Indexical Visualization

Indexical Visualization is about visualizing something by looking at the actual thing.  Most of the time we take the event and turn it into numbers (data), then we take those numbers and create a visualization out of them. The idea of indexical visualization is to skip the numbers part all together.

Here is a great indexical visualization to show how fast olympic swimmer Katie Ledecky was in her 800 meter Freestyle race.  But instead of just giving you the final times, the visualization is actually a recreation of the entire race.

Data Stories podcast presented a variety of wonderful uses of this idea within their fabulous interview with Dietmar Offenhuber about his work with indexical visualization. I really love the idea of removing the middle man, the numbers. How can we describe and visualize the information we need without translating to numbers first?

Below is another great indexical visualization of the microbes on an 8 year old’s hands after playing outside. This visualization was made by Tasha Sturm of Cabrillo College.Lastly, I want to call out the Pinterest board with more great examples of indexical visualizations. The tag line/description they use is, “physical embodiment of information, traces, evidence.”

What examples of indexical visualization can you think of? Is there anything that is easier to understand through indexical visualization? Or perhaps some things that are harder to understand if we don’t translate them into numbers first?


Posted in Art, Communicating Math, Exercise, Nature | Tagged , , , , , , , | 2 Comments

Math in the Media: October 2016

The internet had lots of great and terrible uses of math and mathematical visualizations in October 2016! This is our opportunity to applaud the winners and be confused by the blunders. Here are a few of my favorites:

1. The Gold Star goes too…

Without a doubt, this month’s Gold Star goes to the white house panel of “Math and Movies” that took place on October 28th. I have so much to say about this, that I wrote an entire article. Please check it out! My favorite quote in the whole panel was US Chief Data Scientist, DJ Patel speaking about math:

“It’s about the art. It’s about the humanity of making creativity come alive. It’s not the stodginess of just a set of formulas and equations.” -DJ Patel

Check out my whole article here.

2. Odd Use of “Mathiness”

Usually this is the “terrible use of mathiness” section, but the article I want to feature here isn’t particularly terrible math. I mean, maybe it wasn’t even terrible at all… But it certainly was odd. Very odd.  Chandra Kant Raju is an Indian professor who knows a fair bit about the history of mathematics. His work seems to revolve around crediting the correct person and how the societal pressures of the West stifled and altered the history of mathematics.

I found his current article published to The Wire. The Wire had republished it from The Conversation. And the weirdest part of the story is that The Conversation withdrew the article soon after it was published, citing editing standards. The Wire decided to continue to offer the version they published, but the back and forth of publication, withdraw, republication makes the article an oddity already.

In the actual article C.K. Raju presents arguments behind the bold title: To Decolonise Maths, Stand up to Its False History and Bad Philosophy. The article seems to call for a complete re-write of the history of mathematics, which is rather audacious. And C.K. Raju mostly sites his own publications as evidence. Which, I guess is what you have to do when no one else agrees with you? …But, it’s also something you do when you are really old and famous for a particular topic. So I don’t know what to make of that.

At first, I couldn’t decide if I wanted to feature it in Math in the Media because I wasn’t sure I wanted to provide it more press. But it’s was such a strange experience to read. I couldn’t not write about either. So, if you want to read something strange, then I recommend this article by C.K. Raju.

3. Math Graphic of the Month

My favorite graphic of October is actually the collection of infographics about treats for pets on Facebook by American Veterinary Medical Association. It’s adorable and very meaningful! The infographic does a great job of presenting a relationship that their viewers can relate to. It’s amazing!avma_healthy_weight_infographic

Did you have a favorite experience with math on the internet in October? Share it in the comments below! Until next time, have a mathy November!

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Math and Movies: White House Panel

On October 28, 2016, the white house hosted a panel on “Math and the Movies” where they spoke with DJ Patil, US chief data scientist, Andrea Hariston,  applied mathematician from NSA, Jeremy Irons, cast member from “The Man who knew Infinity”, and Ken Ono, the math advisor on the film. After the panel, they screened, “The Man who knew Infinity.”

“The Man who knew Infinity” is a film chronicling part of the life of Srinivasa Ramanujan, the Indian mathematician who is famous for 3 books of beautiful equations that he wrote down with no proofs. This film focused on Ramanujan’s struggle to break into Western mathematics. Whatever you may feel about the portrayal of mathematicians in this film, there are many things about this film which are to be lauded. For example, one cannot argue with the real struggle to be accepted by the mathematics community. I think this is something the film features quite well. It highlights the different backgrounds of the characters and gives some dimension to their struggles.

“An equation has no meaning unless it expresses a thought of God.” -Srinivasa Ramanujan

The screenwriter/director, Matthew Brown, was especially concerned about presenting the movie from an non-western point of view. In fact, “Colonialism and that white savior ideas are things Dev Patel, Jeremy Irons, and myself wanted nothing to do with, ” said Brown in a Tribeca interview. The panelist talked about this as well. How can we change culture to open up mathematics to more people? How do we draw young people into mathematics?

Ken Ono spoke about a recently created program, The Spirit of Ramanujan Talent Search. They have been searching the planet for undiscovered math talent. Their website is up and taking in applications. In fact, they have already found several young mathematicians to give awards to.

And just between you and I, panelist DJ Patel is amazing. He’s actually my new favorite person because he studied theater in undergraduate just like me! On mathematics, DJ Patel said, “I can’t imagine a more powerful foundation on which you can build so many different things.”  He spoke passionately about how mathematics can teach someone “how to be clever”, how to solve problems in creative ways.

“It’s about the art. It’s about the humanity of making creativity come alive. It’s not the stodginess of just a set of formulas and equations.” -DJ Patel.

Patel spoke passionately about changing the culture of mathematics. For those we see who have an interest in mathematics, Patel says that we should take a moment to say, ‘that’s cool. That’s awesome.’ because “that is going to systemically change the trajectory about how we think about [mathematics]”. I couldn’t agree more.  Let’s celebrate our mathematicians instead of marginalizing their talents. So DJ Patel, if you are reading this, can we be friends?

DJ Patel also presented first math homework problem to be given by the white house. They call it out with the twitter handle: #mathmovies. Actually, this is an amazing hashtag where you can find some fabulous gems like this response to a RedBox post about math movies:


Math culture is unique. There are problems associated with the culture of genius. “One of the biggest misconceptions in mathematics is that you have to be a genius to be a mathematician,” said panelist Andrea Hariston.  There are also problems associated with Western culture of rigor and structure.

“I think I’m a successful mathematician mostly because I’m resilient.” -Andrea Hariston

The panelists also call out the upcoming movie, “Hidden Figures.” Hidden Figures (trailer out now!) is about young female african american mathematicians who work with NASA to get Americans into space. As Andrea states, “Representation Matters.” From a personal stand point, I can not tell you how excited I am for the release of Hidden Figures.

Speaking of films, Jeremy Irons, the actor who portrayed G.H. Hardy in the film said,  “pure mathematics is rather similar to poetry…it’s something you search for.” Irons said he learned this from reading some of Hardy’s essays. And this is something that I find amazing. How an actor can, without any previous interest in mathematics, see the beauty in the math? Or perhaps I should say, he can see the beauty of mathematical thought and passion despite the mathematics. I think this is very similar to Ramanujan’s awe of mathematics, “An equation has no meaning unless it expressed a thought of God.” Those are Ramanujan’s words.


The recording of the panel is also available at

Posted in Communicating Math | Tagged , , , , | 1 Comment

Trade agreements of dining out

Social Mathematics can sometimes be considered Economics. And here is a lovely explanation of how two countries (or people) can share food when dining out. Enjoy!

I am eternally astonished to find not only that many couples I know failed to discuss this key area before they marched up to the altar, but also that many of them still have not developed a joint dining strategy even after 10 or 20 years together. This is madness. You are placing undue stress on your relationship, and you are very probably having a suboptimal dining experience, thereby wasting time and money and missing out on deliciousness. As a romantic economist might put it in a wedding-reception toast, couples have the chance to jointly move to a higher utility curve.

There is all kinds of math about sharing.  Mathematicians have carefully considered how best to fairly split a cake at many levels. We know that between two people the “you cut, I chose” method works fairly. But we also have studied how to cut a cake when it has multiple flavors and multiple people receiving pieces.

Conversing about and understand the various options of dining experiences is important! You want to understand the ordering options before you chose! Enjoy the complete article about dining out on Bloomberg.



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Math in the Media: September 2016

The internet had lots of great and terrible uses of math and mathematical visualizations in September 2016! This is our opportunity to applaud the winners and be confused by the blunders. Here are a few of my favorites:

1. The Gold Star goes too…

Henry Segerman for his amazing 3D mobius transformation henry_segerman_terdragon_smallartwork. In particular, I want to call out his Stereographic projections. A Stereographic projection is a mapping (that is, a function) that projects a sphere onto a plane. In the case of Segerman’s art, you can shine a light from above to make the projection appear.


Stereographic projections have a strong and beautiful connection to moebius transformations. Moebius (or Möbius, like the Moebius band!) transformations have a key part to play in understanding complex analysis. In particular, they encompass a particular type of mappings (functions) that map the complex plane back onto itself. Stereographic projections have the power to make something really complicated appear relatively simple. To really understand why stereographic projections are so meaningful, I recommend the “Moebius Transformations Revealed” by University of Minnesota professor, John Rogness. I promise that there are almost no equations and you’ll probably learn something:

Do you need one of these in your life?

2. Terrible Use of “Mathiness”

My least favorite mathematical graphic from September has to be the Bloomberg BusinessWeek’s visualization on income. Income is a really important topic. And the idea that “everybody thinks they’re middle class” is an important piece of the puzzle to understanding why income inequality is growing over time.  However, I strongly disagree with the way they presented the material.

First, let’s take a moment to wonder at the title: “Everybody thinks they’re middle class.” The visualization gives only 5 individual’s opinions, and yet we presume to call this a representation of “everyone”? In high school I learned that three points make a line… but hopefully we all realize that a 5 person sample out of more than 300 million Americans does not a statistical sample make! And since, with a sample size of 5, it’s not providing scientific insights, then I’m left to believe that it’s goal has to be emotional insights to the nontechnical audience… which is also does poorly.

How well does it communicate with a nontechnical audience member? The article is basically one big visualization with a little bit of text:

bloomberg_middle_class This is a visualization showing the annual income of Americans. Annual income is on the x-axis against the percentage of the population who have that income level on the y-axis. How to read it? Well, one can learn that approximately 10% of the population makes $30k and about 4% of the population make $100.  Basically, this visualization is like a histogram with a million little rectangles… drawn as a continuous line. However, it obfuscates what percentage of the total population are above and below each data point.

Our society has been very focused on the top 1%. So, when I look at this graphic, I want to understand where in this visualization the top 1% is. I’m also curious about other things, like: what is the mean income of this study? It’s really hard to tell when you look at the Bloomberg graphic. What I really want is something more like a Pareto chart or a cumulative distribution curve:

Here we get the histogram AND information about the cumulative values. So I can visually see when I’ve reached 50%, 80%, or 98% of my population sample by reading the right y-axis. Ultimately I think the Bloomberg visualization falls short of providing insights to anyone in both content and visualization. Better luck next time, Bloomberg!

3. Math Graphic of the Month

I’m guessing that if you are reading this website, you probably believe in math and science. Thus, I think you will also appreciate that science (and math!) do not have political agendas. Simply put, mathematics is a tool to learn about and communicate the facts of the world. In our social climate, I think it’s important to remember that there is a division between science and state. They have different goals and different aims. And math, if math had opinions and emotions (which is doesn’t!)… anyways, if it did, Math couldn’t care less about which way you vote. In short, I think this is a beautiful reminder:

You can buy yours here.

Did you have a favorite experience with math on the internet in September? Share it in the comments below! Until next time, have a mathy October!

Posted in Art, Communicating Math, Media | Tagged , , , , , , | 3 Comments

Zodiacs and Earth’s Precession

You may have seen recent posts claiming that NASA published some new results on the zodiac signs. Some of the bigger internet publishers, like Distractify, have popularized the story. The story says there is a 13th sign we haven’t seen before and all the zodiacs need to have their dates adjusted.

For clarity, NASA has no opinions about astrology. They never have and the won’t any time soon.  And there is no 13th sign. Snopes, bless their hearts for their wonderful works, called this out clearly.

However, your sun sign still might not be safe… depending on your level of trust in the basic components of your zodiac’s definition. Traditionally your zodiac, or sun sign, is defined to be the constellation that the sun is in on the morning of your birth. This is reliable from year to year because we travel around the sun once. So the sun cycles through each section of the sky in a big arc.

However, the location of the sun is strongly controlled by the location of earth’s axis in relationship to the rest of the sky. The tilt of earth’s axis rotates every 22,000 years- this is called Precession. Many ancient cultures knew about this and it is well studied and validated. Precession causes the zodiacs to slide backwards through the season at a rate of one sign shift each 1.2kyr. Even the Greeks noticed that the zodiac signs were slowly cycling, or regressing, through the seasons.

The Greeks wanted to stop this from happening so Ptolemy, a famous Astronomer who worked in Alexandria, fixed the astrological signs to the equinoxes and seasonal changes.  From then on, the western notions of zodiac signs are now more tied to local seasonal elements than to the solar systems located many light years away. These are the definitions we use today! This means that there is no way for NASA (or anyone else!) to scientifically reinvent them. Timing of zodiac signs are kind of like math definitions. Humans defined them, so there is no scientific way that NASA could ever prove them wrong. There is nothing to prove, timing of each zodiac is just a definition applied by Ptolemy.

If you are someone who happens to believe deeply in astrology, then it’s deeply unlikely that anything in this article will effect that way you view your zodiac sign. However, I think it’s interesting to note that if you believe in Astrology, then you implicitly believe that the time of year you were born in affects your personality. This belief in zodiac signs could mean you would also appreciate research on seasonal affective disorder. Or you might really want to check out this fabulous article by the Atlantic which discusses recent research showing people born in the summer are more susceptible to mood swings (and many other personality quirks that are seasonally significant!).

This is one situation where science, math, and astrology aren’t always well separated. Science can tell us something about the world around us. And, historically, humans built belief systems around that knowledge (Astrology).  Then we had to update the astrological belief system when the science no longer matched. Because initially, sun sign was a good indicator, until we learned about Precession. So astrology must be adjusted to match the seasons, not the stars. Which, some argue, is what the sun signs were supposed to do anyways. And, while we can understand Precession scientifically, we can’t prove anything about the tenants of astrology. Perhaps it’s only a matter of time until we find more scientific reasons to support certain aspects of Astrology? Certainly much of the internet is ready to believe that NASA should be able to weigh in.


Posted in Communicating Math, Media, Nature | Tagged , , , , , , , | 3 Comments

A Very Mathy Yarn

written by Shannon Paper-Negaard

Knitting and Math

I have two main communities: my math department (I’m a grad student) and my knitting group. When I talk to some people in the knitting community at large, they have the impression that more complex knitting is very “mathy,” because “you have to count stitches” in various patterns.

I don’t really see the counting as mathematical. The person who designed the original pattern used spatial reasoning, and that’s math. When I knit complicated cable or lace patterns, I visualize the pattern part of it, and that’s math. Even something like adjusting gauges (the number of stitches per inch) to make garments of an intended size – that is sometimes math.

And a particular type of yarn that I’m enamored with – so much so, that I’m writing this – is mathematical.

Getting Stripes

For the non-knitting folks: if you’ve ever seen a hand knitted garment with multiple colors, there are a few ways for a knitter to achieve this. The first (more traditional) way is to have several balls of yarn and to knit each stitch with one of these, changing as one goes. That’s not a bad way to do it, if one only has to change colors a few times. But there’s a catch. Every time one length of yarn is broken to let another be tied in, there are now two ends to weave in. If you don’t knit or crochet, ask someone who does how they feel about weaving in ends. You’ll soon learn it’s not the enjoyable part of fibre art.


Another project, with many ends, waiting to be woven in.

The second way to change colors is to buy a ball (or skein or hank) of yarn, which already has multiple colors in it.

Usually, when a ball of yarn has multiple colors, those colors occur at regular intervals. If the yarn was dyed as a 2 yard skein, then every two yards, the same color will appear. So, roughly the same number of stitches will be required to get back to a color.


This makes for a really pretty fabric, if you’re knitting something that has about the same width (or circumference) throughout the project. Socks, mittens, rectangular scarves, and even hats will all look roughly as you would expect them to.


These socks – these socks look great in a yarn that works this way.

However, a lot of the patterns knitters like to make change width as they go. Specifically, triangle shawls and semicircular shawls have been extremely popular in recent years. Here’s how they’re constructed.image008


We cast on just a couple of stitches (usually 4 or 6) to start and then knit back and forth (as indicated by the very “high-tech,” penned arrows above). Each row, we add a few stitches. By the time we get to the outside edge, there are hundreds of stitches in each row.

Now, let’s say our skein had fairly long bands, so we actually get full color stripes all the way to the end of the shawl. In knitting, length of yarn translates to a number of stitches, and stitches translate to area: a block of 100 stitches can be 2 rows of 50 stitches, or it can be 10 rows of 10 stitches, or it can be a single row with 100 stitches. It all depends on the row width. So, if 3 yards of yarn gives 100 stitches, the area of knitted fabric won’t change – but the shape it takes might. Knitting friends, this spatial reasoning was brought to you by mathematical thinking J. You’ve all done math, every time you’ve estimated the yarn needed for a pair of socks or a scarf. No counting needed.

Anyway, in the shawls pictured above, we can clearly see the row sizes are changing and it doesn’t take much mathematical intuition to realize that a stripe that’s 1” deep near the start (the part with short rows) has a lot less area, and therefore a lot fewer stitches, than a 1” stripe near the end (with the long rows). So it takes much less yarn to make that 1” stripe near the center or beginning, than it does to make a stripe of the same width toward the outer edge.

How much difference, exactly? Caterpillar Green Yarns figured that out, and we just get to reverse engineer it.

Finally, the mathy yarn

The shawl below – before Caterpillar Green’s innovation – could only have been constructed by

  • purchasing skeins of all of the unique colors you see below,
  • knitting a few rows in grey, cutting the yarn, tying on purple yarn,
  • knitting a few rows in purple, cutting the yarn, tying on grey yarn,
  • knitting a few rows in grey, cutting the yarn, tying on red yarn,
  • etc,

and then weaving in all of those ends. Again, weaving in that many ends just sucks.


But Caterpillar Green found a way to dye the yarn – all in one skein – with the appropriate lengths of yarn.

Our goal is to understand what lengths and why it works.

We’ll use the first style of triangular shawl as our model. Now, we need a standard unit for the length of our yarn. Let’s use the length it takes to make that first “stripe.” If the whole picture shows the shawl, the darkened triangles, together, make up the first stripe. It’s actually going to be more convenient to think about half of that stripe.

(Yes, the first stripe looks like two little triangles put together. Further on, they’ll actually look more stripes.)


Also, every knitter will make a slightly different size triangle with the same length of yarn, and those will change if they change needle sizes. But most knitters also stay consistent within a project. The same knitter will also be knitting the later rows, so the amount of yarn needed for an equal area will remain the same.

Let’s break the rest of the shawl up. This is somewhat simplified. I only have 4 color stripes, but we’ll get the idea. The sides are also mirrored, so I can show off the stripes and the triangles that are key.


The second stripe is made up of 3 of those triangles, or 1+2. The third stripe has 5 triangles, or 3+2. In fact, if we keep adding two triangles for each new section, we have enough area to all stripes in the pattern. Also, we’re using all the odd numbers.

Doubling that (to account for both sides) doesn’t change the proportions. The second full stripe (both sides together) contains 3 times the area as the first (the two dark triangles put together). So the second color section in our yarn has to have 3 times as much yarn as the first. The third color section has to have 5 times as much as the first. The forth has to have 7 times as much as the first, etc.

This is just one shape, though. Let’s see what else can be knit with this yarn.

The key is to notice that the total amount of yarn – using that first color section as our unit – is as follows:

After 1 color: 1 = 12,

after 2 colors: 1 + 3 = 4 = 22,

after 3 colors: 4 + 5 = 9 = 32,

after 4 colors: 9 + 7 = 16 = 42,

after n colors: n2.

It’s all squares. Or, at least, it’s perfect squares times the yarn that was in that original color section.

It turns out, there are many shapes, for which this same pattern will make nice color stripes. A few of them are drawn below.


Another triangular shape


A circle of radius r, which would have r color sections, is \pi r2. The first ring (actually a small circle) uses the standard amount of yarn in the first color section of the yarn. It happens to make a circle of radius r, with area\pi r2. The next section has three times as much yarn, so it adds an area of 3\pi r2. The total area is now 4\pi r2 =\pi (2r)2, or a circle of radius 2r, which is exactly what we wanted. It means we have a ring around the first circle, of width r. It’s easy to see that adding the next color strand, which contains 5 times as much as the first, adds 5\pi r2 in area, for a total of 9\pi r2 =\pi (3r)2, or a circle of radius 3r. The pattern continues. Even though we can’t break the subsequent rings up, visually, into smaller circles (like we did with the triangles), the areas still work as we would hope.

Similar calculations work for a semicircular shawl or even a quarter circle shawl. They also work for squares, different shaped triangles, and wedges of triangles. The yarn, which worked so well for a triangular shawl, also lends itself well to a variety of shapes. Some example shawls, from the Caterpillar Green Yarns website, are shown below.



Here is a photo of the shawl I designed. It looks a little less triangular, because of the way the increases curve.



There are so many things that can be done with this yarn. I wrote a pattern for it.  Of course, the makers of the yarn at Caterpillar Green Yarns will happily give suggestions. Then, there’s the community at Ravelry, who have found wonderful things to do with a yarn that stripes in a different way than we’ve all been used to.

There aren’t crochet projects yet, but who knows, maybe you’re the industrious crocheter to come up with an equally lovely project in your craft.

If you are a knitter, be sure to check out Ravelry, even if you’re not searching for this yarn. It’s where I get inspiration for almost all of my knitting projects, and when I’ve finished with them, it has been a wonderful place to display pictures. I only post about 70% of what I complete, but that percentage is getting better, especially as it gets easier to upload photos.

I hope you liked learning about this yarn. I’ve already made two lovely shawls and have a third set aside as a fall project. And of course, I shall only work on it after putting in a full day of work on my research and grading (wink).


Shannon_picAuthor’s bio: Shannon Paper-Negaard has a masters and is working toward a PhD, both in mathematics. She is studying dynamical systems at the University of Minnesota. She loves to knit and spin yarn, and she’ll talk your ear off about knitting, math, travel, chocolate, or her bicycle. Find her on Twitter at @ShannonPaper or on Ravelry as paperdolls.

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