I wrote a quite long and substantial piece on AI, basic income, and labor rights regarding data creation. If you worry about AI-related job loss and AI’s impact on the economy, or if you often think about social media systems or the influence of tech companies, you’ll probably find a lot of references and ideas in here: https://theartofresearch.org/ai-ubi-and-data/
This is the latest in a series of unconventional research groups I’ve been a part of, following the Human Advancement Research Community at Y Combinator Research and the Communications Design Group at SAP, inspired by Xerox PARC. I just posted a history of our groups, their influences and the thinking that went into them, and how our current approach draws from our history: https://theartofresearch.org/a-history/
This is our 9th annual Pi Day video!
It’s March 14th, 2019. That’s 3 14 19, which as we all know are the first six digits of pi.
Now this is our 9th annual Pi Day video, I understand by now there’s some of y’all who know a few digits of pi, or maybe you’re the one who gets tasked with checking against a printed copy while that kid in math class tries to break the school record during the annual pi day competition, do all schools have that or is it just me?
Anyway you have the printed copy and they’re like 31415926535979… and you’ll be like wait, did they get that 8? but you don’t have time to think about it because they’re all like 323846264338… and you’re like well that all sounded right, unless they did miss that 8 earlier in which case all of that was wrong because once you skip a digit everything’s shifted, the 9 should be the 8 and the 7 should be the 9 and the 9 should be the 7 and there’s cascading failure, oh except when there’s two 3s in a row one of them is right just by chance, i mean the shifted version will be right whenever there’s two in a row or it’s right twice in a row when you come across three numbers in a row, and at the Feynman point which is 6 9s in a row then 5 of them will be right in a row, but point is once you skip a digit then what you’re reciting might sound like pi but it might as well be random digits.
Then again, the word mispelling is still the word misspelling even if we misspell it and miss the second s, it’s not like “oh no, you missed the s, now the p is wrong and the e and you got one L correct by chance but on the whole it’s a loss. We know what word it is, it still means what it means and we understand it even when it’s technically wrong, and maybe Pi is still Pi no matter what mistakes a Pi reciter might make.
So that’s why Pi Day 2019 has the first six digits of Pi and is thus even more pi-like than 3/14/15 a few years back which only has the first 5 digits. So there’s a typo, so what, let’s not nitpick. We can all empathize with skipping a digit of pi here or there, we’ve all done it, right? And if you don’t know many digits of pi may I suggest that you use digit skipping to your advantage and skip right to the Feynman point like this: 3141 999 999, and there you go, you know ten whole digits of pi, just not all in a row.
In fact, if you want to learn to recite a thousand digits of pi real quick, you can just skip everything that’s not 9s. Check the rules of your local pi day competition, as long as you’re reciting actual real digits of pi maybe it doesn’t matter what order they’re in or if you get them all.
Now you might think you could use this trick to recite infinite selected digits of pi, 9s forever, but no one has found a proof that that’s true. After a certain point you might run out of 9s, or maybe not, no one knows. Yet.
Anyway 31419, pi of the year, skips the 5, so if you do know a lot of digits of pi this year’s pi day challenge is to recite pi but skip all the 5s. It’s a little harder than it sounds, but it’ll help you get to the end faster because now Pi is 10% shorter. Or so we conjecture. And if you don’t know a lot of digits then your challenge is to recite pi but skipping to all the 9s and keep repeating 9 until the word loses all meaning and you have an epiphany that words are just sounds made out of our vibrating meat flaps and it is utterly impossible magic that they ever manage to mean anything. Good times.
Ok happy Pi Day, go eat half a pie, good luck with your life and stuff!
New video about Monkeybread! Secretly part of the Scutoid series, but mostly about Voronoi diagrams, and a little bit about the mathematical features of the Bundt pan. Hopefully you’ll never see baked goods in the same way again.
I wrote a little bit about the process here: https://www.patreon.com/posts/24390752
So say you’re ruining yet another batch of cookies because, who knows, too much butter? Not enough flour? Didn’t chill the dough long enough? Could be anything, there’s too many variables and this is why baking from scratch is hard and I’ll stick with mathematics thank you very much.
But I do know one delicious recipe that’s hard to get wrong. And by the way this video is in VR180 so use a headset or look around by moving your phone or dragging the video because today we’re making Monkeybread.
Monkeybread, aka puzzle bread or pull-apart bread, is a classic american food invented in the 1970s to take advantage of pre-prepared refrigerated biscuit dough for an easy-to-make snack suitable for groups of children and/or adults with no plates or utensils necessary.
I’ll be making the dough bits round to better simulate properties of Voronoi diagrams, but the basic idea is that each ball of dough is like a little cell coated in cinnamon sugar, and large amounts of brown sugar butter. Lots and lots of butter.
In the oven all these spheres of dough will expand and develop facets as they smoosh into each other, so they’re more polygonal and no longer spheres. What kind of shapes would you expect the cells to form?
Let’s go back to my batch of cookie, and I’ll use icing to draw the lines where the cookieblobs hit each other. It looks a lot like a Voronoi diagram, which is a kind of diagram where you start with a bunch of points, or, cookiedough blobs, and then it’s as if each point spreads out until it gets all the area that’s close to it, or at least, closer to it than to any other point.
If you started with points organized into a very efficient cookie packing like this, then the Voronoi diagram would look like a bunch of hexagons, except on the edges where technically the cell includes the slice of space going infinitely off the cookie sheet, not that I have enough dough for infinitely large cookies, which just marks another place where mathematical theory is better than the realities of baking.
But for our more randomly placed cookie blob sheet, the Voronoi cells are irregular polygons, and these look pretty typical for 2D Voronoi cells.
But what about 3D Voronoi cells?
There’s many theoretically perfect way to pack spheres together where they’d expand into perfectly fitting cubes or rhombic dodecahedra or other fun shapes, but when you toss all the dough balls randomly into a bundt pan we’ll get more typical random Voronoi cells. I mean it’s not quite mathematically Voronoi-y because of how dough works and physics but it’s similar enough that our Monkeybread bits will have that distinctive Voronoi flavor.
The Bundt pan, by the way, not only makes genus 0 baked goods into genus 1 baked goods, but the hole in the middle adds surface area, which is not only great for having lots of glaze or crust but essential for Monkeybread in particular so that more cells are on the surface. You eat it by just grabbing a cell and pulling it apart from the bread, and the toroidal shape means you can pick at it from all sides, including inside.
Bundt pans also provide areas of both negative and positive curvature to observe, which helps better simulate a comparison to the formation of epithelial cells, hence the Scutoid connection (more about scutoids next time).
Altogether, Monkeybread is quite the mathematical snack.
I was wondering whether to work, not work, or trick-myself-into-working-through-pretending-it’s-not-real-work today, and remembered that last year on MLK day I had the same conundrum and decided to fake-work by writing a post about a few of my very favourite authors: Helen Oyeyemi, Octavia Butler, and Toni Morrison, who whether through statistical improbability or otherwise all happen to be black women.
One year later, they are still a few of my very favourite authors, though none of them have published anything in the past year to talk about. Oyeyemi at least, who is the living author of the bunch, has Gingerbread coming out this March and I am eagerly awaiting it!
But then I realized that through statistical improbability or otherwise, I have three more great black women authors to talk about this year.
4. Nnedi Okorafor
Here’s another author where I am baffled that no one introduced me to her work before last year, especially because in addition to being a great writer of weird scifi/fantasy she has math bits in her work! I love all these things, why did no one tell me???
Binti is her scifi series with mathy main character. Akata Witch and Akata Warrior are a fantasy duology that is lighter and more accessible, very fun and has some memorable world building. Lagoon is more strange and literary with odd perspectives and oh do I love anyone who can write an alien or nonhuman mind.
I first heard her name connected to Who Fears Death, a book that I’ve heard is great and has lots of critical acclaim and stuff, but I’ve heard is an emotionally tough read, so for both those reasons I’m saving that for later. In the mean time, I’ve read six of her other books and plan to keep nomming up her catalog and whatever she writes next.
5. N.K. Jemisin
Jemisin is known for her Broken Earth trilogy, a wildly popular gritty post-apocalyptic fantasy series, and I’ll admit I have some conflicting feeling about it.
I stopped reading part-way through The Fifth Season despite the excellent writing because, well, it is very cruel and graphic not just in cliche “Look at me I’m a gritty fantasy book!” way, but in creative inventive new ways, and I don’t need creative new cruelties in my head. But people kept recommending it to me so I pushed through, and I’m glad I did because it smoothed out with less of the cruelty and more of the superb writing and worldbuilding.
I will definitely be looking out for whatever Jemisin writes next, and can heartily recommend the Broken Earth trilogy to anyone who can stomach, say, watching game of thrones.
6. Tomi Adeyemi
Woooo! Debut author on scene!
It’s not hard to read through Adeyemi’s entire catalogue because she just published her debut book, Children of Blood and Bone, in 2018. It won a lot of well-deserved awards. Definitely YA fiction, doesn’t have the strangeness or literary quality that would make it a favourite for me, but Adeyemi is an author I’ll be watching, and Children of Blood and Bone is a book I’d recommend to all fans of YA fantasy.
This book follows the standard formula fairly closely, but not stiflingly closely, and is overall well done and light and fun to read. The kind of book capable of winning popularity contests among books (which it did), but it’s not as shallow as most popular YA fantasy, and it has that magical feeling that I love reading.
I’m looking forward to Adeyemi’s next book, and even more so to her next series or single!
Oh my goodness, there are so many great books to read in the universe! And so many great authors! But Helen Oyeyemi is still my fav! Uggghh I am so excited to read all the books!!!!!!
New video, and also a new page about scutoids that is still a work in progress. The video is in VR180 and super fun to watch in a headset, if you have a google cardboard or something!
Script and description:
So you may have heard of… bees. Yes, bees are pretty great at making hexagons, as seen in this honeycomb that I regret picking up because now I can’t touch anything with my stickyfingers but anyway bees make not just hexagons but 3d hexagon pockets with depth, full of bee candy, so maybe they’re like hexagonal prisms, although not exactly cuz the bottoms of the candypockets aren’t flat but kind of faceted, you can see the facets real well in this beeswax candle that just has the bottoms of the cells,
But that’s not the point, the point is that hexagonal cells fit together really good to make a flat sheet, and it’s not just bees that use hexagons see here’s a wasp nest and besides the hexagon thing wasp nests are totally different, they’re made out of paper not wax, they don’t store wasp candy, they don’t even store food in here they’re just wasp pockets for more wasps, anyway, you might start to wonder what other things are made out of hexagons, like what about our cells? What shape are the cells in your own hand?
Usually when people talk about cells they draw this flat diagram that’s all roundy, but of course in real life cells are 3d, wait are eggs a cell? Well let’s just use it as the nucleus of a bigger cell, and then there’s other kinds of cell bits in there so let’s stuff those in, and while some cells might be roundy and floating around in your blood stream or whatever some kinds of cells are snuggled up next to each other with no space inbetween so they get all squished into really shapey shapes, even if they began life wanting to be round.
It’s kind of like monkey bread. You can start with these round cells floating in butterstuff, lots and lots of butterstuff to properly simulate the mathematics of course, and once the cells puff up and all connect to each other then they get less roundy and more faceted like a cell with shapes. Hmm… now that I think of it there’s a lot of mathematics in monkey bread, more on that another time.
Or maybe cells are like a bubble foam. Individual bubbles are all roundy like a free floating cell, but once you stick a bunch together they get all this shape stuff going on.
Anyway if you’ve ever played with beeswax you might notice that a sheet of hexagons is pretty good at rolling up, or bending one way or another, but it doesn’t like to be roundy. And animals have lots of roundy bits, and anti-roundy bits, and most animals aren’t made by bees, and roundy shapes just don’t like to be all hexagons all the time, like a soccerball has 20 hexagons but 12 pentagons too.
so long story short some scientists and mathematicians were wondering what different shapes a sheet of cells might be made out of and they did math to it and discovered that cells can sometimes be like a hexagonal prism, and they can sometimes be pentagonal prisms or other kinds of prisms, or pyramid-like prisms called frustums, but another kind of shape came up, one that didn’t have a name.
This shape has one polygon on one end, in this case a hexagon, and a different one on the other end, such as a pentagon, and to get from one to the other it has this little triangle. And you know how I feel about triangles.
And they named it the Scutoid. And yes, they scoot. scoot scoot scoot.
This shape is so new to the world of human language that I had to get in touch with one of the authors of the paper, Clara Grima, to ask whether scutoids had to be hexagons to pentagons, and it turns out scutoids come in many forms, but point is that scutoids are alive not just in your body but as a new subject of research being done by living mathematicians and computational biologists right now.
Scutoids are a brand new 2018 shape and winner of the 2018 Shape Awards, yaay scutoid, you won a new award we just made up right now, but anyway next time I’ll show you how to make your own scutoids, well, technically you’re making scutoids all over your body all the time but you know what I mean ok bye!
Scutoids are the Best Shape of 2018! This video is VR so if you have a cardboard or other headset you can see the shapes be even shapier in stereo, or looking around in magic window on a phone is pretty cool too. Or you can mouse around the video on a desktop.
Scutoids are officially named for the first time in this paper: “Scutoids are a geometrical solution to three-dimensional packing of epithelia” https://www.nature.com/articles/s41467-018-05376-1
Special thanks to Clara Grima for answering my scutoid questions, and to Laura Taalman and Tom Ruen who contributed to the creation of paper scutoid nets based off of 3D scutoid models by Laura Taalman, who invented the packable pair of identical scutoids (see https://www.thingiverse.com/thing:3024272 ) , and John Peplinski, who created the space-filling cairo tile scutoid design (see https://www.thingiverse.com/thing:3042446 ). More on that next time!
Also thank you Caleb Wright, Albert Wenger, Pat Devlin, David Perryman, Jade Bilkey, Chris Pierik, Donald “Chronos” King, Andew Romaner, Jodi Vezzetti, Carol Ghiorsi Hart, Andrea Di Biagio, Charley Sheets, Yana Chernobilsky, David A Smith, Michael Tiemann, and all of my patrons on Patreon!
You too can support my work here: https://www.patreon.com/vihart
One of the many things I learned from Evelyn Eastmond in my time working with her is this technique for deeply exploring, releasing yourself from preconceptions, thinking in new ways.
This technique is called “Make 50 of Something”, and to do it, you make 50 of something. In one big marathon. Maybe over a few days, but it’s important to do them one after another, exhausting your possibilities to break and illuminate habits. This is in contrast with thing-a-day or thing-a-week marathons, where you’re making habits and have time to think between things.
50 is a LOT, and it takes you on this weird journey of ups and downs and complexifying and simplifying and combining and taking apart. It can be oddly emotional and intense. It gets at something and teaches us something that is hard to find if we space things out or wait for inspiration.
This technique was meant for artists, but my research group has used it for VR stuff and programming language design too. So now to learn python, over the weekend I made 50 implementations of fizzbuzz.
Fizzbuzz is a classic kids game where you take turns counting, but if the number is divisible by 3 you say “fizz”, and if it’s divisible for 5 you say “buzz”, and if it’s divisible by both you say “fizzbuzz”. Now it has become a common programming exercise and interview question.
Fifty is a lot of programs, even if many are just 1 to 5 lines. It was an intense exercise. The first 20 or so are fairly normal programming exercises. But 50 is a lot, and things start to get weird.
Some of my fizzbuzzes output the solution. Some of them are flagrantly wrong. Some write you poetry, some play you songs.
Some favourites you might check out:
#22, which uses a linear regression and proves AI is smarter than humans.
#32, which plays a quite musical auralization of the algorithm.
#40, a chatbot that will listen to you and write you a poem.
#48, a complete text adventure with sound effects and monsters to fight.
Twitter user @quasiben was kind enough to figure out how to host an interactive version with all the dependencies.
To play with it, go here and click Fifty Fizzbuzzes.ipynb (it might take a minute to load).
This week we’re playing “collect what you might have tweeted into a blog.” Here we are:
Last night I had a dream that I dressed as santa clause for halloween
Amazing mace, How sweet the taste That saved a dish for three? It once was husks But now is ground Was bland, but now spicy.
HAVING A LOOPHOLE IN A LAW DOES NOT MEAN THE LAW IS BAD OR THAT ALL LAWS ARE BAD OH MY GOD
Charity, research, and content/media creation all have short term vs long term conundrums. Do you want to do something that’s helpful/useful/popular to the most people in the moment? Or do something that is less useful/popular in the moment, but potentially has a much bigger and lasting positive impact? Luckily we don’t have to stress over the decision, because long term projects are unfundable.
There was a twitter thread about cylinders and spherinders in which someone pointed out the unhelpfulness of the main image on wikipedia in visualizing the 4-dimensional spherinder:
So I made this image to help explain 3d cylinders to 2d beings:
Now I’m wondering what other helpful information can be found in Flatland’s Wikipedia.
I made a video about soup labels. People seem to like it.
So these soups were on sale and I was trying to figure out how to get the most bang for my buck and I noticed something odd. So I went down a little bit of a rabbit hole to see if there were a SOUP CONSPIRACY.
Now, people look at calories on products for all sorts of reasons; some soups are marketed as Light for people who look at 70 or 80 calories and think, that’s good, I want a low number, while some people think 70 calories is not worth the effort of opening the can.
A savvy consumer will look at the fine print and see that’s per serving, and a serving is one cup, and I don’t know if anyone in the universe opens up a can of soup and carefully pours out just half of it, but assuming you eat the whole thing like a normal adult who eats canned soup by themselves because they can’t cook and have no friends and are home alone with only their microwave for company, of course you eat the whole can which has about two servings. And who doesn’t expect to do a little arithmetic in the grocery store, two times 70 is 140, which if you want a low number, is still pretty low.
It’s kind of funny though because calories per serving means total calories divided by number of servings, so multiplying it by the number of servings is really just undividing it. Or maybe it’s division that should be called unmultiplication?
Anyway maybe you want high numbers, life’s too short and budgets too tight, so they market the rich and hearty soups with 150 calories per serving! Also about 2 servings per can, and 300 calories is a number worth opening a can for.
Funny thing though, both serving sizes are one cup but on the light soup one cup is 236 grams and on the rich hearty soup one cup is 253 grams. Do rich and hearty things weigh more? On the one hand it seems intuitive that yes, rich food is heavier, mm look how rich and gloppy, can you tell this video isn’t a paid sponsorship? But on the other hand fat and oil weigh less than water, foods weigh all sorts of things. Luckily we can look at the net weight of the product and huh, it’s the same for both soups.
So does that mean light food isn’t lighter? I don’t know, maybe there’s more of it in the can. But I do suspect there’s something going on with these serving sizes, meaning we can’t really compare these advertised calorie numbers, and we can’t just double the calories per serving to get the calorie count for the can either, so, so much for arithmetic in the grocery store.
See, for light zesty santa fe style chicken, arithmetic says 236 grams times 2 = 472 grams. Which is clearly not helpful as the total grams is 524, not 427. If we want to figure out how this 524 grams happens, we have to multiply 236 grams by “about 2”. So don’t get distracted by the suggestion of 2-ness, we don’t know what this number is… unless we treat it like any other variable and do algebra to it. So let’s just unmultiply this 236g, which means we unmultiply the other side too, and we get “about 2” equals about 2.22. Which is a very 2-ey number, but definitely not to be confused with actual two. So now that we know how many servings are actually in here, we can multiply this 80 calories by about 2 and get 177.6. So if you were expecting 160 calories you’re cheating by 11%. And if you were expecting 33% fewer calories than a leading competitor that has 140 calories, I have bad news.
But what about rich and hearty chicken pot pie style? One cup is now 253 grams. Times 2 equals 506, but times “about 2” equals 524. So once we unmultiply both sides by the serving size we see that this time about 2 equals 2.07, that really is about 2. and 2.07 times 150 calories is 310.5. So if you were expecting 300 calories you’re getting just 3.5% more than you bargained for.
So about 2 can be more than 2, it could be actually about 2, but what about less than 2? Is this can of black bean soup an organic alternative with the same amount of calories per serving as the non-organic soups? Obviously it’s a smaller can but maybe those organic black beans are really just that much more dense and nutritious. Serving size is still one cup, still “about 2” servings per container.
One cup equals 256 grams, times two equals 512 grams of soup with 300 calories. But 256 times “about 2” = 405 grams. That’s a pretty big difference. This “about 2” equals 1.58. This time, about two servings of 150 calories gets you only 237 calories, that’s 79% of the 300 you might be expecting when you read this label.
So according to this soup company “about 2” can mean anything from 1.58 to 2.22 and I wondered whether that was about the legal range of what’s allowed on the can so I went and read the FDA guidelines for nutrition information and learned lots of interesting things. Now between 2 and 5 servings you have to round to the nearest .5, but somehow numbers less than 2 aren’t accounted for here. Wonder how that loophole got in there.
Other fun marketing details: Light is in a little spoon, rich and hearty is in a big spoon. Organic is definitely an entire bowl that is a meal that is organic. Also the chicken soups are inspected for wholesomeness, but steak and beef is USA inspected and passed like a champ. Good job, soup.
But servings aren’t the only numbers with exploitable rounding rules. According to the guidelines any calorie numbers over 50 get rounded to the nearest 10s place, so this 80 might represent 75, or it might really be 84.9, which * 2.22 is 188 calories. And then again, there’s margins of error for how many grams in the can and in a serving and maybe they round the number of ounces first and then convert that number to grams and round again, which means the number of grams isn’t necessarily accurate to three significant figures, and I’m sure there’s margins of error for everything, basically who knows.
Let’s just take a moment to appreciate the layers here. If all we want to do is know how many calories are in this can of soup, we’ve got four strategies. A quick look at the label gets you an answer of 80 calories. “read the fine print and do basic arithmetic,” which sounds like due diligence to me, gets you 160. “read the finer print and solve an algebraic equation,” as ya do in a grocery store, bumps that to 176, and finally, “read the Department of Health and Human Services plus Food and Drug Administration 132 page guidelines plus do an advanced analysis with fuzzy numbers that even I couldn’t do without a special computer program” gets you to “up to 188+ but nobody knows, and that’s assuming their accounting is both correct and within the guidelines”.
I mean, just imagine you’re this soup company with a 188 calorie can of soup that you want to market as low-calorie as possible. You’re not allowed to round 188 down to 180 and you certainly don’t want to have to round up to 190, but by choosing the right amount of soup and the right serving size you can make sure you get a round-downablenumber of calories, ideally a maximally round-downable number like 84.9, and label it 80. So in one way, your number is accurate within 5 calories which is the rule. But on the other hand, you’ve rounded off those 5 calories more than once. Also use a maximally downroundable number of servings and you can shave an additional 10 to 12 % of calories off and there you go.
And careful number wrangling can trick you with other things too, like this reduced sodium soup that if you read the fine print still has 20% your daily value of sodium, times two is 40, plus algebra is 43% your daily sodium in one soup and that’s the reduced sodium soup.
I don’t know how much it matters if your numbers are a little off, most people are pretty far from the recommended daily values of everything anyway, but I find the math interesting and also the politics, like, if companies go out of their way just to tweak the presentation of some numbers by 10 or 20 percent, that to me is a sign of how successful the DHHS and FDA have been. I like that I can go to a store and pretty much trust that the food I buy probably won’t make me sick and that the labels are roughly accurate, so these agencies are a positive force for both public health and consumer trust, which is like food for economies, and economies are food for federal services, at least when digested properly through an educated tax-paying voting public, and that’s the kind of non-zero-sum feedback loop I like to see, everything gets better for everyone. Unless your feedback loop grows parasites who are too small to understand why cutting off their hosts circulation will kill it.
Ooh I just found a supposedly 50 calorie per serving can of french onion, 524 grams divided by 230 grams per cup means there’s 2.28 servings per can, so busted, that should round up to about 2.5 servings per container for the more accurate informational benefit of anyone who doesn’t want to waste 2/3 of their daily sodium quota on something that barely surpasses instant ramen for food content.
Actually, I take that back. This instant ramen is significantly more food-like than this particular canned soup, and somehow less sodium even with seasoning packet because their “about 2” actually equals 2, wow, I thought ramen was just an excuse to eat textured saltwater but this is something else. Ramen pro-tip, gently drop an egg or two in the boiling noodles for the last about 2 minutes, then it’s definitely food, lookit that protein and calories, and I like to mix it all up in the bowl so that the pot doesn’t need much cleaning. Mmmm. Eggs.
This video not sponsored by top ramen or its parent nissen, also not sponsored by… eggs? Whose parent is… chicken. This video is sponsored by viewers like you through patreon!
Anyway, go check out your products and see if you can find some interesting “about 2s” in your life.
Two videos since I last posted here!
so I noticed this way cool spider web in the corner of my couch and
*its making me have math thoughts*
Lookit the shape, its roundy but also antiroundy and now I’m gonna be stuck staring at this spider while my brain goes all
It reminds me of a wormhole, like, if you get sucked into the web what is on the other side? Besides mandibly death?
It reminds me of a pseudosphere, or maybe a hyperboloid, lookit that negative curvature, right? It’s roundy one way around the funnel but unlike a sphere which would be roundy around the same way it’s roundy around the opposite way at the same time.
it reminds me of this sculpture by Anish Kapoor called Vortex which I guess doesn’t photograph very well but trust me in real life it just sucks your mind into this incalculable void and you’re left standing there staring helplessly with your brain all fuzzy.
It makes me imagine the entire room could get sucked into the roundy vortex of spiderness, or at least the bugs in it, like flies are points in a vector field and the vector field is like choochoochoochoo into the funnel, or maybe it’s a spirally kind of vector field where everything spirals in…
I looked it up it’s a funnel web spider, not to be confused with the australian funnel web spider, and its funnel is more a cozy retreat for the spider to hang out in than a place where bugs get stuck, it’s not even sticky, the bugs can fall on any part of it or even just be near the web and the spider comes out and does noms. So its kinda like the web extends smoothly to become the couch and the rest of the universe…
It makes me think we are all already inside the web we just don’t know it yet because the spider hasn’t come to claim us,
it reminds me of a 4d torus stereographically projected into 3d space, there’s this cool way to divide space into like two infinite donuts, whatdya call it, ooh that’s what I’m talking about, thank you Jason, so you think you’ve got a regular donut but it starts growing cuz spiders and soon it’s infinitely big inside and everything is spider territory and there’s this funnel web spider hole but actually surprise, the rest of space that isn’t the donut is actually also a donut that belongs to reverse-spiders or antispiders or something.
I know that’s probably not what the spider had in mind when it made this web in the corner of my couch buuuuut I can’t help wonder when presented with such a shapey shape.
*it’s making me have math thoughts*
Ok postscript, no joke, I went to see if that Anish Kapoor sculpture was still on display at sfmoma and
the 5th floor sculpture gallery is now entirely full of spiders. By Louise Bourgeois because she knows what’s up. So here’s some free promotion for sfmoma, go do art to your brain its good for you bye.
This video is funded by my patrons on patreon, so thank you all of you, and if you want a tau is greater than pi shirt you’ve got one more day before we put in the order.
So some months back Grant Sanderson of the channel 3blue1brown invited me to take part in a collaboration where various science and math youtubers attempt a twist on a classic puzzle where the original goes like this: There’s three houses and three utility stations: gas, water and power. Connect every house to every utility, simple enough, except oh no if you cross two utilities they explode so do it with no crossings. People usually manage pretty well up to line 7 or 8 but number 9 seems impossible.
The timing didn’t work out for me to take part in the original collab, where the same puzzle is printed on a mug, but I finally saw the video, and I was surprised to find that my answer was different from anyone else’s. You see, I wanted to leave room for future infrastructure, so, sure we need to untangle everyone’s gas water and power but we also want internet that connect every house to every other house, and a separate secure network that connects every utility station to every other utility station.
So if you don’t know the original puzzle try that first, and if you do know it then you might want to check out the mug variation first, and if you’ve already figured that one out then you might want to pause and see if you can figure out the Vi Hart Infrastructure Special and make everything connect to everything, that’s 15 lines, three each for gas, water, power, social network, and secure network, without any of them crossing. But on a mug.
And one more thing before we start. The official version of this mug is sold on mathsgear and comes with a dry erase marker so you can make mistakes, but I’m a sharpie person and a lover of beautiful solutions so we need something that makes enough sense that I can draw in all 15 lines using permanent marker without worrying about making a mistake.
Ok so here’s the thought process.
First I look at the puzzle and see what associations come up. If you’re a science or math person you might think, oh, this has to do with graph theory, there’s lines connecting vertices and that’s like the definition of a graph.
For the mug variation, you might think, oh, maybe it has to do with topology, because there’s that joke about how a topologist thinks a mug is the same as a donut because topologically you can transform one into the other, as seen in this piece by Henry Segerman and Keenan Crane.
So the main associations come up, and depending on how much you know about them these secondary associations might come up, or maybe you just start trying things out, or maybe your biggest association is “hey, that’s the classic impossible utilities puzzle,” which might stop you from looking further than that.
The classic puzzle is supposed to frustrate you with your inability to get that last line in there until you either mathematically prove it’s impossible or come up with some other creative answer like that the last line goes through a house, or over a bridge, or through a hole in the notebook paper and around the back, which is kind of what the torus will allow us to do. I mean, mug handle.
So I know I can solve the original three utilities problem if I can solve the four utilities problem. And I’d know it’s possible to connect all six into what we call a “complete” graph if it’s possible to connect seven into a complete graph on the torus. And maybe this bridge of reasoning can land somewhere near the four-color map theorem which is connected to the seven color map theorem for toruses,
And I just happen to have a seven color torus right here to demonstrate. Every color section of this torus touches all six other colors, this is by Susan Goldstine and I’ll link a tutorial in case you’re into mathematical beadwork.
Ok so now we just need to connect the bridge. Each color area here touches every other area, see, it’s like… well we’d better bagel it,
So take the orange area, it touches yellow on one side, red on the other, then next to yellow is green which touches the orange part over here, and then purple, well let’s do pink, yeah so all the colors have two neighbors on their sides and they touch two other colors on one end and two other others on the other end so they all touch all six other colors.
Maybe next time we’ll use colored frosting and do a bundt cake.
Anyway then I just take the dual, which means I make every area a vertex and then I can connect them all to each other, so it’s really handy that I know a lot about dual graphs. And there we go, I can connect 7 dots to each other without crossing, which means I could ignore one vertex to get 6, and then delete the extra lines until we’ve solved the original mug puzzle.
So this went through my head within seconds of seeing the mug because all this math stuff is close to the surface of my brain, I have experience using these facts and making them accessible, but it’s not yet a solution, it’s a sketch of a proof and a method of finding a solution, but we don’t want just any solution but a pretty one that’s simple to draw. And that means I want to take advantage of symmetry.
So when I wanna put six points with symmetry on a torus I’ve got some options, we want something reminiscent of the original puzzle even though on the mug they’re all squashed together. But I’ll keep the idea of having three on top and three on bottom, and with something like this I can see the symmetry and triangulation, we connect these to each other and then we connect the top ones to the bottom, so that’d be like everything is squares if we unwrapped it, and then just do triangles to it, all symmetric like so it’s easy to visualize, every square is the same, I mean it gets squished when you do topology to it but I can visualize it as a pattern that’s all the same, and now we have more than enough lines to connect the utilities to all the houses.
If the mug had the little icons printed around the handle maybe it’d be more obvious how to start mapping it back on, or maybe if the icons were spaced around the mug and if the inside of the mug actually had a hole in the bottom so you could loop through that way, and then we wouldn’t even need the handle. But from here we can scoot over to the handle and see just one more interesting thing that the more common solutions don’t take advantage of and that we’ll need more colors to make clear, so let’s go ahead and do this.
Ok, we’ll start with internet because that’s important and it’s just two straight lines that circle around the mug. Then we’ll triangle up our other utilities between them, and now it’s just three lines left and we’ll just pretend they’re looping through the cup so they’re nice and organized, and then scoot them over to the handle, And now here’s the funnest part.
If just one line is going over the handle it might as well be a flat bridge. But we’ve got three lines, and notice that they don’t match up. On a flat bridge you’d have to worry they’d block each other from connecting, But our solution uses all of the 3d roundiness of the handle by rotating around it so the lines come out apparently in a different order on the other side.
So there’s my answer to the three utilities mug puzzle, and also the four utilities puzzle, see if you can rediscover it yourself without looking, and if you need another mug challenge try adding a fourth house.
“Topology Joke”: https://www.youtube.com/watch?v=9NlqYr6-TpA
Susan Goldstine’s bead crochet demo: https://www.youtube.com/watch?v=4DR7rNty1eQ
Susan has also collected a page of other artistic renditions of the 7-color torus and links to instructions for beadwork, fiber arts, and other further reading: http://faculty.smcm.edu/sgoldstine/torus7.html
This video not sponsored by or affiliated with mathsgear, but here’s where they sell the official version of the mug: https://mathsgear.co.uk/products/utilities-puzzle-mug