Hitomezashi Stitch Patterns - Numberphile

Ayliean MacDonald discusses Hitomezashi Stitch Patterns.
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Inspired by work by Dr. Katherine Seaton… more at doi.org/10.1080/17513472.2023...
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[@numberphile]

Ayliean has kindly hand-drawn a selection of stitch patterns as prizes of Patrons - find out how to win one here: www.patreon.com/posts/59579953

[@vonantero9458]

So when are we getting the shirt with the pattern stitched? ;)

[@asheep7797]

@@vonantero9458 probably in a couple days

[@internetsydney]

For those curious, hitomezashi [一目刺し; ひとめざし] is a Japanese sewing technique. "Hitome" means "one stitch" and "zashi" is a nominalized form of the verb "sasu," meaning to sew or stitch. It's a kind of sashiko [刺し子; さしこ], which, as Ayliean MacDonald states at 10:19, is a traditional form of decoratively mending clothes, and is also used in quilting.

[@niismo.]

How do you know this?

[@MrEst97]

Thank you, I hadn't learned the verb 刺す yet

[@jamtea573]

@@MrEst97 It's the same kanji that you find in Sashimi (刺し身), so you'll often see it there. It can be used meaning slice/cut/stab/sew, there's quite a few meanings depending on the compound/context you find it in.

[@internetsydney]

@@niismo. I speak Japanese and I live in Japan :D

[@TheDirge69]

Thanks mate!

[@AKhoja]

There's something so beautiful about seeing larger patterns emerge when generated by randomness

[@vigilantcosmicpenguin8721]

@@Atamask Yeah, and there's also the psychological instinct where we're comforted by familiarity and excited by novelty, so it's just the right balance of the two.

[@WestExplainsBest]

A blended class of geometry and art should be taught at the secondary level. It would be extremely intriguing and engaging for students!

[@revimfadli4666]

@@WestExplainsBest along with 'practical creativity', showing that technical minds and creativity aren't mutual exclusives

[@motherisape]

no can't see any patern in randomness

[@plackt]

Everyone talking about proofs for two-colourability while I’m here wondering how 0.5 wasn’t the intuitive/obvious answer to “most random pattern”.

[@VinceOfAllTrades]

Same. That's why I came to the comments.

[@alexanderwatson9845]

I agree but I guess that goes to show how different people's intuitions are

[@reidflemingworldstoughestm1394]

Funny, I thought the .5 probability _was_ the obvious pick for most random.

[@PhilBagels]

That's exactly what I was wondering. How could anyone think that a 0.9 probability would be any more random than a 0.1 probability? And likewise, isn't it obvious that a 1 probability isn't random at all (just like a 0 probability)?

[@mr.cheese5697]

If we shift the whole pattern by 1 unit, all on's will be transformed into of's, and other way around, but since it's the same pattern, chaoticity will stay the same→ Chaos([A,B])=Chaos([B,A])→ Chaos([of(.9),on(.1)])=Chaos([on(.1),of(.9)]) In other words there is no difference between those two patterns.

[@captplanet8688]

"I didn't come here to spell" - might be the most mathematician thing I've ever heard

[@tbpotn]

Every region always has only 1 region on its outside, that's why it's always two-colourable. As having two regions around it would require an intersection of 3 dashes, but only 2 dashes intersect at every point.

[@juanignaciolopeztellechea9401]

Good proof!

[@MitkoNikov]

It would be cool to be able to approximate the number of components...

[@martimrocha9067]

That is not a proof. If you placed the lines however you wanted, you could create 3-colourable images, eventhough there are still 2 dashes intersecting at each point. That's because it has nothing to do with the number of dashes intersecting, since each kndicidual shape can touch others in different places

[@juanignaciolopeztellechea9401]

@@martimrocha9067 ohh, right.

[@HarzemTube]

@@martimrocha9067 you can't create a non-2-colorable map if you don't have three dashes intersecting on a point. That's a basic requirement.

[@smallishkae]

Textiles and fibre crafts have a surprising amount of mathematics baked into them and I love seeing maths nerds come across interesting ways it’s been applied.

[@SuLokify]

One of the first "computers" was an automatic loom.

[@WestExplainsBest]

A blended class of geometry and art should be taught at the secondary level. It would be extremely intriguing and engaging for students!

[@Triantalex]

false.

[@trombone_pasha]

Ayliean is very cool! Would be happy to see more videos with her.

[@tqnohe]

During the shut down I learned to knit. I can totally use this to design patterns. Live it.

[@osmia]

I'm a knitter and crocheter and that was my first thought also :-)

[@rfldss89]

Im so eager for my next pair of jeans to rip so i can create a random mending stitch pattern.

[@5thearth]

Saw a thing recently, someone calculated you could knit a Doom installer in about 3300 square feet (~300 square meters)

[@osmia]

@@5thearth what is a doom installer?

[@bordershader]

@@rfldss89 overlay the rip with new fabric (cut to shape if you prefer, use same or contrasting fabric as you prefer) then use the stitching to go through both layers. Use proper silk thread for strength. In Sashiko mending, you use a really long needle so you can do a long line of running stitch on one go, it really does help (look up videos on here). The repair really is pretty tough and looks fab. Enjoy!

[@mtranchi]

9:20 Ah nice little editing trickery. I was like, "that ain't the mirror of the first one." Then through the magic of editing (at 9:42), the last one turned from P(on)=0 to P(on)=1

[@ThisIsStapes7]

Was about to comment the very same.

[@Bronzescorpion]

Spotted it as well.

[@scottmansfield2197]

Was here to say the same.

[@HPD1171]

the proof for two colorable is very simple. in order for the map to require three colors at least one vertex would need three lines connected. since lines only are drawn from two directions at each coordinate and every line segment is always preceded and followed by a space then at most and exactly two line segments can ever touch at any vertex. this means that no region will never touch more then one on its border.

[@ShankarSivarajan]

Huh, that _is_ trivial. Thanks.

[@floyo]

I don't think it's that simple because your first statement is not trivial. It doesn't generalize to higher numbers for example. A map that requires four colors does not necessarily have a vertex with four lines. In the video there are many examples using the hexagonal/triangular patterns.

[@AngryArmadillo]

I don’t think this is correct. The relevant transformation is to create a vertex for each connected region of space, and an edge between two vertices if those regions are adjacent. Also, your first statement is incorrect. There are plenty of 2-regular graphs that are not 2-colorable. Take any odd cycle for example.

[@lonestarr1490]

​@@floyo They never claimed it would generalize to higher numbers or other patterns. And while it's true that a map which needs to be colored in four colors doesn't need to have a vertex where four lines meet, it is in fact the case for three-color-maps. If I'm not completely mistaken, that should be equivalent to the Venn diagram problem (you cannot draw a complete Venn diagram with four circles).

[@lonestarr1490]

@@AngryArmadillo Can you do an odd cycle in Hitomezashi stitch patterns?

[@Luca_5425]

She is back!!! She is soooo coooll, bring her back pleeeassee!!

[@0brokeJaw]

You just got to take the "L" on this one

[@Luca_5425]

@@0brokeJaw wut? Outta nowhere, why that bud?

[@0brokeJaw]

@@Luca_5425 Your profile pic is an "L."

[@Luca_5425]

@@0brokeJaw fair enough

[@Triantalex]

??

[@WolfRose11]

When you catch that she wrote the same thing as the beginning at 9:40 but drew what she intended. They must have noticed because they switched paper a few seconds later to the one with P(on) = 1.

[@SometimesDrawings]

This is an opened door to a branch of knowledge I knew nothing about, and is mild-blowing. Thank you!

[@alecj3454]

Ayliean is such a good explainer. I love her appearances. This was a fun one.

[@user-el4np5xt8c]

Love the accent!

[@Shad0wLucky]

9:43 Sneaky edit (fix) on the mistake of writing P(on)=0 and P(off)=1 ;)

[@AB-Prince]

each vertex has only two edges (in the square arrangmeny) meaning there is no point where three edges meet, thus no point where three regions meet, therefore every arangement must be two-colorable

[@TheAstronomyDude]

I love it! I hide my passwords in plain view as artwork and this would look great. None of my guests have ever asked me if there is any significance to my geometric statutes or my chaotic wallpaper.

[@xyz39808]

does your off‎ice also have secret doors that activate when you pull a certain book?

[@liamogrady5868]

This brings the idea of naming your clothing to a whole new level

[@mouwersor]

That flex of casually writing out tens of digits of pi

[@ShankarSivarajan]

With "May I have a large container of coffee? Thank you …" you too can do ten digits.

[@TranquilSeaOfMath]

@@ShankarSivarajan ☕😀

[@Triantalex]

false.

[@_wetmath_]

this is some vihart stuff right here, amazing

[@KaiKunstmann]

4:03 On an infinite plane, every node in this construction always has exactly two edges (one vertical, one horizontal). Both edges part the local region around a node into two areas. Every string of edges must either be a closed loop, parting the plane into "inside" and "outside", or it must be an infinite string, parting the plane at infinity into "this" and "that". There may be multiple closed loops, multiple infinite strings, and even nested loops, all of which still have exactly two different colored sides of their string of edges. The necessity for three colors can only arises from nodes with an odd number of edges.

[@tudibelle]

Thank you for this, I love patterns and embroidery, it is so nice to see these two brought together, and I love that I can use them as a physical representation of words. Going to have a play with this 🥰

[@ultrakatiz]

I used to doodle those on my notebooks, although with a different method. I would draw a rectangle with two relatively prime numbers as side lengths, and then draw a line making a 45° angle with the sides starting from a corner, going on / off whenever i hit one of the sheet's small lines. It is important that the two numbers are relatively prime, or else you will finish in a corner without passing through all of the diagonals. I never bothered trying to find if each pair of relatively prime numbers give a unique pattern or not, but it could be interesting to prove or disprove.

[@nemecsek69]

Have you got an example somewhere to see? I don't understand the algorithm and very curious about it...

[@wobaguk]

Im wondering what happens to the average of the shape size (if a square or triangle is 1) when the probability changes, is it constant for sufficiently large grids because as you make a big shape you break others? Or is there a peak around max randomness. I guess number of shapes per starting area is basically the same question just the inverse, because fewer shapes mean bigger area.

[@hurktang]

Yes it seems like, given a big enough distribution, the mean size of the less common color will tend toward the more common color when the randomness is maximized. I was observing the same thing.

[@biggiemac42]

Yeah this is the question I'm most left with by this video. During a numerical methods in physics course I took, we had something similar with porous materials. A phase transition when the probability a cell is empty gets high enough, when you go from it being nearly impossible to find a clear path from one edge to its opposite, to it being nearly certain. I imagine something might apply here that is similar

[@SaveSoilSaveSoil]

Yup! It's hard not to think of the Ising Model.

[@lindsey3038]

This would be a really cool way to design a crochet pattern. Kinda like a temperature blanket but more pretty. Now to start another WIP lol

[@_rlb]

Ayliean has instantly become one of my favorite Numberphile persons. Nice shirt by the way, are those the Pleiades?

[@Nickt01010]

A Sierpiński Triangle, sir.

[@_rlb]

@@Nickt01010 ah yes I know, thank you, but the stars look like the Pleiades cluster ;)

[@everydaykronicler9974]

I'm so happy that you started your fourth ISO with the exact same writing as the first and then after you had begun adding the third axis we get a jump cut to a different ISO with the correct writing 9:41 and 9:45

[@TwoCherriesIns]

I love the combination of art and math; this video is amazing.

[@sethgilbertson2474]

Oh man, I LOVE this! I love playing with math in ways like this!

[@NogueiraVini]

Its beautiful! Reminds me of the amazing embroidery of Peruvian elders Shipibo Conibo. Incredibly detailed, handcrafted patterns inspired by shamanic visions of Ayahuasca

[@Syntax753]

What a fantastic video! Thanks for putting this together... will be using this technique in my "game development" hobbying :) I say game development but mean "terrain generators" and leave others to build games on top of that :D

[@matthewgough9533]

The famous brown paper has a Christmasy snowy white tint to it.

[@gizatsby]

Aaa! I've been following her on TikTok for a while. Great to see her on this channel

[@bioZone101]

These patterns are so soothing to look at, and I imagine to draw as well!

[@palibebeh]

'who wouldn't want that stitched on the front of their shirt' Numberphile t-shirts with Hitomezashi Stitch Patterns on the front confirmed.

[@vigilantcosmicpenguin8721]

They have to do it now.

[@michaeldeierhoi4096]

Very cool. This ended up being complex yet with an underlying simplicity.

[@darreljones8645]

I just LOVE Ayilean's Scottish accent!

[@annabarabanna7668]

Wonderful. Thank you.

[@danibee535]

loved this one, makes me want to write a program to draw these! thanks, ayliean and crew!

[@danielemazzali9810]

4:05 It's Always 2 colors because to have the Need of three colors there should be an intersection point with at least three edges. This never happens. For every point there are always exactly two edges. So you'll never need a third color.

[@presbarkeep]

what if its plotted in 3 dimensions, X, Y, Z?

[@danielemazzali9810]

@@presbarkeep Idk, i've only ever studied plain graphs xD I don't even know if this pattern would still be duable. But if it is i'm pretty sure you'll always would have to use only two colors, because to need a third you would have to have an intersection point with four edges, and in the pattern every node would have 3 and only 3 of them.

[@myvh773]

There are graphs which vertices all have degree at most 2 that are not 2-colorable. For instance, an odd cycle.

[@danielemazzali9810]

@@myvh773 Oh yeah. But here we are coloring regions, not nodes. I used a bad terminology i guess...

[@alexchan3287]

9:44 something magical happened

[@mataichi14]

In the colored pattern there is an 8 acting like an odd number in Pi. I drew it out to use as a cutting board pattern and it came out different twice. I thought I was doing something wrong the first time but got the same thing the second time. It’s surprising how much the pattern changes from one line being switched.

[@HandyClock]

Simple proof that it's two-colourable: For every dot (where a stitch ends), it will always have exactly 2 stitches touching it. (Because they alternate over-under-over). Therefore, only 2 "areas" touch at each vertex. Since you never have more than 2 areas touching, you can always alternate from one colour to the other. ...I don't know how parseable that is.

[@GaryDunion]

The iso one looks like it would be three-colourable, is that right? And could you keep generalising that as # of axes = # of colours required?

[@HandyClock]

@@GaryDunion I believe that's right, yes.

[@walterkipferl6729]

I could parse it! Nonetheless, my try to make it more understandable: The only way for this to not be two-colorable, is to have a vertex, one of the points, where three faces or areas meet. With just up to two faces meeting at every corner (and obviously up to two faces meeting at every edge or line) you can always find a two-coloring. Now, every stitch follows the on-off-on-off pattern. So, at every vertex, there is either a line coming from the top or going out the bottom. Similarly, every vertex has either a line to the left or to the right. So, at every vertex, there are exactly two edges (except maybe the start and end, but those don‘t matter here.). Thus, the patterns are two-colorable. QED And this does generalize to the iso version. Three-colorable because at most (or rather exactly) three edges at every vertex.

[@jeremylakeman]

Following a line around the plane, it must either stretch to infinity or form a closed loop. Because if it formed a spiral, there would need to be a vertex with either 1 or 3 line segments at the end, which is impossible.

[@jordanweir7187]

thanks bro im pleasantly surprised its that elegant, nice work

[@sreid32]

Thanks!

[@jasonremy1627]

This is such a beautiful process to watch.

[@BinaryDash]

This is by far one of my favourite channels

[@hedger0w]

9:30 Matt be like: "Hang on a minute!".

[@828burke]

This would be great for designing a quilt pattern. I made one of a Hilbert curve (as the line between colors) using 3 types of 2 by 2 blocks and I could do something similar here using only 2 if I'm thinking it through right.

[@828burke]

@Doc Brown I thought of ONLY doing the line and having it be two pieces soldered along that and was like... maybe if you have magic

[@beesareLameWasps]

I'm not much of a math person after high school, but I love these videos. They're so interesting and well explained. ty!

[@emmaperry4867]

More of these videos with Ayliean!

[@harpalvaghela2798]

lovely nail paint there!!

[@iamavataraang]

Yet another way I can enjoy using spreadsheets

[@fedesartorio]

Great video! I’m no programmer but I’d love to see larger patterns generated like this, anyone has an idea on how to make this for example in Processing or any other free software?

[@macronencer]

This is one of my favourite videos of yours, thank you! I've been interested in steganography (hiding messages in plain sight) for a while now, and I have a design on a mug that encodes my own name, which I made myself in software and ordered from Vistaprint. Hitomezashi stitching is a lovely new steganographic technique to add to my collection :)

[@Cellottia]

How intriguing! I shall look up steganography...

[@00000ghcbs]

Wait but how do you invert the process...?

[@macronencer]

@@00000ghcbs Well, it's easy enough if you can see the whole pattern - you can read the code from the edges directly.

[@00000ghcbs]

@@macronencer I mean it is a way hide binary messages, but I guess you also need to read it using ascii or something

[@macronencer]

@@00000ghcbs Yes, my usual approach is to simply use 7-bit ASCII, though of course the encoding is a matter of personal choice. The idea is to make the message easy to decode, despite being hidden from view at first. It's fun! I have a mug with a design on it that involves a weaving line, turning left for 0 and right for 1. It spells my name on one side, and on the other side is a similar design that simply says "NERD!" :) I call it "Turncode", and I spent a long time getting the software just right so that it optimises the path and makes a nice compact pattern. It was a great project.

[@MisterMobius]

# I enjoyed this video so much that I had to write some python code # to print random Hitomezashi stitch patterns import numpy as np import random def stitch(rows=24, columns=16): pattern = np.empty((rows, 4*columns), dtype='object') horizontal = '' vertical = '' for i in range(columns): horizontal += '_ ' for k in range(rows//2): vertical += '| ' if rows % 2: vertical += '|' horizontal = np.array(list(horizontal)) vertical = np.array(list(vertical)) for l in range(rows): pattern[l] = np.roll(horizontal, random.choice([0, 2])) for m in range(2 * columns): pattern[:, 2 * m + 1] = np.roll(vertical, random.choice([0, 1])) for v in pattern: print(''.join(v)) stitch()

[@marika147]

My bilingual brain cannot accept Y as a consonant. 😂

[@djudjux3936]

Same here. It's a vowel. Period.

[@Einyen]

Yeah, I never heard about Y as a consonant, but then again English is not my 1st language. But in Danish we even have 3 more vowels besides a, e, i, o, u, y which is æ, ø, å. If you do not have a Danish keyboard, they are written by 2-letter combinations: ae, oe and aa.

[@marika147]

@@Einyen Polish also has a few extras! Ą ą, Ę ę, and Ó ó! And don't get me started on all the extra consonants 😂

[@peterbrockway5990]

"y" at the start of a word or syllable in English often marks a sound that is described as a "hard y" or semivowel. Unlike the true consonants it nonetheless clusters with them. For instance we have an antelope, a bear, a cat, a dog, an emu, ..., a yak. Not "an yak".

[@JohnGolden]

Isometric variation and controlling the randomness is AWESOME.

[@rahulmooley3298]

i'm going to draw a polar version of this. it'll be a bunch of concentric circles with an increment of one or half a centimeter in radius. every circle will be divided by radiating lines seperated by 10 degrees. same on/off logic. will post it on reddit and paste the link here. thank you for the idea

[@Projacked1]

Very cool, we learn everyday :)

[@romanski5811]

I wish this video were an hour long with Ayliean drawing random patterns

[@artertemisartzetakis3077]

I love this!

[@mathphysicsnerd]

I see plenty of people have tackled why the square stitchings are always 2-colorable, so I'll generalize the logic a bit to point out why the triangular stitchings are only 4(+)-colorable. Noting that each stitch has a gap both before and after it, on the (infinite) triangular plane we can see that each point must have 3 stitches connected to it, else the pattern would break along one of the axes. The stitches divide the plane into regions. These regions necessarily touch at least 3 points. Each point in isolation is 3-colorable, but since the regions necessarily share points the stitching overall needs at least 4 colors.

[@livedandletdie]

At least it's a finite color map. Unlike 3D and higher space.

[@falpsdsqglthnsac]

and the four color theorem means that it will never need more than 4 colors

[@padenzimmermann1892]

I made one of the golden ratio and my siblings names. Loved it thanks :)

[@HontubeYT]

I used to make these patterns using my friends' name and found some patterns. Now I practically have degree level knowledge in these patterns. Also I can take any pattern of my choice and convert it to hitomezashi code (literally any pattern)

[@maxlindemann1429]

As nice as it is to see the patterns created live on paper, this video really would have benefitted from computer illustrations to easily generate large patterns with different probabilities

[@herosnts]

Beautiful

[@piotrarturklos]

Those are beautiful, they have a kind of a harmony to them even though the inputs may be random.

[@thetommantom]

That looks like my front yard like when I rake the leaves and how they pile up in the wind and the slight hills and like walking steps

[@Nuovoswiss]

I would love to see a follow-up on this, extending these patterns to different dimensions or grid angles.

[@Astromath]

I think this is the first time I saw it live when a Numberphile video released. Usually I'm a few hours late

[@EverlastingPinecone2]

It was very interesting to see the mathematician’s perspective on these patterns. In preparing these patterns for stitching you typically draw a grid like graph paper and as you stitch row by row you decide for the next row to be in or out of phase relative to the previous row instead of relative to the grid itself

[@abuk95]

This is amazing! I have to create a program that visualizes different inputs. It reminds me of Wang tiles. Also, I didn't know that in English language the 'y' is no a vowel.. might learn also something else than math from this channel, nice.

[@venuscarey]

>in English language the 'y' is no a vowel But it is! ...sometimes. Y is a consonant when it makes a "yuh" sound, like in "you"; Y is a vowel when it makes any vowel sound (usually "ee" or when it's present in a diphthong), like "baby." That's why when listing the vowels, you might hear someone say "A, E, I, O, U, and sometimes Y."

[@gumbykevbo]

Indeed. If Y were not sometimes a vowel, then my, by, try, wry, etc. would violate the principal that all english words are required to have at least one vowel.

[@talideon]

There's a difference between orthographic vowels (what you see written) and phonological vowels (the sounds you make). Orthographically, 'y' and 'w' are semivowels, meaning that they represent both phonological vowels and consonants, depending on how they're used. Interestingly, there are some more consonants n that can behave as vowels in the right context, as they have a vowel-like quality to them. These are the liquid consonants, 'l' and 'r'. To give you an example, in many rhotic dialects of English (those that don't drop their 'r's), the 'r' in 'nurse' behaves as a vowel: if you sound it out, there's no 'u' there, and the 'u' is just there as an orthographic 'carrier' for the 'r'.

[@TuberTugger]

You read my mind. I'm also going to start programming the visuals for this. I'd love to see this pattern scrolling sideways with constantly generated numbers.

[@crumble2000]

In the word 'yummy' the letter y is used first as a consonant, then as a vowel at the end

[@davidgillies620]

Of course I now have to go and code this up in Mathematica.

[@belg4mit]

Y is a semi-vowel. In "may" it is a vowel, as it it modifying the sound of the A. In "you" it is a consonant; A, E, I, O, U, sometimes Y and W.

[@erictaylor5462]

For Y, which is sometimes a vowel the way to know when it's a vowel is my how sound. If it sounds like a vowel it is a vowel. So in the word "yes" Y is not a vowel, in the word "my" it is.

[@ramdevgohil4757]

So beautiful! I'll surely try this and teach to my siblings and parents. Thank you for bringing this Math-Art here!

[@jmiquelmb]

I guess that this could be a very wacky numerical system, since with only a triangle and a few lines you can write up to 20 numbers, and it's also scalable. Wonder if you could find some arithmetic properties combining numbers into shapes. Or maybe even find ways to do addition or multiplication?

[@Lykrast]

That glitter nail polish is legit soooooo cool!!

[@Igahwanoiar]

I belief the 50% maximum randomness can be explained by the entropy ( H(X)=-sum(p(x)*log(p(x))) ). Maybe if we'd take a closer look at the joined entropy between the axis, we could predict something about the pattern that emerges

[@TuberTugger]

I don't believe it needs any proving. It is pretty intuitive. If you flip a coin that's weighted to one side, it doesn't matter which side.

[@ezg5221]

@@TuberTugger Math is all about formalizing intuitions. Information theory is scarcely a century old in part because of that dismissive attitude. If you ever feel like you really understand a loosely reasoned argument, go and collect your Fields Medal.

[@TuberTugger]

@@ezg5221 Math is about elegance. Not heavy handedness. Don't try and justify over engineering. That's childish and arrogant.

[@ezg5221]

@@TuberTugger Right, so this is just about the Constructivist vs Intuitionist argument that plagued mathematicians of the 20th century. "Don't be a conspiratorial looney" vs "Don't get lost in the weeds". I'm a programmer, so I must be a conspiratorial looney in situations where deterministic logic is too restrictive, but I must also provide a witness for my ideas so the compiler knows what I'm on about. My only advice to you is to learn to tread water, so you'll never be afraid of drowning. The computer can choke on the symbols for all I care; I just need a language that's precise enough that I don't have to explain myself further. Mathematics is an ancient field, precisely because it is a concentration of natural, logical thought. We use a terse notation, because written English has a habit of saying more than we meant and simultaneously, being too vague to communicate anything specific without expounding further constraints. That's why Socrates held disdain for the written word; you can't ask it to clarify what it meant.

[@peterbrockway5990]

Suppose we came (however unreasonably) to the belief that max randomness occurred at 40%. We might test that not by drawing, but with actual "under/over" stitches with 40% starting with under stitches. Now turn the fabric over (and possibly look at the result in a mirror).

[@JRH2109]

Be great to see good old Matty Henderson get this animated on his computer.

[@WestExplainsBest]

A blended class of geometry and art should be taught at the secondary level. It would be extremely intriguing and engaging for students!

[@jeffreyleonard7210]

Yes yes yes yes! To help non-math kids grow confidence and trust in numbers, and to help numbers kids respect art

[@jeffreyleonard7210]

Keith Haring art, no?

[@travismiller5548]

This kind of reminds me of a drawing I once did... also based on a bit of Japanese culture: the origami crane. I had to do a drawing to help me visualize the careful and specific tearing of paper I would have to do to divide a square into multiple little squares (still joined at diagonally opposite corners) to create a chain of cranes, still joined at their wing tips, from a single square of paper.

[@oogrooq]

I feel like Vi Hart would love this.

[@rcapracp3867]

"I do love when things get chaotic." I'd buy that T-shirt.

[@josephmelnick3446]

Additional randomization sources could include musical notes in a song (is the next note higher or lower - ignoring repeating notes), or a digital image (grab each pixel and use some or all of the bits in some combination) to generate your 1's and 0's.

[@coliander4180]

Surreal stumbling across a fellow Invernessian on this channel!

[@flotador7]

I believe I said this in a previous Ayliean appearance on this channel: Mathematicians: Oh, cool fractal pattern on her clothes. Gamers: Oh, she has a triforce on her clothes! It's dangerous to go alone, take this!

[@addymant]

In this case, Y is definitely a vowel. There is no consonant sound at the end of "may"

[@gurrrn1102]

there is no consonant sound at the end of "through", either, but does that mean g and h are vowels?

[@ragnkja]

It’s a semivowel in that diphthong.

[@vigilantcosmicpenguin8721]

Yes, and I will not stand for Brady's anti-Y-as-a-vowel nonsense!

[@yaseen157]

1:11 "Uhh, I'll have a consonant please (Rachel)" - Ayelian Didn't know we were playing Countdown :DD

[@CamAlert2]

If you substitute for the fibonacci word bit pattern on both axes you get a fractal like pattern

[@MrChefjanvier]

Beautiful. And the patterns were also nice ;)

[@JCCyC]

"I do love it when chaotic things happen" MUST have been a line spoken by the Joker in some Batman story.

[@AngryArmadillo]

A proof of 2-colorability: Construct a graph by taking each connected region as a vertex, and connect two vertices if their regions are adjacent. The claim is that this graph is actually a tree (meaning that it contains no cycles). To see why, choose any vertex v and notice that v’s region partitions the plane into an interior and an exterior. Let u be in the exterior and w in the interior. Clearly there cannot be a path between u and w that doesn’t pass through v. Thus the graph has no 3-cycles. By inducting on the length of the cycle, it is easy to show that the graph cannot contain any cycles whatsoever. Finally, to show that a tree is 2-colorable, first choose an arbitrary vertex to be the root and color it red. Take all vertices which are an odd distance from the root and color them blue. Take all vertices that are an even distance from the root and color them red. That should do it! This was really more of a sketch of a proof but I think this is the meat of it :)

[@ilovethesmellofdbranesinth7945]

Alternatively you can do this: we can complete every curve into a cycle by extending it along the boundary if we have to. It won't matter how we do this. Then since we're in the plane every curve is the boundary of a region. Each point not on a curve shall be colored by the number of regions mod 2 that it's contained in.

[@AngryArmadillo]

@@ilovethesmellofdbranesinth7945 what about on an infinite plane?

[@Krekkertje]

Are we just gonna ignore her incredible skill for making very precise drawings on plain white paper? Also, are we just gonna ignore the lack of brown paper?

[@vcprado]

It isn't plain white, the paper has little dots

[@HarzemTube]

It's a guide paper with dots.

[@Qermaq]

@@vcprado are the dots at least brown? ;)

[@colinberg3342]

The brown paper is under the white paper

[@fusion67]

it’s just reeeeeeaaaaaallllyyy light brown

[@mikosoft]

Y is a vowel in my language! Stop messing with my brain! :D

[@bryanroland9402]

The progression of triangular patterns reminded me of the stages of entropy with the 50/50 pattern corresponding to the random swirls in the partially mixed coffee and cream. Is there any mathematical connection?

[@bb2fiddler]

Probability probably kekw

[@legendarygary2744]

Oh this is definitely becoming an activity when my brain is fried and I need to relax.

[@honkynel]

not entirely sure what i witnessed but really appreciated the skill with which the patterns were drawn. no tipex was used in this video.

[@atikahrostam5778]

I'm a pixel artist and I'm going to try using this in my drawing.

[@PC_Simo]

9:29 *WOW!* Now, *_THAT’S_* an oversight, if *_I’VE_* ever seen one. It’s supposed to be P(on) = 1; and P(off) = 0. 🤯 At least they corrected it @9:49 😮‍💨.