Simple Words, Complicated Math

Part of the up-goer five. Image: Randall Munroe, xkcd.com. Click for full comic.

Part of the up-goer five. Image: Randall Munroe, xkcd.com. Click for full comic.

A couple years ago, xkcd described the Saturn V rocket (Up Goer 5) using only the thousand ten hundred most common English words. Of course, xkcd readers were eager to try it themselves, and geneticist Theo Sanderson created an online text editor for it. Thus tenhundredwordsofscience and upgoeryourphd were born. Both sites feature attempts by people from all sorts of branches of science to describe their work using only those thousand words.

Last month, David Roberts posted a proof of multiple cardinalities of infinity using only one-syllable words to the n-Category Café. Like the up-goer five challenge, the one-syllable exercise is part Oulipo and part math/science communication. The requirements are strictly enforced, leading to circumlocutions that would be clearer with a little more flexibility. (“The small round thing that passes through the sky every night as it moves around us” is not clearer than “moon,” but “moon” is 1809 on the list of most common words, so it doesn’t pass the up-goer 5 test.) You can get away with a little more with the one-syllable challenge, but it’s still tough.

The comments on the post and the related Google+ post have some good examples of mathematics written in one-syllable words or with other constraints: cartoon proofs, proofs in verse, proofs without the letter ‘e,’ and so on. I am also reminded of the (sadly dormant) @ProofinaTweet and @TinyProof Twitter accounts.

The constraints are fun to play with, and they’ve helped me think about the difference between using simple or short words and actually making a concept easier to understand. The Simple English Wikipedia is designed to have articles that are more accessible to children and adults who are learning English than the regular English Wikipedia articles are. There are guidelines, not rules, that help authors make their ideas easier for English learners. Authors are encouraged to use the 850 words on the Basic English list, but they shouldn’t adhere to that limitation if doing so results in more confusion. Flexibility is important for clarity.

When writing about math and science, people with technical backgrounds are often encouraged to avoid jargon, and in general, that’s sound advice. But sometimes, it’s better to explain the word homotopy and then use it in an article than to say “deformation of one thing into another thing without cutting it” twelve times. (By the way, that’s the Simple English Wikipedia explanation of homotopy, and it’s pretty good, isn’t it?) Jargon has a place not only in communication between experts in the same field but also in popular science writing. But it is a hurdle for readers, and I think it’s a good idea to approach it with caution. The up-goer five and one-syllable challenges feel like extreme versions of a no-jargon challenge. (OK, maybe not if you study stacks, sheaves, or schemes.)

After my recent post on higher homotopy groups, a jargon diet is probably a good idea. I didn’t participate in the up-goer 5 challenge, but the one-syllable just short words math challenge task sounds more interesting to me. I haven’t decided what proof I’m going to monosyllable-ize yet, but I will be participating. I’m interested to see what other people do with it as well. Feel free to share your contributions in the comments here, at the n-category Café, or on your own blog.

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Highly Unlikely Triangles and Other Beaded Mathematics

I first encountered Gwen Fisher’s work at the fiber arts exhibit at the 2014 Joint Mathematics Meetings in Baltimore. Fisher has a Ph.D. in math education and is an accomplished mathematical artist who specializes in beading. I featured one of her pieces (a beaded group of order 18) in an article I wrote about the fiber arts show. Since then, I’ve been following her blog at her website gwenbeads. She posts about her mathematically inspired beadwork and often includes explanations of the underlying mathematics.

A beaded "highly unlikely triangle." Image copyright Gwen Fisher. Used with permission.

A beaded “highly unlikely triangle.” Image copyright Gwen Fisher. Used with permission.

The bead that caught my eye most recently is the “highly unlikely triangle,” based on the “impossible triangle,” or “Penrose triangle,” that shows up in many M.C. Escher works. Fisher’s triangles are not actually impossible, but they do seem to twist around in an unlikely way. A link from that post led me to Borromean linked beaded beads and a highly unlikely hexagon! She’s also made beaded beads named in honor of mathematicians Harold Coxeter and John Conway.

A beaded snub tetrapentagonal tiling of the hyperbolic plane. Image copyright Gwen Fisher. Used with permission.

A beaded snub tetrapentagonal tiling of the hyperbolic plane. Image copyright Gwen Fisher. Used with permission.

Hyperbolic geometry enthusiasts (like me!) will probably enjoy Fisher’s post about beaded tilings of the hyperbolic plane. Like crochet, it seems that beading can allow for a slight increase in area around vertices that distributes the negative curvature of the hyperbolic plane in an even—and very visually appealing—way. Fisher has beaded several different tilings of the hyperbolic plane: the {4,5} tiling (5 squares around every vertex) and the rhombitetrahexagonal and snub tetrapentagonal tilings, both of which use multiple shapes. I think the prettiest one is the snub tetrapentagonal tiling made of pink pentagons, yellow squares, and green triangles shown above.

The Genie Bottle at Burning Man. Image copyright Gwen Fisher. Used with permission.

The Genie Bottle at Burning Man. Image copyright Gwen Fisher. Used with permission.

I’ve just finished helping out with a level 2 Menger sponge build as part of MegaMenger, so I’ve also been interested in Fisher’s posts about the Genie Bottle she and her group Struggletent built at Burning Man this year. It was a giant, furnished, climbable sculpture. It was also ephemeral, spectacularly going up in flames at the end of the event. I’m tired from just a few days spent folding business cards for our Menger sponge. I’m in awe of how much effort went into the Genie Bottle!

The Genie Bottle goes up in flames at Burning Man. Image copyright Gwen Fisher. Used with permission.

The Genie Bottle goes up in flames. Image copyright Gwen Fisher. Used with permission.

In addition to the blog, Fisher has an etsy shop where she sells tutorials for many of her designs as well as beads, hats, jewelry, and other items she makes. She also runs a business called beAd Infinitum with fellow mathematician Florence Turnour. All of her sites are interesting if you’re into math, art, and making things!

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Blogging in Math (History) Class

Plimpton 322, the Babylonian mathematical text that started off our math history class. Image: public domain, via wikimedia commons.

Plimpton 322, the Babylonian mathematical text that started off our math history class. Image: public domain, via wikimedia commons.

I am teaching a math history class this semester, and in addition to trying to teach my students math and history, the course satisfies an upper-level writing credit. It’s a lot to try to cram into one three-hour course! With 40 students enrolled at the beginning of the semester (enrollment has dropped a bit since then, but it’s still large), I wasn’t sure how to get my students doing a significant amount of writing, give them meaningful feedback, and let them revise their work without burying myself in a mound of paper every time an assignment was due.

In part because of that concern and in part because I like blogging, I decided to start a class blog. I have a rolling deadline system that keeps the flow of new writing somewhat manageable, and doing everything online means I can easily email comments and suggestions to my students. Now that the semester is about a third of the way through, almost all of my students have written at least one post for the blog, and I think it’s time to share it with you.

The blog is called 3010tangents because the course number for our class is math 3010, and the posts on the blog should be at least tangentially related to topics we cover in class. We started the course talking about how we write numbers, so we have some posts up there about different base systems, including an impassioned plea to switch to dozenal and an exploration of a binary monetary system in the Book of Mormon. (The religious text, not the musical.) Subsequent classes have touched on a lot of different topics, and my students’ posts reflect that. They have written about Euler, Ramanujan, Noether, al-Khwarizmi, and Zhao Shuang. They have also written about art, religion, limits, and women in math. And of course, the perennial question of whether math is invented or discovered has gotten some treatment.

One of the reasons I started the blog was to get students who are interested in math teaching and communication involved in the wider online mathematics community, so I hope some of you will stop by and give them (kind, helpful) comments on their posts or read and share them. You might even learn a little something about math history!

This is my first time running a class blog, and I am keeping track of what goes well and not so well about the experience. I’m sure I’ll write more about blogging in a math class when all is said and done.

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e is for Ebola

A recent NPR blog features a few quotes emphasizing a math word that is lamentably absent from many readers’ vocabularies: “It’s spreading and growing exponentially,” President Obama said Tuesday. “This is a disease outbreak that is advancing in an exponential fashion,” said Dr. David Nabarro, who is heading the U.N.’s effort against Ebola. I can’t help but feel sad that the occasion for someone to learn about the term “exponential” might be directly linked to Ebola. I would hope instead that it would pertain to the growth of their earnings in a bank account, or the growth curve of some formerly endangered species as it recovers.

However, it is exciting to see some coverage of the “basic reproduction ratio”, R_0 , and a plethora of graphs aimed at showing the variety of scenarios that might unfold in West Africa. Amy Greer at Math.Epi.Lab is one of the researchers using the IDEA (incidence decay and exponential adjustment) model to regularly update predictions as to when this outbreak will reach its peak.

Evolution of the basic reproduction ratio and the control parameter d in the IDEA model for incidence

Evolution of the basic reproduction ratio R_0 and the control parameter d in the IDEA model for incidence

Currently, the IDEA estimate is that in December of this year, the number of cases will peak at around 13,000. Dr. Greer sees the total number of cases due to this outbreak as easily reaching 20,000. In one of her posts, she posts the evolving value of R_0, a nice reminder that this is a dynamic parameter that is estimated using Estimation Theory. In this graphic, the control parameter d “controls” the weight of the mitigating measures in reducing incidence.

Data on Ebola has been provided by the World Health Organization to the general public, and Caitlin Rivers, a computational epidemiologist at Virginia Tech, is making this data more accessible. Ms. Rivers titles her Ebola-related posts #HackEbola, and a quick twitter search shows others using the hashtag as well. World Health Organization more accessible. And her most recent post looks at the data concerning follow-ups with those who have come in contact with someone infected with Ebola.

My favorite math and epidemiology blog so far has been Musings on Infection, in which computational epidemiologist David Hartley ponders various infectious diseases, but especially focuses, in his last half a dozen posts and on his twitter feed, on Ebola. In his post “Epidemiology and behavior in the time of Ebola”, Hartley points us at some great articles, and gives some food for thought, including the possibility that Ebola could become endemic due to the distrust between healthcare workers and the local population. It is interesting to see how different estimates are holding up. Looking back, one of the earlier studies that Hartley references on a September 02 post entitled “Why Model Infectious Disease”, the graphic from physicist Alessandro Vespignani’s paper predicted the number of cases today to be between about four and eleven thousand. Indeed, the current number of cases is 6,500 according to the CDC, which is well within Vespignani’s range.

Alessandro Vespignani's research as featured in Science Insider on August 31st

Alessandro Vespignani’s research as featured in Science Insider on August 31st

The ability to isolate currently infected individuals and follow up with those who have been exposed will play a huge role in determining the evolution of the “effective reproductive number” – the number of individuals that are actually infected by each currently infected person. So I leave you with a tool that you might consider using with your students or just play with for your own edification. The game VAX, developed by Phd candidate Ellsworth Campbell, is a great way to get a feel for how disease can spread through a network depending on the connectivity of the network and the ability to vaccinate those who are healthy (not yet a possibility for Ebola) and quarantine those who are infected.

VAX game screenshot

VAX game screenshot

Hope all of this helps to better inform you as to some of the mathematics involved in helping to analyze the current situation in West Africa. Please let me know if you have a favorite blog that discusses mathematics and epidemiology.

Posted in Applied Math, Biomath, Math Education, Mathematics and Computing, people in math, Statistics | Tagged , , , , , , , , , , | Leave a comment

First Impressions of the Second Heidelberg Laureate Forum

Last year, I wrote with some envy about the first annual Heidelberg Laureate Forum. This year, I’m there! I mean, here! Yes, after several flights and a few train delays, I’m finally in Heidelberg, and if the fog clears and drizzle dissipates, the surroundings will be beautiful!

A panorama photograph of Heidelberg. Image: Coolgarriv, via Flickr.

A panorama photograph of Heidelberg. Image: Coolgarriv, via Flickr.

The Heidelberg Laureate Forum is a chance for stodgy old Turing, Abel, Nevalinna, and Fields laureates to get an infusion of fresh ideas from the 200 young math and computer science researchers who graciously agreed to attend. Or maybe it’s a chance for 200 fresh-faced, eager young mathematicians and computer scientists to take in some of the wisdom of the laureates who graciously agreed to attend. Either way, it will be a chance for mathematicians and computer scientists to network not only within their respective groups but across the aisle, as it were.

The opening ceremony for the meeting was Sunday night. I don’t know whether the highlight was the reappearance of the odds-defying saxophone quartet that John Cook wrote about last year or International Mathematical Union president Ingrid Daubechies’ joke that mathematics is a very old profession, but of course not the oldest profession. Two themes of the opening comments by organizers were about speaking slowly to communicate well and about using our powers for good, especially in the case of computer science, where as we have seen recently (hi NSA, what’s up? Nothing to see here…), computer security can have very serious consequences. As Alexander Wolf, president of the Association of Computing Machinery, which awards the Turing Prize, pointed out, these cautions about computer science represent something of a victory: the benefits of computing are so widespread and obvious that they are just taken for granted. It is hard to imagine a world without them.

From the look of the program, we are in for a lot of outstanding talks this week. Personally, I hope to get a better feel for some of the big ideas in computer science from some of the speakers. But we all know that a lot of the real benefit of a meeting is what happens between talks and on excursions. I’m looking forward to making connections during those in-between times!

This year, the meeting has both a German and an Engligh blog. I’ll be blogging (in English) while I’m at the meeting, so you can find me over there. I might even livetweet a few of the talks, so you can also follow me on Twitter if you’re not already. The hashtag for the meeting is #hlf14. Tschüss!

This blog post originates from the official blog of the 2nd Heidelberg Laureate Forum (HLF) which takes place in Heidelberg, Germany, September 21 – 26, 2013. 24 Abel, Fields, and Turing Laureates will gather to meet a select group of 200 young researchers.

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Making Conferences Easier for Families

Most mathematicians want to make mathematics, and especially mathematical academia, more hospitable to women. One way to do that is to help them participate as fully as possible in conferences, even when they have young children. Due to a sometimes inconvenient synchronization of biological and academic clocks, academics often have young children at crucial points in their careers. They may be on the job market or going up for tenure while their children are babies or toddlers, and making it to conferences can make a difference in their careers.

In August, Matilde Lalin wrote a guest post on Terry Tao’s blog about attending conferences with young children focused on the options for nursing mothers. The three main options she identifies are: travel with a caretaker, hire a caretaker at the conference location, or leave the child at home and pump. (The fourth option, not going to the conference at all, is one many families end up choosing, as Kate Owens mentions in the comments, but can come at a cost of opportunities for collaboration and networking.)

In addition to the financial considerations for each option, Lalin writes about scheduling and other logistical concerns, including the choices parents have to make about what to skip when they attend a conference with a child. She includes practical suggestions for conference organizers about how to make conferences easier for nursing mothers and links to several organizations that are helping support academic parents. Of course, many of her suggestions also apply to parents who are traveling with kids but not breastfeeding.

As a non-parent, I must admit that I hadn’t really thought about the burden of child care at conferences until Jordan Ellenberg wrote about it a couple years ago. He argues that the NSF should fund conference daycare. There are quite a few interesting comments to the post about whether or not childcare should be considered a business expense. I think the one that sums it up the best for me is by jenfns, who writes, “I guess the question is whether it should be a cost of the employer to pay for travelling employees’ childcare. Since our society does in fact have a vested interest in successful professional women bearing and raising children, I think that the answer should be yes.” Unfortunately, for federal tax purposes, child care costs are not considered “necessary” expenses, and I assume until that changes, the NSF will not be able to reimburse conference child care costs.

Last month, Laura McLay wrote on her blog Punk Rock Operations Research that the Forum for Women in OR/MS (delightfully acronym’d WORMS) is sponsoring grants to reimburse child care costs for parents traveling to the INFORMS Annual Meeting. And I just saw that the AMS and MAA are also rising to the occasion, with about 40 child care grants available for mathematicians attending the Joint Mathematics Meetings in January. Applications for those grants are open until November 18. The JMM has had subsidized (but still expensive) child care available for several years, but as far as I know, this is the first year they will also have reimbursement grants available.

Better support of child care costs will help women most directly, but it should be noted that all the grants I have seen are available to both men and women, as they should be. Women sometimes shoulder the lion’s share of the child-rearing duties, but the idea that raising children is only “women’s work” is antiquated and devalues the active role many men play in their children’s day-to-day care. Paying for child care at conferences is a way to make life easier for both men and women who are trying to balance their careers and families.

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Regression, Twitter, and #Ferguson

Emma Pierson's analysis of hashtags that have appeared in tweets about Ferguson, Missouri. Click for the image on her website. Image: Emma Pierson.

Emma Pierson’s analysis of hashtags that have appeared in tweets about Ferguson, Missouri. Click for more information on her website. Image: Emma Pierson.

Like many people, I have been following news about the events in Ferguson, Missouri with shock and sorrow for almost two weeks. I have been following these events as a human, not as a mathematician. But there’s a mathematical side to this story, too. I’m not just talking about the statistics on how many people are killed by the police each year (which we don’t even know for sure) and the racial composition of the Ferguson police force versus the people they stop and arrest, although those are both important. I’m talking about Twitter. It’s been a crucial part of how the Ferguson story has become international news, but it’s also a useful source of data about how people are responding to the tragedy.

Emma Pierson is a computational biologist currently working for 23andMe, and her blog, Obsession with Regression, focuses on data analysis, often with Twitter’s data. She writes,

“I am very excited about Twitter because it combines two qualities.

“1. People actually use it. Famous people — it’s become standard for celebrities to say “Follow me on Twitter!” — and more importantly, lots of people.

“2. It makes massive amounts of data available in a way you can process with a computer. 500,000,000 tweets are sent every day and Twitter will give you up to 1% of those. And if I know what 1% I want — for example, only Tweets containing the word “Spock” — it will give me all of them, which means I can actually hear everything that’s being said on a topic by millions of people worldwide. And not just what’s being said, but who’s saying it — how they describe themselves, where they live, who their friends are, and the last few thousand things they said.”

She has been blogging about Twitter data since December 2013, when 23andMe was ordered to stop providing disease risk information to their customers. She wrote a post about who was reacting to the news on Twitter and how they felt about it. Of course, being an employee of the company represents an obvious potential source of bias, so she also included a link to the tweets she analyzed so others could study them. She’s done several other interesting data analyses as well. Earlier this summer, she wrote an interesting analysis of tweets about LeBron James’ most recent career move, and of personal interest to me is her post about gender in the symphony. (Her analysis seems to match my experience. In my four years in the orchestra in college, I think we only had two men in the viola section.)

On Tuesday, Pierson wrote a post about using Twitter to study people’s reactions to current events, focusing on Ferguson. She mined a few hundred thousand tweets about Ferguson and analyzed the diferent hashtags that appeared in tweets with #Ferguson. (Part of the visualization she made is at the top of this post.) She also put her mineTweets program up on Github so others can use it to collect tweets about any topic in real time. She has some ideas for further analysis, particularly about whether the day/night-peace/violence pattern is apparent in tweets, and she’s invited others to contribute either ideas or analyses of their own.

The events in Ferguson have also highlighted the difference between the way Twitter and Facebook work. I’m not the only one whose Twitter feed has been saturated with #Ferguson, while Facebook has been nearly silent on the topic. In a Medium article, Zeynep Tufekci explains how Facebook’s algorithm for deciding what to show us caused this discrepancy and wonders what would have happened to Ferguson without Twitter. “It’s a clear example why net neutrality is a human rights issue; a free speech issue; and an issue of the voiceless being heard, on their own terms,” she writes. “Algorithms have consequences.”

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Medaling Mathematicians

You may consider the Fields Medal a boon to the mathematical community as it showcases amazing young mathematicians and brings math into the limelight. Or you may view the Fields Medal as an unfortunate reinforcement of the notion that mathematics is the work of lone geniuses. Whichever the case, you can blog about it, and you’ll be in good company.

From left to right :, Subhash, Khot, Martin Hairer, Manjul Bhargava,  the South Korean President Park Geun-Hye , Maryam Mirzakhani , the President of the International Mathematical Union (IMU) Ingrid Daubechies, and Artur Avila.

From left to right: Dr. Martin Hairer, Dr. Manjul Bhargava, the South Korean President Park Geun-Hye , Dr. Maryam Mirzakhani , the President of the International Mathematical Union (IMU) Dr. Ingrid Daubechies, and Dr. Artur Avila. Photos Courtesy of Dr. Alina Bucur.

The four winners of the Fields medal are pictured above at the International Congress of Mathematics (ICM).  Hands down, the best summaries of these winners’ contributions as well as the contributions of the Nevalina Prize winner were written by Erica Klarreich and Natalie Wolchover at Quanta Magazine. The accompanying videos provided at the Simon’s Center site are great for showing in a classroom. It’s particularly inspiring to me to see Maryam Mirzakhani drawing diagrams of manifolds on giant pieces of paper in her living room and talking about how her three-year-old daughter probably thinks she’s an artist. Also very interesting is the non-academic background of Artur Avila, who is Brazilian, and likes to think about math problems while he walks down the beach. Short videos like these will likely prove quite inspirational to young kids.

Lucky for you and me, I happen to subscribe to the email list of Women In Number Theory, on which Alina Bucur, a Number Theorist from UCSD, posted some great photos of the ceremony that she agreed to share. The beaming smiles of the recipients are much more lively to me than the official pictures making the rounds. In particular, I like these of Mirkazhani and Bhargava with Ingrid Daubechies, the inventor of wavelets and president of the IMU.

IMG_0269IMG_0241

Because of the Fields Medal ceremonies, there were a few unexpected corners of the web where math showed up this week that actually described the mathematical accomplishments in some detail. Some of the basic ideas behind winner Maryam Mirzakhani’s research were covered in a Business Insider Article and an inforgraphic at a tutoring website (matific).

It was particularly interesting to read the comments on the BI article, in which a large portion of the discussion was based around whether a practical application of her work existed, or whether it mattered.

The Aperiodical blog’s great round-up pointed out that even popular outlets like Elle and Jezebel covered the medal this year. Four posts that I’d like to highlight that have recently come out are:

Jordan Ellenberg’s Slate piece, Math is Getting Dynamic, details the rise of Dynamics, the field of two of the young winners – Maryam Mirkhazani and Avila. He does a great job of making this field relatable and approachable.

Cathy O’Neil at mathbabe expresses her dismay at the tendency of awards such as the Field’s medal to paint a picture of mathematics as non-collaborative by awarding individuals rather than groups.

Izabella Laba’s short post expresses her feelings about attending the awards ceremony.

Mike Lawler’s reflection on how Mirzakhani’s high school experience made him reflect on his own great teachers and mentors.

In closing, it’s disappointing to all of those suave mathematicians out there I’m sure, but Nobel didn’t leave math out of his prizes because his wife slept with a mathematician.  As Evelyn points out in her recent post about how to talk about the Fields Medal at cocktail parties, the inventor of dynamite wasn’t even married.

 

Posted in Events, Math Education, Mathematics and the Arts, Number Theory, people in math, Theoretical Mathematics, Uncategorized, women in math | Tagged , , , | Leave a comment

Alias, Schmalias

While the great line from Romeo and Juliet: “a rose by any other name would smell as sweet” rings true, would a digital rose smell as sweet?  We often think of the digital world as a mere “renaming” of the real world.  But some interesting effects emerge from digitizing, and one of them is commonly known as aliasing.

If you’ve danced in a club with strobe lights, you’ve laughed at the slow motion effect that results from your eyes interpolating between positions. Sampling data can be thought of in much the same way. As a young child I was in love with both dance and mathematics, and I recall my uncle describing to me a David Parson’s piece involving a strobe light. The piece, Caught, involves a man appearing to fly through the air as a strobe light flashes on at just the right moments in a progression of jumps.

While I certainly didn’t think of it this way at the time, the choreographer Parsons was sampling his dancing at the same rate that he jumped. In this way he could appear to be at the same height above the ground (i.e. floating/flying) at every moment. If we think of Parsons (who it doesn’t hurt to think of since he was pretty gorgeous) moving up and down over a sine wave as he jumps, then we are simply sampling at a rate of exactly once per period and essentially leaving out the information that he ever touches the ground. So what made me think of all this again?

Mathalicious’s blog! While Mathalicious’s lessons require a subscription to view, the blog (and a few sample lessons) are free, and the most recent lesson entitled “Spinning your Wheels” is about aliasing, the same phenomenon mentioned above, in which we sample less frequently than is necessary to faithfully reproduce the information (movement) that is occurring. Mathalicious explains the “Wagon Wheel effect” which results in a car’s wheels appearing to stop moving or to spin backwards even as the car (or wagon) is moving forwards.

Chris Lusto's illustration of how a wheel that is spinning can appear still if it's movement is captured at a rate that coincides with it's turning through an angle of rotational symmetry.

Chris Lusto’s illustration of how a wheel that is spinning at a rate of 72 degrees per frame may appear still if consecutive frames are as shown above (where the green spoke would not be colored green in the film).  Any object that undergoes a symmetry in the time between successive frames will appear stationary.

A nice point made at the beginning of the post is that this effect is due completely to the nature of digital media in which the image is sampled. In the same way that we interpolate position when strobe lights are flashing, we interpolate the movement of the wheel as pixels on a screen change color. While a car wheel doesn’t have spokes, its hub caps often have the typical five-fold symmetry described in the blog post, and Chris (the blogger) has put in some great animation widgets to help the reader understand this phenomenon without ever mentioning Nyquist, sampling rate, or the word aliasing. While I applaud his explanation, I also think it would be worth plotting the sine wave traced out by one point on the wheel.  Then the typical definition of aliasing would be more directly connected to this wonderful example. Another great post of Chris’s on the Mathalcious blog that is in the same Digital Signal Processing vein is the Siren Song which illuminates the Doppler Effect and even addresses what happens if the siren travels at more than the speed of light!

Likewise, there are spatial examples of aliasing such as the Moire patterns that emerge from digital photographs of objects that have periodic patterns (like brick walls and textured clothes).

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Also aliasing manifests in music as what we commonly call distortion when higher pitches are aliased down to lower ones. Any analysis of a cyclic phenomenon can be colored by the affects of aliasing. I happened upon the arxiv paper “Could sampling make Hares eat Lynxes?” which discusses the potential of aliasing as an explanation for misinterpretations of cyclic behaviors of populations in the context of the Lotke-Volterra predator-Prey model. What examples of aliasing have you experienced or found interesting?

Hares high-fiving after eating a lynx? :)

Hares high-fiving after eating a lynx? :)

 

Posted in Applied Math, K-12 Mathematics, Math Education, Mathematics and Computing, Recreational Mathematics | Tagged , , , , , , , , , , | 1 Comment

The Funny Pages

Ah, summer! Sleeping in, reading fiction, traveling, and, of course, preparing for fall classes. I’ll be teaching a math history class, which will be fun but is entirely new to me. As I cling to the last few weeks of freedom before the semester starts, when I have the luxury of prepping at a nice leisurely pace, I sometimes find my browser wandering from the Convergence website, with its useful articles about teaching mathematics using history, to a couple of my favorite funny math blogs.

Math Prof 4 Life illustrates the glamorous life of a math professor with the finest animated gifs that imgur can provide. From students who don’t understand how exams work to students who nail a proof at the board, from Joint Meetings exhibit hall candy to interminable committee meetings to those pesky problems you just can’t quit, the author has an appropriate gif for all sorts of awkward and awesome academic occasions.

By the way, undergraduate abstract algebra professor, I’m sorry for that time I thought all finite groups were abelian. Now I know that this is how you felt inside.

Math Professor Quotes is a dangerous blog to visit if you teach math and think you’re funny. (Guilty as charged.) Because you might be disappointed when you refresh it after class and find that the students surreptitiously using their phones under the desk weren’t submitting your wit to the blog for posterity. But maybe you’ll find a few quotes that will spice up your lectures. (Because recycled jokes never feel forced!) Here are a few of my favorites from the site.

“Using the chain rule is like peeling an onion: you have to deal with each layer at a time, and if it is too big you will start crying.”
“Differentiating is like squeezing toothpaste out of the tube. Integrating is like putting the toothpaste back into the tube.”
“There are precisely as many numbers between zero and one as there are between zero and two. #thefaultinourℝ”
And especially for my math 3220 students from last year: “Heine-Borel is the kind of theorem that is essential for your life. I mean, you can handle doing grocery shopping without Bolzano-Weierstrass, but you would never succeed without Heine-Borel.”

Do you have any favorite funny math blogs?

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