Tag: Mathematics

With new technology, mathematicians turn numbers into art

Once upon a time, mathematicians imagined their job was to discover new mathematics and then let others explain it.

Today, digital tools like 3-D printing, animation, and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.

Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.

The atmosphere was electric. Talks ran the gamut, from wildly creative thinkers who apply mathematics in the world of design to examples of pure mathematical results discovered through computer experimentation and visualization. It shed light on how powerful visualization has become for studying and sharing mathematics.

Reimagining math

Visualization plays a growing role in mathematical research. According to John Sullivan at the Technical University of Berlin, mathematical thinking styles can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.

Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles.


Borromean Rings, the logo of the International Mathematical Union.
John Sullivan

Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity.

The “bubble” version of the configuration, shown below, is minimal, in the sense that it is the shortest possible shape where the tubes around the rings do not overlap. It’s as if you were to blow a soap bubble around each of the rings in the configuration. Techniques for proving that configurations like this are optimal often involve concepts of flow: If a given configuration is not the best, there are often ways to tell it to move in a direction that will make it better. This topic has great potential for visualization.

At the workshop, Sullivan dazzled us with a video of the three bands flowing into their optimal position. This animation allowed the researchers to see their ideas in action. It would never be considered as a substitute for a proof, but if an animation showed the wrong thing happening, people would realize that they must have made an error in their mathematics.


In this version of the Borromean Rings, a virtual ‘soap bubble’ is blown around the wire-frame configuration.
John Sullivan

The digital artists

Visualization tools have helped mathematicians share their work in creative and surprising ways – even to rethink what the job of a mathematician might entail.

Take mathematician Fabienne Serrière, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.

Edmund Harriss of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”

Mathematical algorithms power the products made by Nervous System, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist, and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology.

Their first 3-D printed dress consists of thousands of interlocking pieces designed to fit a particular model. In order to print the dress, the designers folded up their virtual version, using protein-folding algorithms. A selective laser sintering process fused together parts of a block of powder to make the dress, then let all the unwanted powder fall away to reveal its shape.

Meanwhile, a delightful collection called Geometry Games can help everyone, from elementary school students to professional mathematicians, explore the concept of space. The project was founded by mathematician Jeff Weeks, one of the rock stars of the mathematical world. The iOS version of his “Torus Games” teaches children about multiply-connected spaces through interactive animation. According to Weeks, the app is verging on one million downloads.

Mathematical wallpaper

My own work, described in my book “Creating Symmetry: The Artful Mathematics of Wallpaper Patterns,” starts with a visualization technique called the domain coloring algorithm.

I developed this algorithm in the 1990s to visualize mathematical ideas that have one dimension too many to see in 3-D space. The algorithm offers a way to use color to visualize something seemingly impossible to visualize in one diagram: a complex-valued function in the plane. This is a formula that takes one complex number (an expression of the form a+_b_i, which has two coordinates) and returns another. Seeing both the 2-D input and the 2-D output is one dimension more than ordinary eyes can see, hence the need for my algorithm. Now, I use it to create patterns and mathematical art.


A curve with pleasing 5-fold symmetry, constructed using Fourier techniques.
Frank A Farris

My main pattern-making strategy relies on a branch of mathematics called Fourier theory, which involves the superposition of waves. Many people are familiar with the idea that the sound of a violin string can be broken down into its fundamental frequencies. My “wallpaper functions” break down plane patterns in just the same way.

My book starts with a lesson in making symmetric curves. Taking the same idea into a new dimension, I figured out how to weave polyhedral solids – think cube, dodecahedron, and so on – from symmetric bands made from these waves. I staged three of these new shapes, using Photoshop’s 3-D ray-tracing capacity, in the “Platonic Regatta” shown below. The three windsails display the symmetries of Platonic solids: the icosahedron/dodecahedron, cube/octahedron and tetrahedron.


A Platonic Regatta. Mathematical art by Frank A. Farris shows off three types of polyhedral symmetry: icosahedral/dodecahedral, cube/octaheral and tetrahedral.
Frank Farris

About an hour after I spoke at the workshop, mathematician Mikael Vejdemo-Johansson had posted a Twitter bot to animate a new set of curves every day!

Mathematics in the 21st century has entered a new phase. Whether you want to crack an unsolved problem, teach known results to students, design unique apparel or just make beautiful art, new tools for visualization can help you do it better.

This article was updated on April 5, 2017 with the full name of Mikael Vejdemo-Johansson.

Frank A. Farris, Associate Professor of Mathematics, Santa Clara University

Photo Credit:  Frank Farris

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This article was originally published on The Conversation. Read the original article.

Momentum isn’t magic – vindicating the hot hand with the mathematics of streaks

It’s NCAA basketball tournament season, known for its magical moments and the “March Madness” it can produce. Many fans remember Stephen Curry’s superhuman 2008 performance where he led underdog Davidson College to victory while nearly outscoring the entire determined Gonzaga team by himself in the second half. Was Curry’s magic merely a product of his skill, the match-ups and random luck, or was there something special within him that day?

Nearly every basketball player, coach or fan believes that some shooters have an uncanny tendency to experience the hot hand – also referred to as being “on fire,” “in the zone,” “in rhythm” or “unconscious.” The idea is that on occasion these players enter into a special state in which their ability to make shots is noticeably better than usual. When people see a streak, like Craig Hodges hitting 19 3-pointers in a row, or other exceptional performances, they typically attribute it to the hot hand.

The hot hand makes intuitive sense. For instance, you can probably recall a situation, in sports or otherwise, in which you felt like you had momentum on your side – your body was in sync, your mind was focused and you were in a confident mood. In these moments of flow success feels inevitable, and effortless.

However, if you go to the NCAA’s website, you’ll read that this intuition is incorrect – the hot hand does not exist. Belief in the hot hand is just a delusion that occurs because we as humans have a predisposition to see patterns in randomness; we see streakiness even though shooting data are essentially random. Indeed, this view has been held for the past 30 years among scientists who study judgment and decision-making. Even Nobel Prize winner Daniel Kahneman affirmed this consensus: “The hot hand is a massive and widespread cognitive illusion.”

Nevertheless, recent work has uncovered critical flaws in the research which underlies this consensus. In fact, these flaws are sufficient to not only invalidate the most compelling evidence against the hot hand, but even to vindicate the belief in streakiness.

Research made it the ‘hot hand fallacy’

In the landmark 1985 paper “The hot hand in basketball: On the misperception of random sequences,” psychologists Thomas Gilovich, Robert Vallone and Amos Tversky (GVT, for short) found that when studying basketball shooting data, the sequences of makes and misses are indistinguishable from the sequences of heads and tails one would expect to see from flipping a coin repeatedly.

Just as a gambler will get an occasional streak when flipping a coin, a basketball player will produce an occasional streak when shooting the ball. GVT concluded that the hot hand is a “cognitive illusion”; people’s tendency to detect patterns in randomness, to see perfectly typical streaks as atypical, led them to believe in an illusory hot hand.

GVT’s conclusion that the hot hand doesn’t exist was initially dismissed out of hand by practitioners; legendary Boston Celtics coach Red Auerbach famously said: “Who is this guy? So he makes a study. I couldn’t care less.” The academic response was no less critical, but Tversky and Gilovich successfully defended their work, while uncovering critical flaws in the studies that challenged it. While there remained some isolated skepticism, GVT’s result was accepted as the scientific consensus, and the “hot hand fallacy” was born.

Importantly, GVT found that professional practitioners (players and coaches) not only were victims of the fallacy, but that their belief in the hot hand was stubbornly fixed. The power of GVT’s result had a profound influence on how psychologists and economists think about decision-making in domains where information arrives over time. As GVT’s result was extrapolated into areas outside of basketball, the hot hand fallacy became a cultural meme. From financial investing to video gaming, the notion that momentum could exist in human performance came to be viewed as incorrect by default.

The pedantic “No, actually” commentators were given a license to throw cold water on the hot hand believers.

Taking another look at the probabilities

In what turns out to be an ironic twist, we’ve recently found this consensus view rests on a subtle – but crucial – misconception regarding the behavior of random sequences. In GVT’s critical test of hot hand shooting conducted on the Cornell University basketball team, they examined whether players shot better when on a streak of hits than when on a streak of misses. In this intuitive test, players’ field goal percentages were not markedly greater after streaks of makes than after streaks of misses.

GVT made the implicit assumption that the pattern they observed from the Cornell shooters is what you would expect to see if each player’s sequence of 100 shot outcomes were determined by coin flips. That is, the percentage of heads should be similar for the flips that follow streaks of heads, and the flips that follow streaks of misses.

Our surprising finding is that this appealing intuition is incorrect. For example, imagine flipping a coin 100 times and then collecting all the flips in which the preceding three flips are heads. While one would intuitively expect that the percentage of heads on these flips would be 50 percent, instead, it’s less.

Here’s why.

Suppose a researcher looks at the data from a sequence of 100 coin flips, collects all the flips for which the previous three flips are heads and inspects one of these flips. To visualize this, imagine the researcher taking these collected flips, putting them in a bucket and choosing one at random. The chance the chosen flip is a heads – equal to the percentage of heads in the bucket – we claim is less than 50 percent.

The percentage of heads on the flips that follow a streak of three heads can be viewed as the chance of choosing heads from a bucket consisting of all the flips that follow a streak of three heads.
Miller and Sanjurjo, CC BY-ND

To see this, let’s say the researcher happens to choose flip 42 from the bucket. Now it’s true that if the researcher were to inspect flip 42 before examining the sequence, then the chance of it being heads would be exactly 50/50, as we intuitively expect. But the researcher looked at the sequence first, and collected flip 42 because it was one of the flips for which the previous three flips were heads. Why does this make it more likely that flip 42 would be tails rather than a heads?

Why tails is more likely when choosing a flip from the bucket.
Miller and Sanjurjo, CC BY-ND

If flip 42 were heads, then flips 39, 40, 41 and 42 would be HHHH. This would mean that flip 43 would also follow three heads, and the researcher could have chosen flip 43 rather than flip 42 (but didn’t). If flip 42 were tails, then flips 39 through 42 would be HHHT, and the researcher would be restricted from choosing flip 43 (or 44, or 45). This implies that in the world in which flip 42 is tails (HHHT) flip 42 is more likely to be chosen as there are (on average) fewer eligible flips in the sequence from which to choose than in the world in which flip 42 is heads (HHHH).

This reasoning holds for any flip the researcher might choose from the bucket (unless it happens to be the final flip of the sequence). The world HHHT, in which the researcher has fewer eligible flips besides the chosen flip, restricts his choice more than world HHHH, and makes him more likely to choose the flip that he chose. This makes world HHHT more likely, and consequentially makes tails more likely than heads on the chosen flip.

In other words, selecting which part of the data to analyze based on information regarding where streaks are located within the data, restricts your choice, and changes the odds.

The complete proof can be found in our working paper that’s available online. Our reasoning here applies what’s known as the principle of restricted choice, which comes up in the card game bridge, and is the intuition behind the formal mathematical procedure for updating beliefs based on new information, Bayesian inference. In another one of our working papers, which links our result to various probability puzzles and statistical biases, we found that the simplest version of our problem is nearly equivalent to the famous Monty Hall problem, which stumped the eminent mathematician Paul Erdős and many other smart people.

We observed a similar phenomenon; smart people were convinced that the bias we found couldn’t be true, which led to interesting email exchanges and spirited posts to internet forums (TwoPlusTwo, Reddit, StackExchange) and the comment sections of academic blogs (Gelman, Lipton&Regan, Kahan, Landsburg, Novella, Rey Biel), newspapers (Wall Street Journal, The New York Times) and online magazines (Slate and NYMag).

The hot hand rises again

With this counterintuitive new finding in mind, let’s now go back to the GVT data. GVT divided shots into those that followed streaks of three (or more) makes, and streaks of three (or more) misses, and compared field goal percentages across these categories. Because of the surprising bias we discovered, their finding of only a negligibly higher field goal percentage for shots following a streak of makes (three percentage points), was, if you do the calculation, actually 11 percentage points higher than one would expect from a coin flip!

Not just an illusion, those hands can be hot.
Athlete image via www.shutterstock.com.

An 11 percentage point relative boost in shooting when on a hit-streak is not negligible. In fact, it is roughly equal to the difference in field goal percentage between the average and the very best 3-point shooter in the NBA. Thus, in contrast with what was originally found, GVT’s data reveal a substantial, and statistically significant, hot hand effect.

Importantly, this evidence in support of hot hand shooting is not unique. Indeed, in recent research we’ve found that this effect replicates in the NBA’s Three Point contest, as well in other controlled studies. Evidence from other researchers using free throw and game data corroborates this. Further, there’s a good chance the hot hand is more substantial than we estimate due to another subtle statistical issue called “measurement error,” which we discuss in the appendix of our paper.

Thus, surprisingly, these recent discoveries show that the practitioners were actually right all along. It’s OK to believe in the hot hand. While perhaps you shouldn’t get too carried away, you can believe in the magic and mystery of momentum in basketball and life in general, while still maintaining your intellectual respectability.

Joshua Miller, Affiliate at IGIER and Assistant Professor of Decision Sciences, Bocconi University and Adam Sanjurjo, Assistant Professor of Economics, Universidad de Alicante

Photo Credit: Shutterstock

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This article was originally published on The Conversation. Read the original article.

How I used math to develop an algorithm to help treat diabetes

When people ask me why I, an applied mathematician, study diabetes, I tell them that I am motivated for both scientific and human reasons.

Type 2 diabetes runs in my family. My grandfather died of complications related to the condition. My mother was diagnosed with the disease when I was 10 years old, and my Aunt Zacharoula suffered from it. I myself am pre-diabetic.

As a teen, I remember being struck by the fact that my mother and her sister received different treatments from their respective doctors. My mother never took insulin, a hormone that regulates blood sugar levels; instead, she ate a limited diet and took other oral drugs. Aunt Zacharoula, on the other hand, took several injections of insulin each day.

Though they had the same heritage, the same parental DNA and the same disease, their medical trajectories diverged. My mother died in 2009 at the age of 75 and my aunt died the same year at the age of 78, but over the course of her life dealt with many more serious side effects.

When they were diagnosed back in the 1970s, there were no data to show which medicine was most effective for a specific patient population.

Today, 29 million Americans are living with diabetes. And now, in an emerging era of precision medicine, things are different.

Increased access to troves of genomic information and the rising use of electronic medical records, combined with new methods of machine learning, allow researchers to process large amounts data. This is accelerating efforts to understand genetic differences within diseases – including diabetes – and to develop treatments for them. The scientist in me feels a powerful desire to take part.

Using big data to optimize treatment

My students and I have developed a data-driven algorithm for personalized diabetes management that we believe has the potential to improve the health of the millions of Americans living with the illness.

It works like this: The algorithm mines patient and drug data, finds what is most relevant to a particular patient based on his or her medical history and then makes a recommendation on whether another treatment or medicine would be more effective. Human expertise provides a critical third piece of the puzzle.

After all, it is the doctors who have the education, skills and relationships with patients who make informed judgments about potential courses of treatment.

We conducted our research through a partnership with Boston Medical Center, the largest safety net hospital in New England that provides care for people of lower income and uninsured people. And we used a data set that involved the electronic medical records from 1999 to 2014 of about 11,000 patients who were anonymous to us.

These patients had three or more glucose level tests on record, a prescription for at least one blood glucose regulation drug, and no recorded diagnosis of type 1 diabetes, which usually begins in childhood. We also had access to each patient’s demographic data, as well their height, weight, body mass index, and prescription drug history.

Next, we developed an algorithm to mark precisely when each line of therapy ended and the next one began, according to when the combination of drugs prescribed to the patients changed in the electronic medical record data. All told, the algorithm considered 13 possible drug regimens.

For each patient, the algorithm processed the menu of available treatment options. This included the patient’s current treatment, as well as the treatment of his or her 30 “nearest neighbors” in terms of the similarity of their demographic and medical history to predict potential effects of each drug regimen. The algorithm assumed the patient would inherit the average outcome of his or her nearest neighbors.

If the algorithm spotted substantial potential for improvement, it offered a change in treatment; if not, the algorithm suggested the patient remain on his or her existing regimen. In two-thirds of the patient sample, the algorithm did not propose a change.

The patients who did receive new treatments as a result of the algorithm saw dramatic results. When the system’s suggestion was different from the standard of care, an average beneficial change in the hemoglobin of 0.44 percent at each doctor’s visit was observed, compared to historical data. This is a meaningful, medically material improvement.

Based on the success of our study, we are organizing a clinical trial with Massachusetts General Hospital. We believe our algorithm could be applicable to other diseases, including cancer, Alzheimer’s, and cardiovascular disease.

It is professionally satisfying and personally gratifying to work on a breakthrough project like this one. By reading a person’s medical history, we are able to tailor specific treatments to specific patients and provide them with more effective therapeutic and preventive strategies. Our goal is to give everyone the greatest possible opportunity for a healthier life.

Best of all, I know my mom would be proud.

Dimitris Bertsimas, Professor of Applied Mathematics, MIT Sloan School of Management

Photo Credit: Shutterstock.com

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This article was originally published on The Conversation. Read the original article.

School bus routes are expensive and hard to plan. We calculated a better way

Here’s a math problem even the brightest school districts struggle to solve: getting hordes of elementary, middle and high school students onto buses and to school on time every day. The Conversation

Transporting all of these pupils presents a large and complex problem. Some school districts use existing software systems to develop their bus routes. Others still develop these routes manually.

In such problems, improving operational efficiency even a little could result in great advantages. Each school bus costs school districts somewhere between US$60,000 and $100,000. So, scheduling the buses more efficiently will result in significant monetary savings.

Over the past year, we have been working with the Howard County Public School System (HCPSS) in Maryland to analyze its transportation system and recommend ways to improve it. We have developed a way to optimize school bus routes, thanks to new mathematical models.

Finding the optimal solution to this problem is very valuable, even if that optimal solution is only slightly better than the current plan. A solution that is only one percent worse would require a considerable number of additional buses due to the size of the operation.

By optimizing bus routes, schools can cut down on costs, while still serving all of the children in their district. Our analysis shows that HCPSS can save between five and seven percent on the number of buses needed.

Route planning

A bus trip in the afternoon starts from a given school and visits a sequence of stops, dropping off students until the bus is empty. A route is a sequence of trips from different schools that are linked together to be served by one bus.

Our goal was to reduce both the total time buses run without students on board – also known as aggregate deadhead time – as well as the number of routes. Fewer routes require fewer buses since each route is assigned to a single bus. Our approach uses data analysis and mathematical modeling to find the optimal solution in a relatively short time.

To solve this problem, a computer algorithm considers all of the bus trips in the district. Without modifying the trips, the algorithm assigns them to routes such that the aggregate deadhead time and the number of routes are minimized. Individual routes become longer, allowing the bus to serve more trips in a single route.

Since the trips are fixed, in this way we can decrease the total time the buses are en route. Minimizing the deadhead travel results in cost savings and reductions in air pollution.

The routes that we generated can be viewed as a lower bound to the number of buses needed by school districts. We can find the optimal solution for HCPSS in less than a minute.

Serving all students

While we were working on routes, we decided to also tackle the problem of the bus trips themselves. To do this, we needed to determine what trips are required to serve the students for each school in the system, given bus capacities, stop locations and the number of students at each stop. This has a direct impact on how routes are chosen.

Most existing models aim to minimize either the total travel time or the total number of trips. The belief in such cases is that, by minimizing the number of trips, you can minimize the number of buses needed overall.

However, our work shows that this is not always the case. We found a way to cut down on the number of buses needed to satisfy transportation demands, without trying to minimize either of the above two objectives. Our approach considers not only minimizing the number of trips but also how these trips can be linked together.

New start times

Last October, we presented our work at the Maryland Association of Pupil Transportation conference. An audience member at that conference suggested that we analyze school start and dismissal times. By changing the high school, middle school and elementary school start times, bus operations could potentially be even more efficient. Slight changes in school start times can make it possible to link more trips together in a single bus route, hence decreasing the number of buses needed overall.

We developed a model that optimizes the school bell times, given that each of the elementary, middle and high school start times fall within a prespecified time window. For example, the time window for elementary school start times would be from 8:15 to 9:25 a.m.; for middle schools, from 7:40 to 8:30 a.m.; and all high schools would start at 7:25 a.m.

Our model looks at all of the bus trips and searches for the optimal combination of school dismissal time such that the number of school buses, which is the major contributing factor to costs, is minimized. We found that, in most cases, optimizing the bell times results in significant savings regarding the number of buses.

Next steps

Using our model, we ran many different “what if?” scenarios using different school start and dismissal times for the HCPSS. Four of these are currently under consideration by the Howard County School Board for possible implementation.

We are also continuing to enhance our current school bus transportation models, as well developing new ways to further improve efficiency and reduce costs.

For example, we are building models that can help schools select the right vendors for their transportation needs, as well as minimize the number of hours that buses run per day.

In the future, the type of models we are working on could be bundled into a software system that schools can use by themselves. There is really no impediment in using these types of systems as long as the school systems have an electronic database of their stops, trips, and routes.

Such software could potentially be implemented in all school districts in the nation. Many of these districts would benefit from using such models to evaluate their current operations and determine if any savings can be realized. With many municipalities struggling with budgets, this sort of innovation could save money without degrading service.

Ali Haghani, Professor of Civil & Environmental Engineering, University of Maryland and Ali Shafahi, Ph.D. Candidate in Computer Science, University of Maryland

Photo Credit: Dean Hochman


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This article was originally published on The Conversation. Read the original article.

3.14 essential reads about π for Pi Day

Editor’s note: The following is a roundup of archival stories. The Conversation

On March 14, or 3/14, mathematicians and other obscure-holiday aficionados celebrate Pi Day, honoring π, the Greek symbol representing an irrational number that begins with 3.14. Pi, as schoolteachers everywhere repeat, represents the ratio of a circle’s circumference to its diameter.

What is Pi Day, and what, really, do we know about π anyway? Here are three-and-bit-more articles to round out your Pi Day festivities.

A silly holiday

First off, a reflection on this “holiday” construct. Pi itself is very important, writes mathematics professor Daniel Ullman of George Washington University, but celebrating it is absurd:

The Gregorian calendar, the decimal system, the Greek alphabet, and pies are relatively modern, human-made inventions, chosen arbitrarily among many equivalent choices. Of course a mood-boosting piece of lemon meringue could be just what many math lovers need in the middle of March at the end of a long winter. But there’s an element of absurdity to celebrating π by noting its connections with these ephemera, which have themselves no connection to π at all, just as absurd as it would be to celebrate Earth Day by eating foods that start with the letter “E.”

And yet, here we are, looking at the calendar and getting goofily giddy about the sequence of numbers it shows us.

There’s never enough

In fact, as Jon Borwein of the University of Newcastle and David H. Bailey of the University of California, Davis, document, π is having a sustained cultural moment, popping up in literature, film, and song:

Sometimes the attention given to pi is annoying. On 14 August 2012, the U.S. Census Office announced the population of the country had passed exactly 314,159,265. Such precision was, of course, completely unwarranted. But sometimes the attention is breathtakingly pleasurable.

Come to think of it, pi can indeed be a source of great pleasure. Apple’s always comforting, and cherry packs a tart pop. Chocolate cream, though, might just be where it’s at.

Strange connections

Of course, π appears in all kinds of places that relate to circles. But it crops up in other places, too – often where circles are hiding in plain sight. Lorenzo Sadun, a professor of mathematics at the University of Texas at Austin, explores surprising appearances:

Pi also crops up in probability. The function f(x)=e-x², where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π.

It’s enough to make your head spin.

Historical pi

If you want to engage with π more directly, follow the lead of Georgia State University mathematician Xiaojing Ye, whose guide starts thousands of years ago:

The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1 – pretty close to the actual value, but still relatively far off.

By the end of his article, you’ll find a method to calculate π for yourself. You can even try it at home!

An irrational bonus

And because π is irrational, we’ll irrationally give you even one more, from education professor Gareth Ffowc Roberts at Bangor University in Wales, who highlights the very humble beginnings of the symbol π:

After attending a charity school, William Jones of the parish of Llanfihangel Tre’r Beirdd landed a job as a merchant’s accountant and then as a maths teacher on a warship, before publishing A New Compendium of the Whole Art of Navigation, his first book in 1702 on the mathematics of navigation. On his return to Britain he began to teach maths in London, possibly starting by holding classes in coffee shops for a small fee.

Shortly afterwards he published “Synopsis palmariorum matheseos,” a summary of the current state of the art developments in mathematics which reflected his own particular interests. In it is the first recorded use of the symbol π as the number that gives the ratio of a circle’s circumference to its diameter.

What made him realize that this ratio needed a symbol to represent a numeric value? And why did he choose π? It’s all Greek to us.

Jeff Inglis, Editor, Science + Technology, The Conversation

Photo Credit: Yelp Inc.

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Episode II: Math Jedi Matt Joins Black Jedi Don and Jamar on the BRBD Show

By TSS Admin

This Saturday evening, March 11th, at 6 pm central standard time on Twin Cities News Talk, Matt Johnson, our Editor-in-chief and mathematician, will be making his second guest apprearance on the Black Republican/Black Democrat show (BRBD).

Here’s the link to Matt’s first appearance on BRBD:

He will be joining co-hosts Donald Allen (R) and Jamar Nelson (D), and roving reporter Preya Samsundar from Alpha News, on the Black Jedi Radio Network to discuss Minneapolis economics and politics, why the presidential election polls and forecasts weren’t wrong, and Bill Nye “The Science Guy” and Tucker Carlson’s now infamous climate change exchange. This is sure to be a light-saber blazing event with a large audience.

Speaking of a large audience, the Black Republican/Black Democrat show has blown up on social media since Matt’s last visit on February 11th of this year. During Matt’s first visit, BRBD had 1,535 followers on their Facebook page. Since then, the Black Jedi Radio Network has gained nearly 5,000 followers; and so this time around, the Math Jedi Matt Johnson will have a much larger audience to share the gospel of mathematics with, while dueling with republicans and democrats.

Where can you listen?

For our Twin Cities’ readers, just simply turn the terrestrial dial to AM 1130 or FM 103.5. For our national readers, just download the iHeartRadio app or you can listen LIVE via the world-wide web by going to www.TwinCitiesNewsTalk.com, which is an iHeartRadio station. For our readers who would like to call into the show, dial (612) 986 – 0010.

We’ll see you Saturday night!

 

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Mathematics is beautiful (no, really)

For many people, memories of maths lessons at school are anything but pretty. Yet “beautiful” is a word that I and other mathematicians often use to describe our subject. How on earth can maths be beautiful – and does it matter?

For me, as a mathematician, it is hugely important. My enjoyment of the beauty of mathematics is part of what motivates me to study the subject. It is also a guide when I am working on a problem: if I think of a few strategies, I will choose the one that seems most elegant first. And if my solution seems clumsy then I will revisit it to try to make it more attractive.

I’ve just finished marking a pile of homework from my second-year mathematics undergraduates. I am struck by two students’ contrasting solutions to one problem. Both solutions are correct, both answer the question. And yet I much prefer one to the other. It’s not just that one is longer than the other, or that one is explained better than the other (both are described well, in fact).

The longer one doesn’t quite get to the heart of the matter, it’s a bit cluttered with unnecessary distractions. The other uses a different approach, which captures the essence of the ideas – it helps the reader to understand why this piece of mathematics works this way, not just that it does. For a mathematician, the “why” is critical, and we are always looking for arguments that reveal this.

Some cases of mathematical beauty are clear. Fractals, for example, are mathematical sets of numbers – corresponding to shapes – that have striking self-similarity and that have inspired numerous artists.

Less is more

But what about less obvious cases? Let me try to give you an example. Perhaps you recognise the sequence of numbers 1, 3, 6, 10, 15, 21, 28, … This is a sequence that students often encounter at school: the triangular numbers. Each number in the sequence corresponds to the number of dots in a sequence of triangles.

The six first triangular numbers: 1, 3, 6, 10, 15, 21.

Can we predict what the 1000th number in the sequence will be? There are many ways to tackle this question, and in fact unpicking the similarities and differences between these approaches is in itself both mathematical and enlightening. But here is one rather beautiful argument.

Imagine the 10th number in the sequence (because it’s easier to draw the picture than for the 1000th!). Let’s count the dots without counting the dots. We have a triangle of dots, with 10 in the bottom row and 10 rows of dots.

If we make another copy of that arrangement, we can rotate it and put it next to our original triangle of dots – so that the two triangles form a rectangle. This shape of dots will have 10 in the bottom row and 11 rows, so there are 10 x 11 = 110 dots in total (see figure below). Now we know that half of those were in our original triangle, so the 10th triangular number is 110/2 = 55. And we didn’t have to count them.

The 10th triangular number x2.

The power of this mathematical argument is that we can painlessly generalise to any number – even without drawing the dots. We can do a thought experiment. The 1000th triangle in the sequence will have 1000 dots in the bottom row, and 1000 rows of dots. By making another copy of this and rotating it, we get a rectangle with 1000 dots in the bottom row and 1001 rows. Half of those dots were in the original triangle, so the 1000th triangular number is (1000 x 1001)/2 = 500500.

For me, this idea of drawing the dots, duplicating, rotating and making a rectangle is beautiful. The argument is powerful, it generalises neatly (to any size of a triangle), and it reveals why the answer is what it is.

There are other ways to predict this number. One is to look at the first few terms of the sequence, guess a formula, and then prove that the formula does work (for example by using a technique called proof by induction). But that doesn’t convey the same memorable explanation behind the formula. There is an economy to the argument with pictures of dots, a single diagram captures everything we need to know.

Here’s another argument that I find attractive. Let’s think about the sum below:

The harmonic series.

This is the famous harmonic series. It turns out that it doesn’t equal a finite number – mathematicians say that the sum “diverges”. How can we prove that? It sounds difficult, but one elegant idea does the job.

The harmonic series with grouped terms.

Here each group of fractions adds up to more than ½. We know that ⅓ is bigger than ¼. That means (⅓) + (¼) is bigger than (¼) + (¼), which equals ½. So by adding enough blocks, each bigger than ½, the sum gets bigger and bigger – we can beat any target we like. By adding an infinite number of them we will get an infinite sum. We have tamed the infinite, with a beautiful argument.

A waiting game?

These are not the most difficult pieces of mathematics. One of the challenges of mathematics is that tackling more sophisticated problems often means first tackling more sophisticated terminology and notation. I cannot find a piece of mathematics beautiful unless I first understand it properly – and that means it can take a while for me to appreciate the aesthetic qualities.

I don’t think this unique to mathematics. There are pieces of music, buildings, pieces of visual art where I have not at first appreciated their beauty or elegance – and it is only by persevering, by grappling with the ideas, that I have come to perceive the beauty.

For me, one of the joys of teaching undergraduates is watching them develop their own appreciation of the beauty of mathematics. I’m going to see my second years this afternoon to go over their homework, and I already know that we’re going to have an interesting conversation about their different solutions – and that considering the aesthetic qualities will play a part in deepening their understanding of the mathematics.

School students can have just the same experience: when they’re given the opportunity to engage with rich questions, when they can play with mathematical ideas, when they have the chance to experience multiple strategies to the same question rather than just getting the answer in the back of the textbook and moving on. The mathematical ideas do not have to be university level, there are beautiful problems that are perfect for school students. Happily, there are many maths teachers and maths education projects that are helping students to have those experiences of the beauty of mathematics.

The Conversation

Vicky Neale, Whitehead Lecturer at the Mathematical Institute and Supernumerary Fellow at Balliol College, University of Oxford

Photo Credit: Ankush Sabharwal – CC BY-SA

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This article was originally published on The Conversation. Read the original article.