Yesterday was my last day as Mathemusician-in-Residence in Pfoho. I had to catch an afternoon bus, which meant I had the entire morning to try and figure out how to make something with these awesome plastic swords I got at the dollar store. Together, we came up with this:
With 45 more swords, this pattern could be extended into a full icosadodecahedron-type shape, where the sword blades make the triangles and the handles make the pentagons. With rigid swords, the shape would even hold itself together with no wire or glue. As it is, it makes a very nice hat:
And then it was time to go. I had a wonderful time, met many wonderful people, and had a big wonderful space to play in. Here's what it looked like as of yesterday morning:
I want to emphasize the improvisatory and collaborative nature of the week. Most of the time, I had some sort of vague plan/inspiration, and it was really through the help and input of other people that things ended up being awesome.
I'd say "Hey, I got these things. Let's try connecting them like this and see what happens. Then maybe we can hang it somewhere or something." We'd try it out, see what potential problems were, and do some bugfixing. The result was an amazing week and a big room full of fun stuff.
I'm interested in finding a longer residency of this sort. And looking forward to hopefully returning to Pfoho next year!
Yesterday I found some more laundry baskets, and we tried making something bigger. Here it is!
It's made of 32 baskets arranged like the faces of a truncated icosahedron (that's the shape of a soccer ball). The 20 "hexagons" are in shades of blue and green in a 5-color pattern. The 12 red baskets are like the black pentagons on a standard soccer ball. It's a little imposing to sit under. Here's how the dining hall looks, as of last night:
Yesterday I did this crazy workshop called "Playing Math" where I set up ten big tables with a pile of things on each, and after a very brief talk I had people get started playing. For example, one table had pirate coloring books and another had stove liners.
There were clothes hangers, dolls, clipboards, and protractors.
Here's some vertically-oriented photos.
There were also sponges, plastic containers, crayons, and wooden spoons.
The people at the crayon table were extra diligent, staying longer to figure out how to make a crayon-tip polyhedron and how to use the wrappers to fold intersecting tetrahedra.
Inspired by the three-fold symmetry clothes-hanger arrangement someone had made, I later grabbed some help and made this:
Saturday, February 19, 2011 at 06:14PM EST
Really cool stuff happened yesterday. In the afternoon, I met with the research group of my friend Erez Lieberman Aiden, and we made a truncated icosahedron out of balloons. He also took this fantastic photo:
Then it was umbrella time! I made the residents start unpacking the umbrellas, and went to get some materials. I know that my mathemusician-in-residenceship has had some effect, because when I got back, this is what greeted me:
Awesome. Then we got started on connecting the umbrellas into a hyperbolic plane. Between us all, we figured out a good way to build and display it. Then something magical happened. Some students had the idea of writing poetry on the umbrellas, so we got out some white-board markers and turned the hyperbolic umbrella surface into a big collaborative writing/drawing space.
I love collaborative sculpture! I'm thrilled with the result. It would have never happened without the creative input of many people.
Yesterday was Laundry Day in Pfoho. In the afternoon, I found these great laundry baskets in the dollar store:
I decided they should be a giant dodecahedron. I guess I'm at that point in my Mathemusician-In-Residenceship that I can gesture vaguely and people will know what I'm talking about, because after explaining just a little, I was able to sit back and drink juice while everyone else did the work (I didn't even have to take the photos! These are by the lovely Janice He).
The dining hall is beginning to shape up rather nicely.
The Harvard Crimson also put out this nice video covering the events from the balloon workshop the night before:
Thursday, February 17, 2011 at 01:45PM EST
I thought the Pfoho dining hall was in need of some redecoration. It was time for a mathematical balloon workshop.
Various groups of people wandered in and out of the dining hall throughout the evening, and we made a bunch of different things, including some snub pfolyhedra in Pfoho colours.
It came together nicely! Thank you to everyone who helped, whether it was by constructing, hanging, or cheering us on.
There's also a photo of me demonstrating some balloon techniques on the front page of the Harvard Crimson. I also just bought a ton of supplies for the workshop I'm giving on Friday, which should be pretty crazy, so don't forget to come by if you're around.
Wednesday, February 16, 2011 at 05:41PM EST
I'm at Harvard this week, as Pforzheimer House Mathemusician-in-Residence. Last night I hung out with a bunch of Pfoho residents and led Cut Stuff In Half Day, where we made all sorts of things out of paper, and then cut them in half. We started with the usual suspects: cutting a Mobius strip in half (and then in half again, or cutting it in thirds), and cutting a double loop in half (made from making a paper cross and taping opposite sides into loops; I first heard of this as a visualization exercise, but it's fun to do for real!).
If you take your paper cross (which you can cut from a square in a single straight cut, if you fold it in half three times) and put a mobius twist in each of the loops, the result is quite different! I first heard of this trick from Mike Caputo (who has an article about it in the Jan/Feb 1990 issue of TOMT), and there's a nice YouTube video by Daniel Keogh demonstrating the process.
Soon we moved on to exploring new shapes to cut in half. I especially like this variation, by Anne Goetz: start with an eight pointed asterisk, and connect opposite ends with a twist (be sure to twist them all the same way). The result was very surprising to me. We explored more, and it seems to generalize to all Mobius asterisks with 6 or more points. I'll let you try it for yourself, rather than ruining the surprise.
Another student, Carlos Rodriguez-Russo, found a variation that when cut in half yields a trefoil knot. And if that's not enough to keep you entertained all night, here's one last puzzle. What asterisk shape, twisted how, gives the result below when cut in half?
I'm quite excited by all the cool cut-in-half things we discovered last night. It's going to be a fun week.
At the Joint Mathematics Meetings last week, I gave a workshop on mathematical balloon twisting as part of a series of fun mathematical workshops presented by the Museum of Mathematics. I had everyone make a unit out of a balloon, and we put them all together into an awesome hyperbolic plane! Luckily Cindy Lawrence took some lovely photos:
I've been wanting to make this particular tiling for a while now, and as far as I know this is the first physical version of this tiling ever made out of any medium. Each vertex is surrounded by exactly three triangles, one square, and one pentagon. Needless to say, I was delighted with the result.
After the workshop, I added one last piece with Glen Whitney, MoMath executive director, and together we hoisted it up onto the top of the MoMath booth where it lived for the remaining duration of the conference:
Thursday, January 13, 2011 at 10:36PM EST
A couple weeks ago I gave a workshop to a pretty cool group of people at Brookhaven National Labs. We made what was, to my knowledge, the world's largest penny Sierpinki's Triangle, 8 feet to a side! I was invited by my friend Cindy Lawrence of The Museum of Mathematics, who also took a couple great photos:
It was pretty awesome.
Saturday, January 01, 2011 at 09:09PM EST
I was at my dad's place last night, and had a chance to check out his new makerbot. "What should we print?" he asked. Apollonian Gaskets were still on my mind due to my most recent video, and I thought doodling one out of plastic would be awesome! To take advantage of the 3d-ness of 3d printing technology, I decided each circle should be a hemispherical crater into a cylinder. George whipped up an .stl file for exactly that, and then we hit print!
This 3d printer is a robot that spits out plastic layer by layer. In the first photo it's about 3/4 of the way up. The second photo shows the finished piece. I think it's super awesome to be able to think of, design, and then print out a 3-dimensional mathematical sculpture during the course of dessert!
Saturday, December 04, 2010 at 11:54PM EST
An oil painting of mine titled "Hyperbolic Planes Take Off!" has just been accepted into the 2011 Joint Mathematics Meetings Exhibition of Mathematical Art:
"What does it look like when you crease the hyperbolic plane? This painting is an attempt at visualizing simple origami done with hyperbolic paper. Each plane has a mountain and valley fold perpendicular to each other. Done with your average Euclidean sheet of paper, it would be impossible to have both creases folded at a non-zero angle, but the hyperbolic plane can fold both ways at once. The creased plane can then be manipulated into different 'birds', or so I imagine."
I will also be speaking at JMM, giving a talk of the same title. You can find me somewhere in the vast program. If you're attending JMM, be sure to come say hi!
Wednesday, November 17, 2010 at 08:53PM EST
I wanted to do something mathematical with candy corn, seeing as they're such an iconic Halloween candy (and you know how I feel about Halloween) in such an iconic shape (and you know how I feel about shapes). They make a very nice Sierpinski's Triangle:
I built this starting from the top with one piece, and following the simple rule that for each layer, put a new piece below the spots where there is exactly one candy corn bottom corner. The process is explained with diagrams in a new page on candy corn.
When I was shopping for a pumpkin to carve into a dodecahedron, I came across a big bag of lollipops that boasted 12 different flavours. Continuing the spirit of Halloween-inspired math, there was really only one obvious way to proceed from there:
There are 60 lollipops arranged with icosahedral symmetry, five each of 12 flavours. The first photo shows the whole ball, looking down at one of the 3-fold axes. The bulkiness of the wrappers makes it difficult to see the structure from the outside, but an inside view centered on one of the 5-fold axes reveals that the sticks are carefully arranged.
Each of the twelve pentagons has one of the twelve flavours circling it. I still have a bunch of extra lollipops, so there may be some more lollipolyhedra in the future... but there's so much other Halloween candy to mathematically explore as well!
I've been studying group theory, and I thought: what better way to visualize the graph of a group than by making it out of balloons?
I started by making the dihedral group of order 6, D3. Just building the graph is not enough—I then used a sharpie to draw the direction of each edge. I might do more of this in the future, but I have to learn more first!
Friday, September 03, 2010 at 07:56PM EDT
The new page has some new photos and results of a couple fruit-drying experiments, such as slicing the apply from top to bottom, or removing the skin before drying. There is still much work to be done in this delicious field of research; let me know if you have any ideas!
I had some fun truncating apple Platonic solids, and added a page with lots of photos and mathematical description to the mathematical food index. This was the first time I got to literally truncate some polyhedra, so it was a lot of fun for me!
The large roundity of melonness made this easier than carving the apple, and I'd also had more practice! The lack of core also made it easy to carve out the faces to achieve Leonardo-style edible polyhedreality. For now I snuck these photos in at the bottom of the fruit polyhedra page, but I'm considering doing more advanced melon-carving in the future... I also know what I'm doing with my pumpkins this Halloween.
New Pages: Applatonic Solids and The Mathematical Food Index
I put up a page with instructions and pictures of making the Platonic solids out of apples. More importantly, this new sub-page is part of a new index of mathematical food projects, where I will be collecting not just my own forays into edible math but also linking to other cool math food on the web (let me know if you know any!). The page is currently under construction but I hope to get more up soon, with pictures and descriptions on the links.
Wednesday, August 25, 2010 at 09:41PM EDT
Since drawing the hyperbolic ghost, the image has been haunting my mind. Tonight I managed to catch her on camera.
I wanted a light and airy hyperbolic plane that would float in the breeze. I decided to try making one out of plastic bags. I cut them into heptagons, ironed them together, and voila! The ironing was not easy, but it worked. I took some pictures of the process and plan on putting together a webpage soon.
To simulate the ghost drawing in a photo, I dressed all in black, wrapped the hyperbolic plane around my waist, and belted it into place. All I need now are proper lighting, camera, and a wind tunnel!
I've been continuing to sketch, visualize, and explore the hyperbolic plane in all its beauty. I've been playing with giving the hyperbolic plane forms that are a little less abstract, perhaps even representational. This one started in my head as the ghost of a lady in white, though it would be better if you too could see the dress flow and ripple as she floats along her favourite haunting grounds.
After my first two hyperbolic figure studies, I wanted to imagine more complex forms, forms that loop and fold in a sort of hyperbolic origami. In doing this, I accidentally re-discovered surfaces like Schwarz's P surface, which as it turns out can be made by rolling up a hyperbolic plane. While both those things are already known, I'm still going to give major points to mathematical art because they just as easily might not have been. Hopefully after a few more series of sketches I'll be entering new mathematical territory.
I'm sure there's others of you out there who have, or at least understand, the compulsion to make all five platonic solids once you've made just one. They're currently sitting in the fridge, waiting for the sun to come back out so that I can cut them up some more and take more photos, and then put together a savvy webpage.
In other news, scientific testing of the hyperbolic properties of dried apple slices is progressing. More on that soon.
If you thought that perhaps I had taken a break from the hyperbolic plane, you didn't realize that hyperbolic planes are at the cutting edge of mathematical apple-slicing research. After seeing one of my hyperbolic figure studies, Patrick Desjardins, a brilliant and observant friend of mine, noted that one of them looked like a dried apple slice. This was a revelation to me. A couple google image searches later I knew we were on to something. I dried some apple slices of my own, to confirm.
This observation brings up two very important questions. Firstly, why do apple slices do this? There's some transformation that is turning a flat Euclidean slice into a hyperbolic one. One would think that in the drying process all cells would get uniformly smaller, which would make the whole slice simply shrink into a smaller flat slice. I'm currently conducting some more research to try to gain insights into this process, which will soon turn into a webpage.
Secondly, why is there nothing on the whole googlable internet about this? Is it really that no one has noted this before, despite that we see hyperbolicness every day in dried food such as potato chips? I can begin to answer this by asking why I myself never noticed this before, and by asking why people have in fact noticed the negative curvature of Pringles, which are relatively hyperbolic.
I think the answer lies in that people are not used to the many forms the hyperbolic plane can curl into, when embedded in Euclidean space. I myself didn't see these "wild" hyperbolic planes as what they were until I started playing with, visualizing, and drawing hyperbolic planes in different configurations, and even then it took someone else pointing it out to me, based on one of those unconstrained drawings. People saw it in the Pringle because the Pringle takes a more standard and recognisable saddle-shape, yet missed the connection that Pringles are an idealized version of potato chips, which have similar geometry. This speaks of the value of mathematical art, and collaboration between fields and people.
I've been wanting to slice up some cuboid fruit for a while, and today I finally got to live that dream. The puzzle is: how do you slice a cube so that the cross-section is a regular hexagon? I've documented the process in a new webpage, so that you can try it yourself!
When writing this music, I tried to capture the repetitive rotational feel in the process of both glassblowing and this particular instance of paper folding, as well as sneaking in elements of symmetry and mathematics. I see this video as a collaboration and communication between art, glass, music, math, paper, and people.
I may have developed a thing for the hyperbolic plane. After making enough physical models, I got a better intuitive sense of the kind of shapes that sections of hyperbolic plane can curl into, and decided to draw some shapes out of my head using chalk pastels (mathematical accuracy not guaranteed):
One neat thing about this is that by drawing freehand I have no limitations except for my own ability to visualize hyperbolicness; it tests my visualization skills and allows for forms that might not be found by other methods. I hope to take the concept further once I develop my drawing skills and hyperbolic intuition a bit more.
The Hyperbottlic Plane and Bottlonic Solids
After talking to Nick Sayers and seeing his wonderful mathematical art made with found objects, I realized I needed to find something that could function as a giant bead. I immediately thought of Mario Marin's method of putting soda bottles together. I varied the method to leave a hole through the bottles, used some ribbon to thread them together, and voila:
The first photo shows the three platonic solids that have triangular faces, and the second photo is of a hyperbolic tiling that has seven triangles around every vertex. I plan eventually to put a webpage together with more details and instructions. Mostly I'm glad that I had a stockpile of empty bottles lying around!
I meant to let this project sit for a bit, but this morning I found some better beads and couldn't help but make another hyperbolic plane, this time by tiling three heptagons around each vertex. It was so much easier than the one I made yesterday, and so fun to play with, that I decided instructions had to be put online immediately!
I put up a new webpage on hyperbolic beading so that you can make your own. So far the page only contains diagrams for the above model, but I hope in the future to try some other tilings and patterns.
To take a break from balloon hyperbolic planes, I made this one out of beads. Each bead represents the edge of a pentagon in the uniform tiling of four pentagons around each point. Beadwork hyperbolic planes have been made by others, though I couldn't find any existing examples of a close-knit uniform tiling like this. It's quite nice to play with. If anyone knows of any other examples of hyperbolic planes made out of beads, let me know!
I consider this a rough test for future mathematical beadwork, hopefully with nicer beads, colour patterns, and artistry. I was inspired to pursue mathematical beadwork by some of the very beautiful models I saw at BRIDGES this year, notably the works by Laura Shea and Bih-Yaw Jin. There are also some connections between computational balloon twisting and computational beadwork (if you wanted to minimize the length of the path the string takes).
Mathematical Balloon Workshop at BRIDGES 2010
BRIDGES this year was super super fun! I held a workshop on mathematical balloon twisting where participants learned a few models, and how to apply balloon twisting to teaching geometry and graph theory in the classroom. We made a large Sierpinski's Tetrahedron, as you can see in the photo. Thanks to everyone who attended and helped to construct it! (I hope to have more photos soon.)
I also put up a small page on Sierpinski's Tetrahedron. I'll be leading a workshop where we make a large one at the BRIDGES conference in two weeks. But first, I'm off to Paris to speak at the ESMA conference!
Martin Gardner (1914–2010) had a profound influence on my life. I have read and enjoyed his work, yet most of his influence was second hand. Many of those who taught me the wonder and beauty of mathematics—my father chief among them—were inspired by Martin's works, especially his column "Mathematical Games" in Scientific American. And the Gathering for Gardner, a bi-annual event in his honor, has kept me amazed and enthralled with the incredible things that us humans are doing and have done, since I first attended when I was 17. Though I am sad about his death, even more than that I am happy in celebration of his long and full life; that he could have such a huge indirect influence on me shows that his legacy will continue, and I am glad to be a part of it.
To get a sense of the kind of wonderful things he's inspired, check out reports from this year's Gathering for Gardner, such as this article by Bob Crease in The Wall Street Journal, or more recently, this one by Alex Bellos in The New Scientist. The building of the balloon snub dodecahedron mentioned in that article is shown at right, and the print version of contains a rather large picture of myself holding a balloon octahedron in front of my face.
The BRIDGES conference is always a lot of fun, but this year was especially so. Not only did I meet and hang out with lots of great people, give a talk, host music night as well as jam with lots of other mathematical musicians, and act in theater night, but I also got tessellated.
Yes, one of the many fun people I hung out with was Mike Naylor, the mastermind behind Naked Geometry. A group of us got together and tried doing some human geometry. Here is a cube we made with ourselves, and a lovely tessellation that Mike created out of me.
For comparison, you can listen to the theme in piano from the Harry Potter Septet. This section is inspired by Dumbledore's speech at the end of Book 4, and spans from the theme entrance to the end of the movement.
The Math Midway premiered last Sunday to great success! The Organ Function Grinder, for which I wrote the music, was a lot of fun to play with. I designed an algorithm which would transform an input melody, creating a mathematical piece of music dependent on how the dials are set and what ticket is input. This algorithm was then programmed into Mathematica by George Hart. I hope to get up video footage soon so that you can hear the results!
I also spent the day making magic balloon octahedra for people. Here are some pictures from the event:
A girl sets the dials on the organ function grinder, while her friend inserts a number ticket.
Making a balloon octahedron wand (orange), watched by a girl who just received one (light blue). In the background, people ride square-wheeled tricycles.
I wrote over 12,000 pieces last month! They were commissioned by the Math Factory for the Math Midway, an interactive mathematics exhibit which will be part of the World Science Festival Street Fair on Sunday June 14 in New York City, at Washington Square Park, from 10AM to 6PM. I'll be there!
The 12,000 pieces are for an "organ fuction grinder." You choose a number to input, and set some dials to various mathematical transformations. Not only does the number get put through the transformations, but a musical theme also! For example, if you add three to your number, the melody gets transposed up three steps. It's very cool to play with the Mathematica version, programmed by George Hart, and I plan to get an online version up in the future.
The rest of the Math Midway exhibit promises to be really fun too. There will be a tricycle with square wheels which, as you can try for yourself, gives a perfectly smooth ride! There will also be giant puzzles and other cool stuff.
Related to this paper is a seven-movement piece in which each movement is inspired by and contains the symmetry of each of the seven frieze patterns. The piece will ultimately be a piano trio, but three movements of the piano sketch are now available: