"I'll take famous fractals for $500..."
October 30, 2022 9:31 AM Subscribe
Wikipedia is notoriously complicated when discussing math-related topics but sometimes the concepts themselves are actually not that difficult when described in plain language. Let's talk about fractals, specifically the Cantor Set, the Sierpinski carpet, and the Menger sponge. Here's how they relate to one another, mathematically.
- The Cantor Set is named for Georg Cantor but it was discovered by Henry Smith. Take a line, remove the middle third. Then go remove the middle third from the two resulting lines. Keep going. What you wind up with is kind of weird, mathematically speaking. Here's a blog post that describes some of these weird aspects more generally.
- The Sierpinski Carpet, named for Waclaw Sierpinski and invented by Stefan Mazurkiewicz, is kind of the same thing except with a square instead of a line. It's also got some weird properties, particularly when you try to calculate its area after infinite iterations.
- The Menger sponge, named for Karl Menger but occasionally called Sierpinski's sponge, pops this out into a third dimension, with a cube instead of a square or a line. It lends itself to artistic endeavors. It's also incredibly complex but a basic one can be built with Lego.
- The Cantor Set is named for Georg Cantor but it was discovered by Henry Smith. Take a line, remove the middle third. Then go remove the middle third from the two resulting lines. Keep going. What you wind up with is kind of weird, mathematically speaking. Here's a blog post that describes some of these weird aspects more generally.
- The Sierpinski Carpet, named for Waclaw Sierpinski and invented by Stefan Mazurkiewicz, is kind of the same thing except with a square instead of a line. It's also got some weird properties, particularly when you try to calculate its area after infinite iterations.
- The Menger sponge, named for Karl Menger but occasionally called Sierpinski's sponge, pops this out into a third dimension, with a cube instead of a square or a line. It lends itself to artistic endeavors. It's also incredibly complex but a basic one can be built with Lego.
Portland didn't get much snow when I lived there. When it did, people made things. One time, a neighbor from a snowier region, who had lots of practice with snow, managed to make a full-sized snowperson with a mere dusting of snow. Another time, after a week of heavy snow, I had a nice sit on a couch made from snow. And then there was the time I walked past cortex's house. Menger sponge.
posted by aniola at 9:46 AM on October 30, 2022 [11 favorites]
posted by aniola at 9:46 AM on October 30, 2022 [11 favorites]
I like that you started the menger sponge post out with a complaint about wikipedia math jargon that seemed to me to describe a fractal.
posted by aniola at 10:07 AM on October 30, 2022 [1 favorite]
posted by aniola at 10:07 AM on October 30, 2022 [1 favorite]
Josh.
posted by bondcliff at 10:16 AM on October 30, 2022 [1 favorite]
posted by bondcliff at 10:16 AM on October 30, 2022 [1 favorite]
We've got a Serpinski Tetrahedron in Breckenridge MN!
posted by AzraelBrown at 10:42 AM on October 30, 2022 [4 favorites]
posted by AzraelBrown at 10:42 AM on October 30, 2022 [4 favorites]
Pathological monsters! Just in time for Halloween!
posted by tspae at 10:46 AM on October 30, 2022 [3 favorites]
posted by tspae at 10:46 AM on October 30, 2022 [3 favorites]
Wikipedia is a reference, not a textbook.
I read a lot of peer-reviewed science articles. These are thick with jargon. If they weren't, the articles would be 10 times longer and super irritating for most readers. For example, if an article about fractures kept saying "the smaller bone than runs down the outside of the lower leg" instead "the fibula".
posted by neuron at 11:17 AM on October 30, 2022 [3 favorites]
I read a lot of peer-reviewed science articles. These are thick with jargon. If they weren't, the articles would be 10 times longer and super irritating for most readers. For example, if an article about fractures kept saying "the smaller bone than runs down the outside of the lower leg" instead "the fibula".
posted by neuron at 11:17 AM on October 30, 2022 [3 favorites]
Isn't this related to Zeno's Paradox about fleet-footed Achilles never catching the tortoise? I asked myself. A quick red-face google check "Zeno's paradox Menger sponge" yielded a nice PDF on the intersect Maths Sparks 72 Geometric Series and Infinite Games passing by Hilbert's Hotel. One of the leads for the Maths Sparks project is the charismatic Aoibhinn Ni Shuilleabhain: a fine role model for women in STEM.
posted by BobTheScientist at 11:31 AM on October 30, 2022 [1 favorite]
posted by BobTheScientist at 11:31 AM on October 30, 2022 [1 favorite]
Sierpinski triangle paintings by Ann Erpino, Silent Night and Wings.
posted by brambleboy at 11:32 AM on October 30, 2022 [1 favorite]
posted by brambleboy at 11:32 AM on October 30, 2022 [1 favorite]
When I first learned about the Cantor Set in a real analysis course, I remember getting a weird feeling in the pit of my stomach.
It was my first experience with math that didn’t really make sense to me.
posted by wittgenstein at 11:34 AM on October 30, 2022 [3 favorites]
It was my first experience with math that didn’t really make sense to me.
posted by wittgenstein at 11:34 AM on October 30, 2022 [3 favorites]
Perhaps my favorite example of the Sierpinski triangle is the reflections within a tetrahedral stack of shiny ball bearings. Sweet, Ott, & Yorke got a Nature paper out of it (paywalled); it's easy to reproduce in real life or by ray-tracer (not paywalled); and playing with ray-tracer parameters can produce striking images (not paywalled).
posted by brambleboy at 12:07 PM on October 30, 2022 [2 favorites]
posted by brambleboy at 12:07 PM on October 30, 2022 [2 favorites]
I feel complete. Thanks, Jess.
posted by cortex at 12:45 PM on October 30, 2022 [9 favorites]
posted by cortex at 12:45 PM on October 30, 2022 [9 favorites]
If you like fractals, sets and other recursive things I'm happy to report that the fractal program Fractint is alive and well.
And then there was the time I walked past cortex's house. Menger sponge.
This is even more amusing when you think about the fact that snowflakes are already natural fractals and you're just mashing up fractals to make human-made approximations of fractals.
When I first learned about the Cantor Set in a real analysis course, I remember getting a weird feeling in the pit of my stomach.
I've experienced this sensation many times. It's like the mathematical version of thalassophobia, the fear of large or deep bodies of water but instead it's weird or complex math or complexity from simplicity and recursion.
Things like prime numbers, pi or Fibonacci strings don't seem to be quite the same kind of weird as simple recursive rules leading to ridiculously endless amounts of complex-yet-organized behavior.
The book Godel, Escher and Bach: An Eternal Golden Braid is basically a concentrated form of this sensation, and, in many ways, isn't a nice or fun book. If you experience this weird feeling it's almost more of a horror story with math.
Another moment that I remember is when I first realized how Julia sets and Mandlebrot sets were basically the same thing and sort of the inside-out and geometrically and topologically translated versions of each other.
I also experienced this really profoundly when I first learned that the 2D plots of fractals were really just a slice or focal plane through an n-dimensional structure of the total sum of a given set, and these could be represented as 3D objects, or animated as a plane slicing through them to reveal more details about the overall structure, and even further that you could slice through these 3D plots at different angles and how that related to how we were visualizing them and that different slices of the same values of an iterated Julia or Mandlebrot set were also self-referential and mapped topologically to each other.
When I was in my dirty raver era in the 90s I spend I don't know how many hundreds of hours in profoundly altered states staring at Fractint pondering fractals and the nature of reality. I also spent a lot of time staring at a DIY laser show I built playing with smoke and Brownian motion by using mirrors and motors to slice a sheet of laser light through incense smoke or fog to get that "liquid sky" effect you may know from various science fiction movies and TV shows, which is perhaps the organic analog version of pondering fractals.
If you've ever seen this optical effect in, say, the opening scenes of one of the Doctor Who series or other shows or movies and thought to yourself "Oh, that looks so cool!" imagine having that on tap whenever you wanted it, sober or otherwise. I messed around with staring at that sort of thing bathing my head in fractals for more than a few years. I'm pretty sure this did some very strange things to me and how I see the natural world around me, but in good ways. I think it gave me a rather plastic and flexible brain and an ability to adapt to new information as well as recognize recursive patterns in unique and unusual ways.
Even today every so often I will be around a fire with some friends at a party or something and I also happen to have a laser pointer in my pocket. And without warning I'll light it up and wiggle the beam through the campfire smoke and light up the sky with a sheet of fractals pouring out of my hand. I love the part where someone says "WOOOOAH COOOOOL" because Science!
And I'm digressing even further away from the FPP, but if you ever want to make your own "liquid sky" toy and play with this optical effect it's ridiculously easy and cheap to do today, and might even make for a fun science project to entertain weird kids.
It can be as simple and minimal as a cheap low powered toy laser pointer, a small motor, perhaps with a gear or pulley on the shaft, a common AA battery, a bit of wire, a small piece of mirror, and some glue or epoxy.
Glue or epoxy the small piece of mirror on the shaft of the motor roughly and vaguely perpendicular to the motor shaft, gear or pulley, mirror side out. Don't worry about making it perfectly straight, you want some slight amount of offset, and there always will be some wobble unless you have a really nice machine shop for some reason. You want about 3-6 degrees off of square so the mirror wobbles, and a laser beam gets scanned in a cone or circle when bounced off the mirror.
Now arrange the laser pointer and motor so the laser bounces off the mirror at some suitable viewing angle. You can do this by hand just holding all of the parts together, or set it up with a stack of books, or some Lego bits, or modeling clay, or tape, or whatever.
Connect the battery to the motor. Turn on the laser. Aim it at the moving, wobbling mirror. Beam the now rotating, scanning cone-shaped through some smoke, steam or fog.
Bam, magic! Liquid holographic fractals floating in the air!
You can also experiment with variations of how to scan a laser with a moving mirror. If you have good glue or epoxy you can try to place the mirror on edge on the shaft parallel to the rotation so instead of a cone of light, it casts a sheet of light, which topologically is just a really flat cone.
I find that the cone-shaped scanner is better for low powered lasers so the scan path is shorter through a given small volume of convective smoke like a stick of incense and you get more light and persistence of vision effects in that given area.
The slightly more advanced version of this can involve things like a battery holder, a switch, maybe a variable resistor to control the speed of the motor and scan rate, more permanent ways to hold and mount the motor and laser, higher quality laser emitters, front surface mirrors to keep the beam tighter, etc.
For increasingly fine results and more refined visual effects and details - use a higher quality and higher powered laser with good collimation, switch to a front surfaced mirror and so on. The thinner and brighter the beam the better and more detailed the effect is.
You can even do a variation of this without a motor at all. A bit of mirror attached to a a rubber band or spring with a laser bouncing off the mirror with a very similar effect, but lacking in the structure, periodicity and more linearly animated effect of a more rigid beam path through smoke or vapor.
You can also glue a bit of mirror on a small speaker and make a laser reactive to audio, especially if you incorporate it with a motor and mirror combo scanning it that reflected beam around. IE, bounce the laser off of a mirror on a speaker, first, then the mirror on the motor, which will give you an oscilloscope-like effect where the cone pulses to the waveform modulated by the mirror on the speaker.
If this is all too complicated, well, simply wave a laser pointer around in the air in some smoke or fog. I like to use a rubber band or clip to hold down the button on the pointer, and then simply wiggle the laser pointer in a straight line between my fingers to get a roughly fan-shaped beamline.
You can also go to a hardware store and buy one of those fancy new laser-guided leveling tools that project a beamline using moving mirrors or slit optics, but that's not nearly as educational or as much fun putting the whole system together, because this concept of scanning a laser beam cone or sheet through a source of Brownian motion illustrates some really cool facets of science, physics, math and geometry.
But, wait, there's even more. I just realized I can tie this all back to recursion, complexity, and fractal-like behavior.
If you take the concept of laser + mirror + motor and expand it by adding another mirror and motor so the laser bounces twice off of two mirrors rotating off-axis at differential speeds and angles you get a Lissajous curve, the same way different periodic functions on the X-Y axis in an oscilloscope can generate these complex shapes and curves, either project on free air or on a screen, which would be an intersectional plane of that projection.
It's like an optical Spirograph and it's lots of fun.
Add a third or forth mirror, motor and bounce, add speed controls or variable resistors to the motors and you can end up with shapes that vaguely resemble Lorenz systems/attractors and something that accidentally and chaotically resembles an entirely useless and unscientific analog computer.
Basically like this, or this.
(Note to people who understand math: Yes, you won't get a proper Lorenz attractor system out of the laser scanners I'm describing, only Lissajous curves and related patterns, because it's missing at least one geometric axis and component, but they can get really close to projecting shapes and curves that are the precursor to Lorenz attractors, as though you're mapping them from a limited axis or projection.)
posted by loquacious at 1:57 PM on October 30, 2022 [11 favorites]
And then there was the time I walked past cortex's house. Menger sponge.
This is even more amusing when you think about the fact that snowflakes are already natural fractals and you're just mashing up fractals to make human-made approximations of fractals.
When I first learned about the Cantor Set in a real analysis course, I remember getting a weird feeling in the pit of my stomach.
I've experienced this sensation many times. It's like the mathematical version of thalassophobia, the fear of large or deep bodies of water but instead it's weird or complex math or complexity from simplicity and recursion.
Things like prime numbers, pi or Fibonacci strings don't seem to be quite the same kind of weird as simple recursive rules leading to ridiculously endless amounts of complex-yet-organized behavior.
The book Godel, Escher and Bach: An Eternal Golden Braid is basically a concentrated form of this sensation, and, in many ways, isn't a nice or fun book. If you experience this weird feeling it's almost more of a horror story with math.
Another moment that I remember is when I first realized how Julia sets and Mandlebrot sets were basically the same thing and sort of the inside-out and geometrically and topologically translated versions of each other.
I also experienced this really profoundly when I first learned that the 2D plots of fractals were really just a slice or focal plane through an n-dimensional structure of the total sum of a given set, and these could be represented as 3D objects, or animated as a plane slicing through them to reveal more details about the overall structure, and even further that you could slice through these 3D plots at different angles and how that related to how we were visualizing them and that different slices of the same values of an iterated Julia or Mandlebrot set were also self-referential and mapped topologically to each other.
When I was in my dirty raver era in the 90s I spend I don't know how many hundreds of hours in profoundly altered states staring at Fractint pondering fractals and the nature of reality. I also spent a lot of time staring at a DIY laser show I built playing with smoke and Brownian motion by using mirrors and motors to slice a sheet of laser light through incense smoke or fog to get that "liquid sky" effect you may know from various science fiction movies and TV shows, which is perhaps the organic analog version of pondering fractals.
If you've ever seen this optical effect in, say, the opening scenes of one of the Doctor Who series or other shows or movies and thought to yourself "Oh, that looks so cool!" imagine having that on tap whenever you wanted it, sober or otherwise. I messed around with staring at that sort of thing bathing my head in fractals for more than a few years. I'm pretty sure this did some very strange things to me and how I see the natural world around me, but in good ways. I think it gave me a rather plastic and flexible brain and an ability to adapt to new information as well as recognize recursive patterns in unique and unusual ways.
Even today every so often I will be around a fire with some friends at a party or something and I also happen to have a laser pointer in my pocket. And without warning I'll light it up and wiggle the beam through the campfire smoke and light up the sky with a sheet of fractals pouring out of my hand. I love the part where someone says "WOOOOAH COOOOOL" because Science!
And I'm digressing even further away from the FPP, but if you ever want to make your own "liquid sky" toy and play with this optical effect it's ridiculously easy and cheap to do today, and might even make for a fun science project to entertain weird kids.
It can be as simple and minimal as a cheap low powered toy laser pointer, a small motor, perhaps with a gear or pulley on the shaft, a common AA battery, a bit of wire, a small piece of mirror, and some glue or epoxy.
Glue or epoxy the small piece of mirror on the shaft of the motor roughly and vaguely perpendicular to the motor shaft, gear or pulley, mirror side out. Don't worry about making it perfectly straight, you want some slight amount of offset, and there always will be some wobble unless you have a really nice machine shop for some reason. You want about 3-6 degrees off of square so the mirror wobbles, and a laser beam gets scanned in a cone or circle when bounced off the mirror.
Now arrange the laser pointer and motor so the laser bounces off the mirror at some suitable viewing angle. You can do this by hand just holding all of the parts together, or set it up with a stack of books, or some Lego bits, or modeling clay, or tape, or whatever.
Connect the battery to the motor. Turn on the laser. Aim it at the moving, wobbling mirror. Beam the now rotating, scanning cone-shaped through some smoke, steam or fog.
Bam, magic! Liquid holographic fractals floating in the air!
You can also experiment with variations of how to scan a laser with a moving mirror. If you have good glue or epoxy you can try to place the mirror on edge on the shaft parallel to the rotation so instead of a cone of light, it casts a sheet of light, which topologically is just a really flat cone.
I find that the cone-shaped scanner is better for low powered lasers so the scan path is shorter through a given small volume of convective smoke like a stick of incense and you get more light and persistence of vision effects in that given area.
The slightly more advanced version of this can involve things like a battery holder, a switch, maybe a variable resistor to control the speed of the motor and scan rate, more permanent ways to hold and mount the motor and laser, higher quality laser emitters, front surface mirrors to keep the beam tighter, etc.
For increasingly fine results and more refined visual effects and details - use a higher quality and higher powered laser with good collimation, switch to a front surfaced mirror and so on. The thinner and brighter the beam the better and more detailed the effect is.
You can even do a variation of this without a motor at all. A bit of mirror attached to a a rubber band or spring with a laser bouncing off the mirror with a very similar effect, but lacking in the structure, periodicity and more linearly animated effect of a more rigid beam path through smoke or vapor.
You can also glue a bit of mirror on a small speaker and make a laser reactive to audio, especially if you incorporate it with a motor and mirror combo scanning it that reflected beam around. IE, bounce the laser off of a mirror on a speaker, first, then the mirror on the motor, which will give you an oscilloscope-like effect where the cone pulses to the waveform modulated by the mirror on the speaker.
If this is all too complicated, well, simply wave a laser pointer around in the air in some smoke or fog. I like to use a rubber band or clip to hold down the button on the pointer, and then simply wiggle the laser pointer in a straight line between my fingers to get a roughly fan-shaped beamline.
You can also go to a hardware store and buy one of those fancy new laser-guided leveling tools that project a beamline using moving mirrors or slit optics, but that's not nearly as educational or as much fun putting the whole system together, because this concept of scanning a laser beam cone or sheet through a source of Brownian motion illustrates some really cool facets of science, physics, math and geometry.
But, wait, there's even more. I just realized I can tie this all back to recursion, complexity, and fractal-like behavior.
If you take the concept of laser + mirror + motor and expand it by adding another mirror and motor so the laser bounces twice off of two mirrors rotating off-axis at differential speeds and angles you get a Lissajous curve, the same way different periodic functions on the X-Y axis in an oscilloscope can generate these complex shapes and curves, either project on free air or on a screen, which would be an intersectional plane of that projection.
It's like an optical Spirograph and it's lots of fun.
Add a third or forth mirror, motor and bounce, add speed controls or variable resistors to the motors and you can end up with shapes that vaguely resemble Lorenz systems/attractors and something that accidentally and chaotically resembles an entirely useless and unscientific analog computer.
Basically like this, or this.
(Note to people who understand math: Yes, you won't get a proper Lorenz attractor system out of the laser scanners I'm describing, only Lissajous curves and related patterns, because it's missing at least one geometric axis and component, but they can get really close to projecting shapes and curves that are the precursor to Lorenz attractors, as though you're mapping them from a limited axis or projection.)
posted by loquacious at 1:57 PM on October 30, 2022 [11 favorites]
Wikipedia is a reference, not a textbook.
I get it, mathematicians tell me Wikipedia is pretty reliable for math. But from my perspective, what good is a reference if you can't refer to it?
posted by aniola at 2:30 PM on October 30, 2022
I get it, mathematicians tell me Wikipedia is pretty reliable for math. But from my perspective, what good is a reference if you can't refer to it?
posted by aniola at 2:30 PM on October 30, 2022
Wikipedia is an encyclopedia, it needs to be accessible to both experts as well as laypeople. But achieving that requires putting substantial work and thought into how the entries are organized and written.
posted by polymodus at 3:11 PM on October 30, 2022 [2 favorites]
posted by polymodus at 3:11 PM on October 30, 2022 [2 favorites]
The Mandelbrot set was first described in a recognizable form by Robert Brooks and J Peter Matelski in 1978, the year before Mandelbrot published. Matelski has been active (including commenting on my backwater of a blog) in making sure that people know that the paper wasn't formally published until a couple of years after Mandelbrot's, but had been available as a pre-print. Some more deets here from StackExchange: Grid spacing, iterations used in the 1978 first published rendering of the Mandelbrot set?
posted by scruss at 3:14 PM on October 30, 2022
posted by scruss at 3:14 PM on October 30, 2022
Q: What does the B. stand for in "Benoit B. Mandelbrot"?
A: "Benoit B. Mandelbrot"
posted by Harvey Kilobit at 3:23 PM on October 30, 2022 [10 favorites]
A: "Benoit B. Mandelbrot"
posted by Harvey Kilobit at 3:23 PM on October 30, 2022 [10 favorites]
I feel complete. Thanks, Jess.
posted by cortex
Of course, cortex feeling complete doesn't feel complete without missing fractal cubes...
posted by Ghidorah at 5:20 PM on October 30, 2022
posted by cortex
Of course, cortex feeling complete doesn't feel complete without missing fractal cubes...
posted by Ghidorah at 5:20 PM on October 30, 2022
Alright, math undergrad major here, so the Cantor Set us uncountable because if I squish a subset of an uncountable set a countable number of times, it remains uncountable?
posted by indexy at 6:38 PM on October 30, 2022
posted by indexy at 6:38 PM on October 30, 2022
My favorite weird fact about the Cantor set is that there's only one Cantor set - if you remove two intervals at each step or intervals of different sizes or if you start with a square and remove a cross at each step to get four smaller squares, it's still topologically the same as Cantor's original construction.
There's something similar for the Sierpinski carpet and the Menger sponge, too: IIRC, if you start with a disc and punch infinitely many holes in it so that no part of the disc is left unpunctured, then you always get something topologically equivalent to the Sierpinski carpet.
posted by ectabo at 7:56 PM on October 30, 2022 [1 favorite]
There's something similar for the Sierpinski carpet and the Menger sponge, too: IIRC, if you start with a disc and punch infinitely many holes in it so that no part of the disc is left unpunctured, then you always get something topologically equivalent to the Sierpinski carpet.
posted by ectabo at 7:56 PM on October 30, 2022 [1 favorite]
We've got a Serpinski Tetrahedron in Breckenridge MN!
That is seriously cool.
posted by biogeo at 9:50 PM on October 30, 2022 [1 favorite]
Neat, I don’t think I’d heard of the Smith–Volterra–Cantor set. The fact that it’s topologically equivalent to the normal Cantor set means you can continuously deform a set of points with measure zero into a set with positive measure, and it’s easy to see how to recursively map any point from one into the other.
posted by mubba at 8:17 AM on October 31, 2022
posted by mubba at 8:17 AM on October 31, 2022
"that fractals are infinitely self similar" factoid actually just statistical error. average fractal has 3 zoom levels. Cantor Georg, who lives in cave & constructs infinite wonders each day, is an outlier adn should not have been counted”
posted by Ned G at 8:53 AM on October 31, 2022 [3 favorites]
posted by Ned G at 8:53 AM on October 31, 2022 [3 favorites]
Square does to doctor, says I’m so depressed, feel like I’m empty inside, doctor says I know just the thing, great fractal Sierpinski carpet is town, topologically thrilling, go see it, will cheer you up
Square bursts into tears, “but doctor…”
posted by cortex at 9:27 AM on October 31, 2022 [3 favorites]
Square bursts into tears, “but doctor…”
posted by cortex at 9:27 AM on October 31, 2022 [3 favorites]
"The Cantor Set" would be a great name for a klezmer band.
posted by biogeo at 10:05 AM on October 31, 2022
posted by biogeo at 10:05 AM on October 31, 2022
"Uncountably infinitely talented! Almost surely not playing near you!"
posted by biogeo at 10:07 AM on October 31, 2022
posted by biogeo at 10:07 AM on October 31, 2022
(if i were to open a flooring biz, i'd almost certainly call it "Sierpinski Carpet")
posted by The_Auditor at 10:07 AM on October 31, 2022
posted by The_Auditor at 10:07 AM on October 31, 2022
I found Helen, an Everything Menger Crouton blinking in the metafilter garden. And now we know who drew the croutons.
posted by aniola at 3:42 PM on November 2, 2022 [1 favorite]
posted by aniola at 3:42 PM on November 2, 2022 [1 favorite]
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Recursion [ri-'k Ur-zh Un] n See “Recursion”
posted by lalochezia at 9:40 AM on October 30, 2022 [8 favorites]