Original image
Wikimedia Commons

William Jones: The First Math Teacher to Use Pi

Original image
Wikimedia Commons

Eighteenth-century mathematician William Jones had a problem with pi—namely, it didn’t exist yet. At the time of his working and teaching in the field of mathematics, there was no term for the ratio of a circle’s circumference to its diameter, despite the value’s importance to even the most basic study of geometry. In his 1706 book, Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics, he made a modest proposal: that the universal constant be known as pi, and thus was mathematical history born.

Before being ascribed a modern name, pi existed under the guise of a bulkier, more antiquated phrase: quantitas in quam cum multiflicetur diameter, proveniet circumferencia—Latin for “the quantity which, when the diameter is multiplied by it, yields the circumference.” While descriptive, the collection of words required to denote pi before “pi” did not lend itself to clear or efficient discussion of the concept. Prior to Jones publishing his bold decision, fractions like 22/7 or 355/113 often served to fill in for the mysterious constant, but gave the erroneous impression that the number was a rational one, which can be fully expressed by one whole number divided into another—an assumption that had not yet been disproved, but with which Jones firmly disagreed. For this reason, only an idealized symbol would suffice to represent the concept, and so the Welshman turned to the Greek alphabet.

π, written in Roman letters as “pi,” is the Greek equivalent to our letter ‘p’. For this reason, 17th-century mathematician William Oughtred used π to denote the “periphery,” or the circumference of any given circle—a value that changed as the circle changed. Jones borrowed this earlier logic and applied it to his theory of an irrational, but universal constant value for the circle’s circumference-to-diameter ratio. Johann Lambert’s definitive proof in 1761 that π was an irrational number justified Jones’s earlier instinct, and once Swiss mathematician Leonhard Euler began to use and widely disseminate the symbol π in correspondence with his contemporaries, π was here to stay.

In honor of William Jones, unsung hero of pi, throw on one of these two new mental_floss t-shirts and have a moment of silence for the man responsible for turning pi from a letter into a number.

Original image
iStock // Ekaterina Minaeva
Man Buys Two Metric Tons of LEGO Bricks; Sorts Them Via Machine Learning
May 21, 2017
Original image
iStock // Ekaterina Minaeva

Jacques Mattheij made a small, but awesome, mistake. He went on eBay one evening and bid on a bunch of bulk LEGO brick auctions, then went to sleep. Upon waking, he discovered that he was the high bidder on many, and was now the proud owner of two tons of LEGO bricks. (This is about 4400 pounds.) He wrote, "[L]esson 1: if you win almost all bids you are bidding too high."

Mattheij had noticed that bulk, unsorted bricks sell for something like €10/kilogram, whereas sets are roughly €40/kg and rare parts go for up to €100/kg. Much of the value of the bricks is in their sorting. If he could reduce the entropy of these bins of unsorted bricks, he could make a tidy profit. While many people do this work by hand, the problem is enormous—just the kind of challenge for a computer. Mattheij writes:

There are 38000+ shapes and there are 100+ possible shades of color (you can roughly tell how old someone is by asking them what lego colors they remember from their youth).

In the following months, Mattheij built a proof-of-concept sorting system using, of course, LEGO. He broke the problem down into a series of sub-problems (including "feeding LEGO reliably from a hopper is surprisingly hard," one of those facts of nature that will stymie even the best system design). After tinkering with the prototype at length, he expanded the system to a surprisingly complex system of conveyer belts (powered by a home treadmill), various pieces of cabinetry, and "copious quantities of crazy glue."

Here's a video showing the current system running at low speed:

The key part of the system was running the bricks past a camera paired with a computer running a neural net-based image classifier. That allows the computer (when sufficiently trained on brick images) to recognize bricks and thus categorize them by color, shape, or other parameters. Remember that as bricks pass by, they can be in any orientation, can be dirty, can even be stuck to other pieces. So having a flexible software system is key to recognizing—in a fraction of a second—what a given brick is, in order to sort it out. When a match is found, a jet of compressed air pops the piece off the conveyer belt and into a waiting bin.

After much experimentation, Mattheij rewrote the software (several times in fact) to accomplish a variety of basic tasks. At its core, the system takes images from a webcam and feeds them to a neural network to do the classification. Of course, the neural net needs to be "trained" by showing it lots of images, and telling it what those images represent. Mattheij's breakthrough was allowing the machine to effectively train itself, with guidance: Running pieces through allows the system to take its own photos, make a guess, and build on that guess. As long as Mattheij corrects the incorrect guesses, he ends up with a decent (and self-reinforcing) corpus of training data. As the machine continues running, it can rack up more training, allowing it to recognize a broad variety of pieces on the fly.

Here's another video, focusing on how the pieces move on conveyer belts (running at slow speed so puny humans can follow). You can also see the air jets in action:

In an email interview, Mattheij told Mental Floss that the system currently sorts LEGO bricks into more than 50 categories. It can also be run in a color-sorting mode to bin the parts across 12 color groups. (Thus at present you'd likely do a two-pass sort on the bricks: once for shape, then a separate pass for color.) He continues to refine the system, with a focus on making its recognition abilities faster. At some point down the line, he plans to make the software portion open source. You're on your own as far as building conveyer belts, bins, and so forth.

Check out Mattheij's writeup in two parts for more information. It starts with an overview of the story, followed up with a deep dive on the software. He's also tweeting about the project (among other things). And if you look around a bit, you'll find bulk LEGO brick auctions online—it's definitely a thing!

Original image
Name the Author Based on the Character
May 23, 2017
Original image