Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

plantearygears chords shapes

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

tumblr_nm56rdMlvl1uppablo1_r3_400 tumblr_noqxoi8EsC1uppablo1_400 tumblr_nolvf9dSt61uppablo1_400

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

doublecircles eyes
 fish  hearts

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

circle_prog_1B_AnimeYou can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

Julia Robinson

Julia Robinson

Julia's sister, Constance Reid

Julia’s sister, Constance Reid

Julia and Constance as young girls.

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. Bon appetit!

Continents, Math Explorers’ Club, and “I use math for…”

Welcome to this week’s Math Munch!

stevestrogatz

Steven Strogatz.

All of our munches this week come from the recent tweets of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused London’s Millennium Bridge to wobble on its opening day?”

On top of his research, Steve is great at sharing math with others. (This week I learned one great piece of math from him, and then another, and suddenly there was a very clear theme to my post!) Steve has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whose voice and vision can tell us about science’s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

NMFLogo_Horiz_RGB_300DPI2This Saturday, Steve will be presenting at the first-ever National Math Festival. The free and fun main event is at the Smithsonian in Washington, DC, and there are related math events all around the country this weekend. Check and see if there’s one near you!

Here are a few pieces of math that Steve liked recently. I liked them as well, and I hope you will, too.

First up, check out this lovely image:

tesselation1-blog480It appeared on Numberplay and was created by Hamid Naderi Yeganeh, a student at University of Qom in Iran. Look at the way the smaller and smaller tiles fit together to make the design. It’s sort of like a rep-tile, or this scaly spiral. And do those shapes look familiar? Hamid was inspired by the shapes of the continents of Africa and South America (if you catch my continental drift). Maybe you can create your own Pangaea-inspired tiling.

If you think that’s cool, you should definitely check out Numberplay, where there’s a new math puzzle to enjoy each week!

Next, up check out the Math Explorers’ Club, a collection of great math activities for people of all ages. The Club is a project of Cornell University’s math department, where Steve teaches.

The first item every sold on the auction site eBay. Click through for the story!

The first item every sold on the auction site eBay. Click through for the story!

One of the bits of math that jumped out to me was this page about auctions. There’s so much strategy and scheming that’s involved in auctions! I remember being blown away when I first learned about Vickrey auctions, where the winner pays not what they bid but what the second-highest bidder did!

If auctions aren’t your thing, there’s lots more great math to browse at the Math Explorer’s Club—everything from chaos and fractals to error correcting codes. Even Ehrenfeucht-Fraïssé games, which are brand-new to me!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAM—the Society for Industrial and Applied Mathematics—has just the video for you. They asked people attending one of their meetings to finish the sentence, “I use math for…”. Here are 32 of their answers in just 60 seconds.

Thanks for sharing all this great math, Steve! And bon appetit, everyone!

The Colorspace Atlas, allRGB, and Hyperbolic Puzzles

Welcome to this week’s Math Munch!

Update: A few weeks ago we met Dearing Wang, mathematical artist and creator of Dearing Draws. Now you can read a Math Munch Q&A with Dearing Wang.

OK, first up in this week’s post, do you remember when we talked about the six dimensions of color and the RGB color system? Well either way, consider this:

color-4

Artist Tauba Auerbach (one of my absolute favorite contemporary artists) made a book that contains every possible color!!! Tauba calls it “The RGB Colorspace Atlas.” The book is a perfect 8″ by 8″ by 8″ cube, matching the classic RGB color cube.

RGB_Cube_Show_lowgamma_cutout_aThe primary colors of light (red, blue, and green) increase as you move in each of the three directions. This leaves white and black at opposite corners of the cube, and all the wonderful colors spread around throughout the cube, with the primary and secondary colors on the other corners. You can read more here, if you like.

The book shows cross-sections moving through a single axis, so Tauba really had 3 choices for how the pages should flip through the cube. In fact, she made all three books!  Jonathan Turner made simulations of all three axes however, so we can see each one if we like. Can you tell which one is open in the pictures above?

That’s the Red Axis. Compare that to the Green Axis and Blue Axis.

For computer graphics, RGB color codes are ordered triples of numbers like (120, 15, 28). Each number says how much of each color should be included in the mix.  There are 256 possible values for each one, with values from 0 to 255. [Examples: (0,0,0) is black. (255,255,255) is white.  (255,0,0) is red. (127,0,0) is a red that’s half as bright.] Since there are only so many number combinations, computers have exactly 16,777,216 possible colors. That’s where allRGB comes in.

starry-night

Starry Night

hilbert

Hilbert Coloring

escher-reptiles

Escher LIzards

As they say, “The objective of allRGB is simple: To create images with one pixel for every RGB color (16777216); not one color missing, and not one color twice.” AllRGB is a bounded concept, since there are only finitely many ways to rearrange those 16777216 pixels. But of course there are a HUUUUGGGEEEE number of ways to rearrange them, so there’s lots to see. (In fact if you wrote a 1 with 100 million zeroes after it, that number would still be smaller than the number of allRGB pictures!! And that’s only part of the story)  Click the pictures above for zoomable versions as well as descriptions of their creation.

hyperbolic maze 1 hyperbolic maze

We’ve posted a little before about hyperbolic geometry. Very very briefly, the hyperbolic plane is a 2D surface where some of our usual intuition gets a little warped. For example, two lines can be parallel to the same line but not parallel to each other, which seems a little awkward. Click the images above to really experience what it’s like to walk through a hyperbolic world. David Madore created these hyperbolic “mazes,” which give you a birds eye view as you walk through a strange new land.

Finally, you might enjoy this old Numberplay puzzle with a hyperbolic feel, based on the movements of whales.

Gary Antonick asks "What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row."

Gary Antonick asks “What is the fewest-bun path between the two white buns? (The two white buns are the first and last — or 40th — buns in the top row.”

What do buns have to do with whales and hyperbolic geometry? You’ll just have to click and find out.

Have a great week and bon appetit!

Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

Wooden sphericonIt was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:How to make a sphericon

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

Sphericon pattern

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

ncbmastertitleI found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

epicycloidaThere are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String cardioid

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

Pi Digit, Pi Patterns, and Pi Day Anthem

pivolant1

Painting by Renée Othot for Simon Plouffe’s birthday.

Welcome to this week’s Math Munch!

It’s here—the Pi Day of the Century happens on Saturday: 3-14-15!

How will you celebrate? You might check to see if there are any festivities happening in your area. There might be an event at a library, museum, school, or university near you.

(Here are some pi day events in NYC, Baltimore, San Francisco, Philadelphia, Houston, and Charlotte.)

 

John Conway at the pi recitation contest in Princeton.

John Conway at the pi recitation contest in Princeton.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.

plouffe

Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

Wikipedia observes:

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon's math art!

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz and Peter's patterns.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

Bon appetit!

Sequence Day, Penguins, and a little folding

Welcome to this week’s Math Munch! And… happy Sequence Day!

Sequence day

If you didn’t know that today was Sequence Day, don’t feel bad– I didn’t know until I ran across this article written by Aziz Inan, an electrical engineering professor at the University of Portland. Why is today Sequence Day? Well, because all of the digits in the sequence 0, 1, 2, 3, 4, 5 appear in today’s date– 3-4-2015!

This particular Sequence Day isn’t super special. There will be another one with the exact same sequence on April 3 (4-3-2015). But, according to Aziz, April 3 will be the last Sequence Day of this year– and the last until 2031! Aziz made this chart of all the Sequence Days that will happen this century. There are 48 all together– and if you look carefully at the chart, you may notice some interesting patterns.

Sequences days this century

See how the Sequence Days mostly occur at the beginnings of decades, in the first half of the month, and never later in the year than June? Why do you think that might be? Also, the last Sequence Day of the 21st century is in 2065. That means we’ll have to wait almost 40 years for the next Sequence Day after that– until 2103! But, in the scheme of things, this actually isn’t so bad– there were no Sequence Days at all in the last century. (Why might that be?)

We all know about days like Pi Day (coming up soon on 3-14-15 — and it’s a special one because we’ve got those two additional digits this year!), but, as Aziz likes to show, lots of days can be mathematical holidays– if you just look carefully enough. Maybe you’ll find a mathematical holiday of your own! If you do, let us know. We love any excuse to have a party!

Next up, you think penguins are cute, right? Well, take a look at this:

You may have heard the narrator say, “Something more organized is going on.” Well, several mathematicians wondered what that more organized thing was… and it turns out to be very mathematical!

How many penguins do you see?

Francois Blanchette, a mathematician at the University of California, Merced, had the idea to use math to study how penguins keep warm while watching penguin movies like this one. He studies the math of something called fluid dynamics, which, basically, is how things like water and air flow. Francois and several other mathematicians at the University of Erlangen-Nuremberg in Germany noticed that when one penguin in a huddle moves just a little bit, it triggers a chain reaction in which all of the other penguins move in an organized way to keep warm. Their tiny movements cause the huddle to organize into the best shape for all penguins to keep warm during the cold of winter.

Huddle up, little guy!

Scientists and mathematicians are only now realizing all of the amazing ways that math comes into play in the lives of animals, especially in large groups. It seems that penguins are only the beginning! To learn more about the organization of large groups of animals, I suggest you check out this awesome PBS documentary about animal swarms.

Finally, we haven’t heard from Vi Hart in a while. If you’ve been feeling the need for some math art fun in your life, check out this video I dug up from the archives. Origami meets Pythagorean Theorem– what could be better?

Stay warm, and bon appetit!

 

SquareRoots, Concave States, and Sea Ice

Welcome to this week’s Math Munch!

The most epic Pi Day of the century will happen in just a few weeks: 3/14/15! I hope you’re getting ready. To help you get into the spirit, check out these quilts.

American Pi.

American Pi.

African American Pi.

African American Pi.

There’s an old joke that “pi is round, not square”—a punchline to the formula for the area of a circle. But in these quilts, we can see that pi really can be square! Each quilt shows the digits of pi in base 3. The quilts are a part of a project called SquareRoots by artist and mathematician John Sims.

John Sims.

John Sims.

There’s lots more to explore and enjoy on John’s website, including a musical interpretation of pi and some fractal trees that he has designed. John studied mathematics as an undergrad at Antioch College and has pursued graduate work at Wesleyan University. He even created a visual math course for artists when he taught at the Ringling College of Art and Design in Florida.

I enjoyed reading several articles (1, 2, 3, 4) about John and his quilts, as well as this interview with John. Here’s one of my favorite quotes from it, in response to “How do you begin a project?”

It can happen in two ways. I usually start with an object, which motivates an idea. That idea connects to other objects and so on, and, at some point, there is a convergence where idea meets form. Or sometimes I am fascinated by an object. Then I will seek to abstract the object into different spatial dimensions.

simstrees

Cellular Forest and Square Root of a Tree, by John Sims.

You can find more of John’s work on his YouTube channel. Check out this video, which features some of John’s music and an art exhibit he curated called Rhythm of Structure.

Next up: Some of our US states are nice and boxy—like Colorado. (Or is it?) Other states have very complicated, very dent-y shapes—way more complicated than the shapes we’re used to seeing in math class.

Which state is the most dent-y? How would you decide?

3fe3acace86442e7a0ddf5c7369f14dc.480x480x351

West Virginia is pretty dent-y. By driving “across” it, you can pass through many other states along the way.

The mathematical term for dent-y is “concave”. One way you might try to measure the concavity of a state is to see how far outside of the state you can get by moving in a straight line from one point in it to another. For example, you can drive straight from one place in West Virginia to another, and along the way pass through four other states. That’s pretty crazy.

But is it craziest? Is another state even more concave? That’s what this study set out to investigate. Click through to find out their results. And remember that this is just one way to measure how concave a state is. A different way of measuring might give a different answer.

Awesome animal kingdom gerrymandering video!

Awesome animal kingdom gerrymandering video!

This puzzle about the concavity of states is silly and fun, but there’s more here, too. Thinking about the denty-ness of geographic regions is very important to our democracy. After all, someone has to decide where to draw the lines. When regions and districts are carved out in a way that’s unfair to the voters and their interests, that’s called gerrymandering.

Karen Saxe

Karen Saxe.

To find out more about the process of creating congressional districts, you can listen to a talk by Karen Saxe, a math professor at Macalester College. Karen was a part of a committee that worked to draw new congressional districts in Minnesota after the 2010 US Census. (Karen speaks about compactness measures starting here.)

Recently I ran across an announcement for a conference—a conference that was all about the math of sea ice! I never grow tired of learning new and exciting ways that math connects with the world. Check out this video featuring Kenneth Golden, a leading mathematician in the study of sea ice who works at the University of Utah. I love the line from the video: “People don’t usually think about mathematics as a daring occupation.” Ken and his team show that math can take you anywhere that you can imagine.

Bon appetit!

Reflection sheet – SquareRoots, Concave States, and Sea Ice