...$

A variant of the same #fractal is clearly an eagle.
micahtillman.com/lsviz/?f&f=f1g… $An image of a fractal. It l...$

Another multi-form fractal (it cycles through several different general shapes depending on the iteration). It's got dollar signs in it, so it's probably not to be trusted $An image of a fractal. The ...$

The link for that fractal is: micahtillman.com/lsviz/?f&f=f1f…

New fractal just dropped:
micahtillman.com/lsviz/?f&f=frb… $An image of a fractal. The ...$

Another new fractal. Looks like a nebula to me:
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Found another fractal in the "void tentacles" family:
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. Two ...$

I think I've seen this fractal before, but not sure. It's based on the first three digits of pi, but looks more like a pine tree than a pie: $An image of a fractal, most...$

The link for the above fractal, where you can see the L-system that generates it and tweak that system to your heart's content:
micahtillman.com/lsviz/?f&f=flf…

A slight modification to the L-system for the previous fractal generates this, which I shall call the "Wicker Man":
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, draw...$

Another small tweak turns the fractal into a combination of the Sierpinski Triangle and the Koch Snowflake:
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Another small tweak turns the fractal into a conglomeration of floofy pups made of stars -- a kind of Star Cerberus, if Cerberus were a Samoyed or Newfie, (And if you watch the fractal develop, you'll see it contains many five-pointed-star-shaped holes)
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$ $A copy of the fractal with ...$

A related fractal to the ones above:
micahtillman.com/lsviz/?flflflf… $A an image of a fractal, mo...$

Still tweaking the L-system from yesterday. Found a new one:
micahtillman.com/lsviz/?f&f=flf…
Looks like some "globus crucigers" (en.wikipedia.org/wiki/Globus_cr…) to me, if the "globus" were to be made of flowers. $An image of a fractal. The ...$

Sometimes you hit a vein in the fractal mines. Here's another one, related to the ones above. But this time it also looks like the Chaos Cookie Monster fractal I found a while back (though this time I see laughing dragonfly faces)
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

A few more tweaks to the L-system that generates the above fractals, and we get the Koch Snowflake, but where each "triangle" is wearing fancy priestly vestments, as if ready to say mass
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

You go to the fractal store and ask if they have anything in an owl that's also a hummingbird. They offer you this:
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

This fractal has very concerned crabs in it
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

tfw you had a Koch Snowflake fractal, but then some prankster trimmed all the points off
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal, show...$

You never know what's going to turn up when yer huntin' fractals
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. It a...$

Today's new fractal. A bit like a brain, perhaps?
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

Adjust the angles, and you get a cleaner version of the same fractal:
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. It l...$

I like the internal details in this fractal:
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

Here's another of the bird-esque fractals, but this time it looks a little more like a bat
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. It a...$

Same fractal as the previous, just with different angles. This version looks like coral to me
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

Today's fractal is a weird "one-point perspective" scene with two rows of octagonal trees. Lower version shows how smaller copies are embedded in the larger figure. Looks like the scale factor for this one is 9, but its fractal dimension is about 1.8.
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

Oh snap, here's the same fractal, using different angles. The fractal dimension of this one is supposedly closer to 1.7, which I find surprising. (Maybe there's a bug in the way I coded the box counting method for measuring fractal dimension.)
micahtillman.com/lsviz/?f&f=fpf… $An image of a fractal. The ...$

What if you were an arrowhead made of arrowheads? Or a super surprised owl face with small wings? That's something this fractal doesn't have to ask itself. It knows.
(Scale factor: 20; fractal dimension: 1.6ish, supposedly)
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Pretty sure I've seen a fractal like this before, but I particularly enjoyed watching it develop.
micahtillman.com/lsviz/?f&f=frf… $An image of a fractal. The ...$

Your Friday Fractal. It's yours. I named it for you.
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. It l...$

A flower pyramid fractal. It appears to be a space-filling-style curve (a la the Hilbert Curve) that deliberately leaves regular gaps. Don't know what to call that.
micahtillman.com/lsviz/?f&f=frf… $An image of a fractal. The ...$

Today's fractal. It has two forms, which alternate depending on how many iterations you have run. That is, this sort of fractal is like an infinite series that doesn't converge. (Some fractals go through a cycle of more than two forms, furthermore):
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, made...$

A sort of lopsided hexagonal fractal (a bit like a shield)? Scale factor of 4.
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, made...$

This fractal reminds me of a landscape from a From Software game, covered in monuments and tombstones at all weird angles. The first image is the fractal after 5 iterations. The 2nd is after 6.
micahtillman.com/lsviz/?f&f=frg… $An image of a fractal. It a...$ $An image of a fractal, that...$

Lookit this weird fractal
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Today's fractal. Trying to figure out how to describe these things for the alt text is sometimes very difficult.
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Hey, I got you a Koch Snowflake made of Koch Snowflakes.

"Isn't that what the Koch Snowflake fractal is?"

No. A full Snowflake's *sides* are made of copies of its *sides*. This one is (well, contains) a full Snowflake made of full Snowflakes

micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Today's fractal:
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal. The ...$

Oh snap! Another fractal for today. Looks like it's in the Dragon Curve family:
micahtillman.com/lsviz/?f&f=frf… $An image of a fractal. The ...$

I should hire someone to write the alt text for these fractals.
micahtillman.com/lsviz/?f&f=frg…
This is a . . . cauliflower-headed spider with cross-masks for knees hanging upside down from the ceiling? $An image of a fractal, draw...$

Same fractal, but with different angles:
micahtillman.com/lsviz/?f&f=frg… $An image of a fractal in th...$

Same fractal again, with different angles. This time it looks like a chibi owl?
micahtillman.com/lsviz/?f&f=frg… $An image of a fractal. The ...$

Starting a new band/company? Let fractals design your logo. I mean, let L-systems create a fractal for you and then use that for your logo. Yeah.
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, made...$

Not quite a Koch Snowflake fractal. Rather, this is what would happen if each point of a six-pointed star were itself a six-pointed star whose points were six-pointed stars, etc. Also, I like what happens when you increase the dot size.
micahtillman.com/lsviz/?frfrfrf… $An image of a fractal. The ...$

That is, I like the visual effects produced by Unity's particle system shader (plus whatever texture I'm using for the dots; I don't remember) when you increase the particle size. Different rendering systems would give this a different look.

https://twitter.com/MicahTillman/status/1511078654528339979

Accidentally broke the fractal thread. It continues here:

https://twitter.com/MicahTillman/status/1511078654528339979

But now, back on the main thread, I think I found my new favorite fractal. Call it "Tumbling Triangles" for now. (The triangles are trying to be Sierpinski Triangles, but aren't quite succeeding)
micahtillman.com/lsviz/?f&f=frb… $An image of a fractal, made...$

Another view of the previous fractal, showing how the figure grows from iteration to iteration:
micahtillman.com/lsviz/?f&f=frb… $An image of a fractal, made...$

The app informs me that this is a fractal. I . . . well. Okay, app. (And yes, I'm the one who programmed the app. But still)
micahtillman.com/lsviz/?f&f=fr&…

Changing the angle makes it look more fractal-y. You can see that this star contains smaller versions of itself that contain smaller versions of themselves.
micahtillman.com/lsviz/?f&f=fr&…

tfw you're a fractal who just saw your first Batman movie and you want to be the new logo
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal drawn...$

Not sure what to call today's fractal
micahtillman.com/lsviz/?f&f=frf… $An image of a fractal made ...$

This fractal belongs to a family of similar figures. This version looks to me like a cosmic space moth
micahtillman.com/lsviz/?f&f=frg… $An image of a fractal, draw...$

The Dragon Curve got a new cousin today
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, at f...$

A cousin of yesterday's Dragon Curve cousin looks like a leaf, of perhaps a dragon scale? Or a topographical map.
micahtillman.com/lsviz/?f&f=flf… $An image of a fractal, made...$

https://twitter.com/MicahTillman/status/1521880302842437633

Whoops, fractal thread broke again:

https://twitter.com/MicahTillman/status/1521880302842437633

The structure of today's fractal is clearer when you focus on the nodes, rather than the lines connecting the nodes.
micahtillman.com/lsviz/?f&f=flf… $Two images of a fractal. Th...$

Today's fractal has something robotic about it
micahtillman.com/lsviz/?F&F=FlG… $An image of a fractal, draw...$

Today's fractal belongs to a family you see a lot in the fractal mines. But I found this particular version particularly interesting.
micahtillman.com/lsviz/?g&f=gfl… $Two images of a fractal, co...$

Sometimes you leave the L-System Visualizer running while you take the dog for a walk, and you return to find this:
micahtillman.com/lsviz/?f&f=lfg… $An image of a fractal. It l...$

Haven't found one of these in a while. Fractals are nice.
micahtillman.com/lsviz/?f&f=ffr… $An image of a fractal. The ...$

Some sort of swallow-tailed bird? A swallow-winged bird? #fractal
micahtillman.com/lsviz/?f&f=rff… $An image of a fractal in th...$

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