This coming Saturday is “Pi Day of the Century”. The date 3/14/15 in month/day/year format is like the first digits of π=3.1415… And at 9:26:53.589… it’s a “super pi moment”.

Between *Mathematica* and Wolfram|Alpha, I’m pretty sure our company has delivered more π to the world than any other organization in history. So of course we have to do something special for Pi Day of the Century.

One of my main roles as CEO is to come up with ideas—and I’ve spent decades building an organization that’s good at turning those ideas into reality. Well, a number of weeks ago I was in a meeting about upcoming corporate events, and someone noted that Pi Day (3/14) would happen during the big annual SXSW (South by Southwest) event in Austin, Texas. So I said (or at least I thought I said), “We should have a big pi to celebrate Pi Day.”

I didn’t give it another thought, but a couple of weeks later we had another meeting about upcoming events. One agenda item was Pi Day. And the person who runs our Events group started talking about the difficulty of finding a bakery in Austin to make something suitably big. “What are you talking about?” I asked. And then I realized: “You’ve got the wrong kind of pi!”

I guess in our world pi confusions are strangely common. Siri’s voice-to-text system sends Wolfram|Alpha lots of “pie” every day that we have to specially interpret as “pi”. And then there’s the Raspberry Pi, that has the Wolfram Language included. And for me there’s the additional confusion that my personal fileserver happens to have been named “pi” for many years.

After the pi(e) mistake in our meeting we came up with all kinds of wild ideas to celebrate Pi Day. We’d already rented a small park in the area of SXSW, and we wanted to make the most interesting “pi countdown” we could. We resolved to get a large number of edible pie “pixels”, and use them to create a π shape inside a pie shape. Of course, there’ll be the obligatory pi selfie station, with a “Stonehenge” pi. And a pi(e)-decorated Wolfie mascot for additional selfies. And of course we’ll be doing things with Raspberry Pis too.

I’m sure we’ll have plenty of good “pi fun” at SXSW. But we also want to provide pi fun for other people around the world. We were wondering, “What can one do with pi?” Well, in some sense, you can do anything with pi. Because, apart from being the digits of pi, its infinite digit sequence is—so far as we can tell—completely random. So for example any run of digits will eventually appear in it.

How about giving people a personal connection to that piece of math? Pi Day is about a date that appears as the first digits of pi. But any date appears somewhere in pi. So, we thought: Why not give people a way to find out where their birthday (or other significant date) appears in pi, and use that to make personalized pi T-shirts and posters?

In the Wolfram Language, it’s easy to find out where your birthday appears in π. It’s pretty certain that any mm/dd/yy will appear somewhere in the first 10 million digits. On my desktop computer (a Mac Pro), it takes 6.28 seconds (2π?!) to compute that many digits of π.

Here’s the Wolfram Language code to get the result and turn it into a string (dropping the decimal point at position 2):

Now it’s easy to find any “birthday string”:

So, for example, my birthday string first appears in π starting at digit position 151,653.

What’s a good way to display this? It depends how “pi lucky” you are. For those born on 4/15/92, their birthdate already appears at position 3. (Only about a certain fraction of positions correspond to a possible date string.) People born on November 23, 1960 have the birthday string that’s farthest out, appearing only at position 9,982,546. And in fact most people have birthdays that are pretty “far out” in π (the average is 306,150 positions).

Our long-time art director had the idea of using a spiral that goes in and out to display the beginnings and ends of such long digit sequences. And almost immediately, he’d written the code to do this (one of the great things about the Wolfram Language is that non-engineers can write their own code…).

Next came deploying that code to a website. And thanks to the Wolfram Programming Cloud, this was basically just one line of code! So now you can go to MyPiDay.com…

…and get your own piece of π!

And then you can share the image, or get a poster or T-shirt of it:

With all this discussion about pi, I can’t resist saying just a little about the science of pi. But first, just why is pi so famous? Yes, it’s the ratio of circumference to diameter of a circle. And that means that π appears in zillions of scientific formulas. But it’s not the whole story. (And for example most people have never even heard of the analog of π for an ellipse—a so-called complete elliptic integral of the second kind.)

The bigger story is that π appears in a remarkable range of mathematical settings—including many that don’t seem to have anything to do with circles. Like sums of negative powers, or limits of iterations, or the probability that a randomly chosen fraction will not be in lowest terms.

If one’s just looking at digit sequences, pi’s 3.1415926… doesn’t seem like anything special. But let’s say one just starts constructing formulas at random and then doing traditional mathematical operations on them, like summing series, doing integrals, finding limits, and so on. One will get lots of answers that are 0, or 1/2, or . And there’ll be plenty of cases where there’s no closed form one can find at all. But when one can get a definite result, my experience is that it’s remarkably common to find π in it.

A few other constants show up too, like *e* (2.1718…), or Euler gamma (0.5772…), or Catalan’s constant (0.9159…). But π is distinctly more common.

Perhaps math could have been set up differently. But at least with math as we humans have constructed it, the number that is π is a widespread building block, and it’s natural that we gave it a name, and that it’s famous—now even to the point of having a day to celebrate it.

What about other constants? “Birthday strings” will certainly appear at different places in different constants. And just like when Wolfram|Alpha tries to find closed forms for numbers, there’s typically a tradeoff between digit position and obscurity of the constants used. So, for example, my birthday string appears at position 151,653 in π, 241,683 in *e*, 45,515 in , 40,979 in ζ(3) … and 196 in the 1601th Fibonacci number.

Let’s say you make a plot that goes up whenever a digit of π is 5 or above, and down otherwise:

It looks just like a random walk. And in fact, all statistical and cryptographic tests of randomness that have been tried on the digits (except tests that effectively just ask “are these the digits of pi?”) say that they look random too.

Why does that happen? There are fairly simple procedures that generate digits of pi. But the remarkable thing is that even though these procedures are simple, the output they produce is complicated enough to seem completely random. In the past, there wasn’t really a context for thinking about this kind of behavior. But it’s exactly what I’ve spent many years studying in all kinds of systems—and wrote about in *A New Kind of Science*. And in a sense the fact that one can “find any birthday in pi” is directly connected to concepts like my general Principle of Computational Equivalence.

Of course, just because we’ve never seen any regularity in the digits of pi, it doesn’t mean that no such regularity exists. And in fact it could still be that if we did a big search, we might find somewhere far out in the digits of pi some strange regularity lurking.

What would it mean? There’s a science fiction answer at the end of Carl Sagan’s book version of *Contact*. In the book, the search for extraterrestrial intelligence succeeds in making contact with an interstellar civilization that has created some amazing artifacts—and that then explains that what they in turn find remarkable is that encoded in the distant digits of pi, they’ve found intelligent messages, like an encoded picture of a circle.

At first one might think that finding “intelligence” in the digits of pi is absurd. After all, there’s just a definite simple algorithm that generates these digits. But at least if my suspicions are correct, exactly the same is actually true of our whole universe, so that every detail of its history is in principle computable much like the digits of pi.

Now we know that within our universe we have ourselves as an example of intelligence. SETI is about trying to find other examples. The goal is fairly well defined when the search is for “human-like intelligence”. But—as my Principle of Computational Equivalence suggests—I think that beyond that it’s essentially impossible to make a sharp distinction between what should be considered “intelligent” and what is “merely computational”.

If the century-old mathematical suspicion is correct that the digits of pi are “normal”, it means that every possible sequence eventually occurs among the digits, including all the works of Shakespeare, or any other artifact of any possible civilization. But could there be some other structure—perhaps even superimposed on normality—that for example shows evidence of the generation of intelligence-like complexity?

While it may be conceptually simple, it’s certainly more bizarre to contemplate the possibility of a human-like intelligent civilization lurking in the digits of pi, than in the physical universe as explored by SETI. But if one generalizes what one counts as intelligence, the situation is a lot less clear.

Of course, if we see a complex signal from a pulsar magnetosphere we say it’s “just physics”, not the result of the evolution of a “magnetohydrodynamic civilization”. And similarly if we see some complex structure in the digits of pi, we’re likely to say it’s “just mathematics”, not the result of some “number theoretic civilization”.

One can generalize from the digit sequence of pi to representations of any mathematical constant that is easy to specify with traditional mathematical operations. Sometimes there are simple regularities in those representations. But often there is apparent randomness. And the project of searching for structure is quite analogous to SETI in the physical universe. (One difference, however, is that π as a number to study is selected as a result of the structure of our physical universe, our brains, and our mathematical development. The universe presumably has no such selection, save implicitly from the fact the we exist in it.)

I’ve done a certain amount of searching for regularities in representations of numbers like π. I’ve never found anything significant. But there’s nothing to say that any regularities have to be at all easy to find. And there’s certainly a possibility that it could take a SETI-like effort to reveal them.

But for now, let’s celebrate the Pi Day of our century, and have fun doing things like finding birthday strings in the digits of pi. Of course, someone like me can’t help but wonder what success there will have been by the next Pi Day of the Century, in 2115, in either SETI or “SETI among the digits”…

Pictures from the Pi Day event:

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For those of you who are interested, Wolfram|Alpha possesses a wealth of sports stats so that you can get all the cold, hard facts about the Patriots and the Seahawks.

And if you can’t wait for Sunday to get your next football fix, or find yourself suffering withdrawal afterward, VICTIV is doing very cool things with the Wolfram Language to run a fantasy sports league. Earl Mitchell delves into the step-by-step process for new users on his blog, The Rotoquant.

But some of you are probably just plain old tired of all this “Deflatriots” business and of having your television occupied by football games, news, talking heads, and commercials from September through February, because after a while, the teams start to blur together. Fortunately, with the help of the Wolfram Language, you can pick your team out of the crowd using this `Graph` of NFL logos we created by pulling the images from our Wolfram Knowledgebase and using `Nearest` to organize them by graphical similarity.

If you’re one of those who are weary of all the football hoopla, then let us soothe your soul with a time-honored and longstanding tradition of cuteness: Animal Planet’s Puppy Bowl XI.

With celebrities such as Katty Furry performing in the halftime show, it promises to be the most adorable sports game you’ll watch all year. The competition will be fierce, with 57 shelter-donated puppies—all up for adoption!—fighting for the honor to be the Bissel MVP (Most Valuable Puppy).

Past MVPs have included Max and Abigail, both Jack Russell Terriers, and the last MVP, Loren, was a Brittany, a breed not present in 2015′s lineup.

It’s not unlikely that one of the eight Labrador Retrievers will take home the prize for the first time ever. Again using the Wolfram Language, here’s the breakdown of Puppy Bowl breeds:

But who knows, one of those Beagles could come out of the end zone and snatch the victorious touchdown from right under their wet noses. Are you ready for some puppy ball?

Now you may think it was trivial to add this functionality given that indefinite integrals already have steps, but there are many tricky cases to consider: before we even begin to integrate, the continuity of the function is examined. If there are discontinuities in the integration domain, the domain is split and the integral is evaluated separately over each domain.

We must determine if the integral is proper or improper.

Absolute values need to be handled carefully.

Symmetries can be exploited.

Simplification of radicals and logarithms must be done very carefully.

Finally, the fundamental theorem of calculus requires that the antiderivative is continuous over the integration domain (see this blog post for more information). Therefore, we need to be careful when finding the indefinite integral, and always ensure the result will be continuous. One way to do this is to detect when we will have a discontinuous antiderivative and split the integration domain up.

Integration is an extremely nontrivial problem, so we hope these Step-by-step solutions will help you learn how they can be done. Be sure to check out Step-by-step solutions for other topics too.

]]>Summer has drawn to a close, and so too have our annual internships. Each year Wolfram welcomes a new group of interns to work on an exciting array of projects ranging all the way from Bell polynomials to food science. It was a season for learning, growth, and making strides across disciplinary and academic divides. The Wolfram interns are an invaluable part of our team, and they couldn’t wait to tell us all about their time here. Here are just a few examples of the work that was done.

Paco Jain

Wolfram|Alpha Scientific Content,

Wolfram|Alpha

Wolfram|Alpha Scientific Content,

Wolfram|Alpha

“This summer, I worked on adding scientific content to the physical systems domain in Wolfram|Alpha. While there is a lot to learn, everyone I worked with seemed enthusiastic to help me get up to speed, and I was able to form several valuable mentoring relationships. I also felt that I was given the resources and responsibility I needed to allow me to make meaningful contributions to the Wolfram|Alpha product. The experience has me already thinking about pursuing a full-time position at Wolfram!”

Daniel McDonald

Wolfram|Alpha Scientific Content,

Bell Polynomials and Recursive Algorithms

Wolfram|Alpha Scientific Content,

Bell Polynomials and Recursive Algorithms

“This summer at Wolfram|Alpha I worked as the Special Functions Intern. My primary project was reading mathematical literature in order to extract and verify formulas that could be useful for The Wolfram Functions Site as well as for possible *Mathematica* implementation. The most interesting part of my work involved creating a compendium of information about *Mathematica*‘s `BellY` function that computes various types of Bell polynomials, which are used in Faà di Bruno’s formula for computing arbitrary derivatives of the composition *f*(*g*) (as well as in generalizations of this formula for computing arbitrary derivatives of compositions of arbitrary depth). I devised an original functional recurrence that suggested a quick recursive algorithm for computing generalized Bell polynomials; as this algorithm ran much faster than *Mathematica*‘s at the time, it was implemented into *Mathematica* 10.0.1. This recurrence and thus the algorithm (with different base cases) can be applied in a more general environment, and I am currently drafting a paper to submit to an algorithms journal.”

Mark Peterson

Scientific Information Group,

Wolfram Demonstrations Project

Scientific Information Group,

Wolfram Demonstrations Project

“During my internship in the Scientific Information Group at Wolfram Research, my work has primarily been centered on the Wolfram Demonstrations Project. Essentially, Demonstrations are self-contained programs written in the Wolfram Language that are designed to appeal to the user in a highly intuitive and interactive way. Whether working on the Project directly or on alternate applications for its material, my time has been spent developing this sort of content.”

Jake Wood

*Mathematica* Algorithms R&D,

*Mathematica* `GeoGraphics`

“Joining the Wolfram team earlier this summer was an exciting professional milestone for me. I am a big fan of not only the software that has come from Wolfram, but also the mission and ambition to proliferate and nurture big ideas. My patient mentor explained that I was to figure out how to make the generated maps in `GeoGraphics` (new in *Mathematica* 10) move around and update from mouse clicking and dragging. Additionally, the maps needed to be zoomable, similar to maps online used for navigation. Right now my prototypes deal with the maps themselves instead of the verbose layers of graphics data that *Mathematica* is capable of imbuing. In the future, though, who knows. Getting the panning and zooming to work proved a difficult task; however, the brunt of the summer was spent on improving the performance speed. No one wants to use an interactive map that is insufferably unresponsive. The utility of this application is pretty clear, as it is similar to programs that people already use daily.”

Jessica Zhang

User Experience,

Wolfram*Tones*

User Experience,

Wolfram

“People would think as a User Experience Designer I would only be designing detailed features within a product or workflow. However, at Wolfram, I not only got to do those things, I also got to take part in the bigger decision-making design processes, even as an intern. I was given the opportunity to learn a variety of skills that are important and also at the cutting edge of the field. Technical skills include wireframing, wireflowing, diagramming, and interface design. Oh, and also using the espresso machine!”

Andrew Blanchard

Wolfram|Alpha Scientific Content,

Named Physical Effects

Wolfram|Alpha Scientific Content,

Named Physical Effects

“For my internship with Wolfram Alpha, I assembled a list of named physical effects. A typical effect provides a link between measurable physical quantities, which are already incorporated into Wolfram|Alpha. Thus, making information about known physical effects computable enables the exploration of relationships between measurable quantities. In addition, the searchable data provides a window into the relationship between the discovery of new effects and advances in the field of physics. By making scientific information searchable, Wolfram|Alpha is providing a wonderful service for researchers, students, and anyone curious about exploring science.”

Surojit Ganguli

Wolfram|Alpha Socioeconomic Content,

Computational Capabilities

“I was part of the team that was involved in increasing the computational capabilities of Wolfram|Alpha in the domain of vehicle dynamics. As a Computational Science and Engineering Minor at UIUC, the opportunity to explore the various ways in which computations are being performed at Wolfram was in itself a rewarding experience. As an additional bonus, I definitely improved in the area of functional programming by using *Mathematica*.”

Ying Qin

Wolfram|Alpha Scientific Content,

Food Data

Wolfram|Alpha Scientific Content,

Food Data

“I’ve been working on expanding food-related information in the Wolfram Knowledgebase. Among other things, this included the characterization and classification of food; I did research involving USDA data and other data sources. I was also working on expanding the food glossary, which gives a more detailed description of the available content. Furthermore, using my knowledge as a Food Science student, I was able to do things like classify fatty acids into groups. My advice to prospective interns is that you shouldn’t hesitate to apply even though your major is not computer science or engineering. As a Food Science major, I was happy to get involved here, and felt like it was a truly valuable experience.”

It’s been an amazing summer all around, and we couldn’t be happier with the contributions our interns have made. While we are sad to see some of them go, we are excited by the new talent that has been added to our team and can’t wait to see what next year will bring!

]]>Our live-tweets will feature facts, queries, and commentary about the show. We’ll take a look at things like the math behind siphoning electricity and insightful statistics based on character names. We’re happy to see another installment in the good fight for the nerd revolution and can’t wait to watch as a bunch of brainiacs save the world. Join us for a night of science and suspense and send us your own Wolfram|Alpha Scorpion queries to @wolfram_alpha!

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