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Math Food for Pi Approximation Day

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July 22 is Pi Approximation Day because pi's fractional approximation is 22/7. To celebrate, here are some math foods Miss C rounded up last year.

Food is often used to illustrate shapes, and recipes are used to introduce children to measurement and proportions. But if you really care about your mathematics -and your food- there are ways to take the marriage of math and food to the next level!

Pi Pie

The simplest way to incorporate math into food is to bake a Pi Pie. Meg made this lovely Pi Pie to celebrate Pi Day in 2004. It's a normal fruit pie with the pi symbol carved into the top crust. Then the numerals of pi were cut from pie crust and added to the rim of the pie. Yummy!

Möbius Strip Bagel

George Hart, who I've run across before, has turned breakfast into a puzzle with a bagel that has been sliced into two halves that are linked together! The secret is that the cut itself is a two-twist Möbius strip. If you can't quite visualize that, there are explicit instructions for this project. This is sure to impress your mother-in-law, one way or another.

Sierpinski Carpet Cookies

Sierpinski Cookies-11

Lenore at Evil Mad Scientist Laboratories made cookies in the fractal pattern of the Sierpinski Carpet. These are made with butter cookie dough, a portion of which was colored with delicious cocoa powder. The dough is shaped, stacked, and stretched, then stacked and stretched again, and when the fractal is at your limit, the dough is sliced to make cookies. Illustrated instructions are included at the laboratory link.

Fractal Snowflake Cupcakes - 24

Lenore also gave us instructions for making this beautiful fractal cupcake in the Koch Snowflake pattern. This uses the same method of stacking and rolling out iterations of fondant, this time tinted blue or left white.

Menger Sponge Gingerbread House

This fractal gingerbread house resembles the Sierpinski carpet a bit, but a Menger sponge contains holes rather than cocoa cookie dough.

A Menger sponge is a 3D fractal that starts with a cube. In each iteration the surface of each cube is divided into 9 sections and the middle section is pushed out.

In the next iteration, you look at the smaller subcubes of the original cube and repeat the same process.

in the picture below you can see the same process. With this gingerbread fractal we'll be making a level 2 Menger sponge and using decoration to make it look like a level 3.

The actual making of the gingerbread is easier than the explanation of the fractal. Understanding fractals can be plenty intuitive when you've seen them built.

Sierpinski Hamantaschen

Hamantaschen is a Jewish holiday pastry that comes in a triangle shape filled with prunes, poppy seeds, or other sweet filling. Deborah Gardner took her family's traditional recipe and made "Sierpinskitaschen," or Sierpinski Hamantaschen. This is not pieced together like the cookies, but it begins with a large pastry which is then subdivided as may times as you can.

Apple Chain

Marius van Voorden enjoys food sculpting. He was inspired by Neil Fraser's soap carvings of links from a single solid, and decided to do the same with apples. You won't find step-by-step instructions, but I suppose it's about as difficult as cutting the bagel above.

Hyperbolic Dried Fruit

Vi Hart has a gift for making mathematics into something enjoyable, even if you can't keep up with the math. She wrote on The Hyperbolic Nature of Dried Fruit Slices because she found it strange that when sliced fruit dried, it not only shrunk but changed shape completely, in a predictable manner. For example, as it dries, an apple slice curls up into a hyperbolic plane! Hart did some experiments and found that it made no difference whether she cut the apple vertically or horizontally, or whether the peel was removed. The same effect can be seen in potato chips, which are fried instead of dried. See Hart's other posts on slicing apples into mathematical shapes.

Cutting the Cheese

High school senior Luyi does many things with geometric shapes. Recently she cut the cheese into a multitude of shapes. A solid cube of cheese became a hexagonal prism, then with some intricate-but-understandable steps, Luyi was left with "one funky-shaped polyhedron and six small congruent tetrahedra." Then the fun starts, as you can find the volume of your irregular polygon by calculating the volume of the shapes you cut off! Or, you could just get the crackers out and finish off this experiment.

Also check out Luyi's Dark and white chocolate Sierpinski triangle brownies. Now every time I come across a Sierpinski triangle, I will start to feel hungry.

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

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Library of Congress
10 Facts About the Tomb of the Unknown Soldier
May 29, 2017
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Library of Congress

On Veterans Day, 1921, President Warren G. Harding presided over an interment ceremony at Arlington National Cemetery for an unknown soldier who died during World War I. Since then, three more soldiers have been added to the Tomb of the Unknowns (also known as the Tomb of the Unknown Soldier) memorial—and one has been disinterred. Below, a few things you might not know about the historic site and the rituals that surround it.


Wikimedia Commons // Public Domain

To ensure a truly random selection, four unknown soldiers were exhumed from four different WWI American cemeteries in France. U.S. Army Sgt. Edward F. Younger, who was wounded in combat and received the Distinguished Service Medal, was chosen to select a soldier for burial at the Tomb of the Unknowns in Arlington. After the four identical caskets were lined up for his inspection, Younger chose the third casket from the left by placing a spray of white roses on it. The chosen soldier was transported to the U.S. on the USS Olympia, while the other three were reburied at Meuse Argonne American Cemetery in France.


One had served in the European Theater and the other served in the Pacific Theater. The Navy’s only active-duty Medal of Honor recipient, Hospitalman 1st Class William R. Charette, chose one of the identical caskets to go on to Arlington. The other was given a burial at sea.


WikimediaCommons // Public Domain

The soldiers were disinterred from the National Cemetery of the Pacific in Hawaii. This time, Army Master Sgt. Ned Lyle was the one to choose the casket. Along with the unknown soldier from WWII, the unknown Korean War soldier lay in the Capitol Rotunda from May 28 to May 30, 1958.


Medal of Honor recipient U.S. Marine Corps Sgt. Maj. Allan Jay Kellogg, Jr., selected the Vietnam War representative during a ceremony at Pearl Harbor.


Wikipedia // Public Domain

Thanks to advances in mitochondrial DNA testing, scientists were eventually able to identify the remains of the Vietnam War soldier. On May 14, 1998, the remains were exhumed and tested, revealing the “unknown” soldier to be Air Force 1st Lt. Michael Joseph Blassie (pictured). Blassie was shot down near An Loc, Vietnam, in 1972. After his identification, Blassie’s family had him moved to Jefferson Barracks National Cemetery in St. Louis. Instead of adding another unknown soldier to the Vietnam War crypt, the crypt cover has been replaced with one bearing the inscription, “Honoring and Keeping Faith with America’s Missing Servicemen, 1958-1975.”


The Tomb was designed by architect Lorimer Rich and sculptor Thomas Hudson Jones, but the actual carving was done by the Piccirilli Brothers. Even if you don’t know them, you know their work: The brothers carved the 19-foot statue of Abraham Lincoln for the Lincoln Memorial, the lions outside of the New York Public Library, the Maine Monument in Central Park, the DuPont Circle Fountain in D.C., and much more.


Tomb Guards come from the 3rd U.S. Infantry Regiment "The Old Guard". Serving the U.S. since 1784, the Old Guard is the oldest active infantry unit in the military. They keep watch over the memorial every minute of every day, including when the cemetery is closed and in inclement weather.


Members of the Old Guard must apply for the position. If chosen, the applicant goes through an intense training period, in which they must pass tests on weapons, ceremonial steps, cadence, military bearing, uniform preparation, and orders. Although military members are known for their neat uniforms, it’s said that the Tomb Guards have the highest standards of them all. A knowledge test quizzes applicants on their memorization—including punctuation—of 35 pages on the history of the Tomb. Once they’re selected, Guards “walk the mat” in front of the Tomb for anywhere from 30 minutes to two hours, depending on the time of year and time of day. They work in 24-hour shifts, however, and when they aren’t walking the mat, they’re in the living quarters beneath it. This gives the sentinels time to complete training and prepare their uniforms, which can take up to eight hours.


The Tomb Guard badge is the least awarded badge in the Army, and the second least awarded badge in the overall military. (The first is the astronaut badge.) Tomb Guards are held to the highest standards of behavior, and can have their badge taken away for any action on or off duty that could bring disrespect to the Tomb. And that’s for the entire lifetime of the Tomb Guard, even well after his or her guarding duty is over. For the record, it seems that Tomb Guards are rarely female—only three women have held the post.


Everything the guards do is a series of 21, which alludes to the 21-gun salute. According to

The Sentinel does not execute an about face, rather they stop on the 21st step, then turn and face the Tomb for 21 seconds. They then turn to face back down the mat, change the weapon to the outside shoulder, mentally count off 21 seconds, then step off for another 21 step walk down the mat. They face the Tomb at each end of the 21 step walk for 21 seconds. The Sentinel then repeats this over and over until the Guard Change ceremony begins.