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Neil Armstrong's Giant Leap

Neil Armstrong -- astronaut, engineer, professor, Navy pilot, and first man on the moon -- has died at the age of 82. He is best known for the words he spoke just after he set foot on the moon. Contrary to popular belief, Armstrong said (emphasis added): "That’s one small step for a man, one giant leap for mankind." That word "a" was garbled in the satellite feed heard by the world. Regardless of our ability to hear him, Armstrong was a man of powerful words. Here are a few more to remember him by.

The Moon Plaque

Apollo 11 plaque

Armstrong and fellow astronaut Buzz Aldrin left a plaque on the ladder of their moon lander, Eagle. The plaque read: "Here men from the planet Earth / First set foot upon the Moon / July 1969 A.D. / We came in peace for all mankind." It bore the signatures of the Apollo 11 crew members and President Nixon. He also left a small silicon disc bearing tiny messages of goodwill from various world leaders, as well as the names of various American dignitaries. You may enjoy this video of Armstrong placing the plaque and then reading its text to the world (his reading starts around 1:30):

In addition to that plaque, the messages of goodwill on the disc were mixed. Most were fairly bland messages of congratulations. But the message from Poland made it clear that the Cold War was in full swing:

"Although we are not suggesting any message from the Polish Head of State, please be assured that the achievements of the U.S. astronauts are followed by us with great interest, appreciation and best wishes for the success in their endeavor."

Sincerely,
Jorzy Michalowski
Ambassador, Poland

And in this short clip at the Apollo 11 40th anniversary celebration in 2009, Armstrong discusses how the space race functioned politically. "I'll not assert that it was a diversion which prevented a war, but nevertheless, it was a diversion."

The Congressional Gold Medal

Armstrong was awarded a Congressional Gold Medal on July 21, 2009. In this video, he shares some memories of the journey. He starts the speech: "Thank you, Mr. Chairman. I am in the position of a pilot without his checklist, so I'll have to wing it. ... [Prior to the Apollo missions,] no one knew what kind of person could be persuaded to take the trip. Prisoners were suggested. Soldiers could be ordered. Photographers could take pictures -- and they're expendable. Doctors understood the limits of human physiology. Finally, both sides picked pilots." Watch the rest for an explanation of how the Apollo missions worked.

Tranquility Base & "About to Turn Blue"

Armstrong suits up for the Apollo 11 launch on July 16, 1969Eagle's touchdown was tricky. The autopilot was sending the lander into a crater that Armstrong didn't like the looks of, so he took manual control and steered the vehicle to a new location, which he dubbed Tranquility Base -- apparently the first time Mission Control had heard the name. With seconds of fuel remaining (and people around the world holding their breath), Armstrong landed and announced the name of the first place where humans set foot on the Moon. Capcom Charlie Duke was audibly relieved, and just a bit flustered. The exchange:

Armstrong: "Houston, Tranquility Base here. The Eagle has landed."

Duke: (Momentarily tongue-tied) "Roger, Twan...(correcting himself) Tranquility. We copy you on the ground. You got a bunch of guys about to turn blue. We're breathing again. Thanks a lot."

You can read the full lunar landing transcript from NASA, including several audio clips in MP3 format.

The BBC Interview

In 1970, Armstrong was interviewed by the BBC about what it was like to be on the Moon. "I'm quite certain that we'll have such [lunar] bases in our lifetime, somewhat like the Antarctic stations and similar scientific outposts, continually manned."

The 60 Minutes Interview

Armstrong was a very private man. Here's a rare profile at age 75 on 60 Minutes. Highlights: he got his pilot's license at 15 -- before his driver's license; video showing his last-minute ejection from a near-fatal test flight (after which he walked back to his office and finished some paperwork); the dicey last-second landing of Eagle (and laughing with Walter Cronkite remembering that landing).

The biography mentioned in the video above is First Man: The Life of Neil A. Armstrong.

NASA's Remembrance

NASA has posted an obituary, including this quote from Apollo 11 astronaut Michael Collins (the man who did not get to walk on the Moon during that mission):

"He was the best, and I will miss him terribly." -- Michael Collins, Apollo 11 command module pilot.

NASA also posted this image showing Armstrong on the Moon. Most of the iconic astronaut-on-the-Moon photos are actually of Buzz Aldrin, taken by Armstrong. But this is the man himself:

Neil Armstrong on the Moon

A Glamour Shot

Here's Neil Armstrong in a Gemini G-2C training suit. Photo courtesy of NASA, via Wikipedia.

Neil Armstrong in a Gemini G-2C training suit

New York Times Archival Coverage

The New York Times has posted archival images and text from their coverage of the first moonwalk, with its famous, gigantic "MEN WALK ON MOON" headline. You may recall The Onion's profane-but-true spoof (warning: curse words, lots of 'em!) of that page. Here's a nice bit from the real NYT coverage:

Tentative Steps Test Soil

Mr. Armstrong's initial steps were tentative tests of the lunar soil's firmness and of his ability to move about easily in his bulky white spacesuit and backpacks and under the influence of lunar gravity, which is one-sixth that of the earth.

"The surface is fine and powdery," the astronaut reported. "I can pick it up loosely with my toe. It does adhere in fine layers like powdered charcoal to the sole and sides of my boots. I only go in a small fraction of an inch, maybe an eighth of an inch. But I can see the footprints of my boots in the treads in the fine sandy particles.

After 19 minutes of Mr. Armstrong's testing, Colonel Aldrin joined him outside the craft.

The two men got busy setting up another television camera out from the lunar module, planting an American flag into the ground, scooping up soil and rock samples, deploying scientific experiments and hopping and loping about in a demonstration of their lunar agility.

They found walking and working on the moon less taxing than had been forecast. Mr. Armstrong once reported he was "very comfortable."

And people back on earth found the black-and-white television pictures of the bug-shaped lunar module and the men tramping about it so sharp and clear as to seem unreal, more like a toy and toy-like figures than human beings on the most daring and far-reaching expedition thus far undertaken.

Armstrong Smiles After a Walk on the Moon

Finally, here's a photograph taken by Aldrin of Armstrong, after they returned from their walk on the Moon. That grin is infectious -- you can see the exuberant sense that "we did it" is on his face, along with the weariness of how hard it was. But we went to the Moon not because it was easy, but because it was hard. Rest in tranquility, Neil Armstrong.

Armstrong after Lunar EVA

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Animals
New Plankton Species Named After Sir David Attenborough Series Blue Planet
John Phillips, Getty Images for Tourism Australia
John Phillips, Getty Images for Tourism Australia

At least 19 creatures, both living and extinct, have been named after iconic British naturalist Sir David Attenborough. Now, for the first time, one of his documentary series will receive the same honor. As the BBC reports, a newly discovered phytoplankton shares its name with the award-winning BBC series Blue Planet.

The second half of the species' name, Syracosphaera azureaplaneta, is Latin for "blue planet," likely making it the first creature to derive its name from a television program. The single-cell organisms are just thousandths of a millimeter wide, thinner than a human hair, but their massive blooms on the ocean's surface can be seen from space. Called coccolithophores, the plankton serve as a food source for various marine life and are a vital marker scientists use to gauge the effects of climate change on the sea. The plankton's discovery, by researchers at University College London (UCL) and institutions in Spain and Japan, is detailed in a paper [PDF] published in the Journal of Nannoplankton Research.

"They are an essential element in the whole cycle of oxygen production and carbon dioxide and all the rest of it, and you mess about with this sort of thing, and the echoes and the reverberations and the consequences extend throughout the atmosphere," Attenborough said while accepting the honor at UCL.

The Blue Planet premiered in 2001 with eight episodes, each dedicated to a different part of the world's oceans. The series' success inspired a sequel series, Blue Planet II, that debuted on the BBC last year.

[h/t BBC]

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5 Ways You Do Complex Math in Your Head Without Realizing It
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The one thing that people who love math and people who hate math tend to agree on is this: You're only really doing math if you sit down and write formal equations. This idea is so widely embraced that to suggest otherwise is "to start a fight," says Maria Droujkova, math educator and founder of Natural Math, a site for kids and parents who want to incorporate math into their daily lives. Mathematicians cherish their formal proofs, considering them the best expression of their profession, while the anti-math don't believe that much of the math they studied in school applies to "real life."

But in reality, "we do an awful lot of things in our daily lives that are profoundly mathematical, but that may not look that way on the surface," Christopher Danielson, a Minnesota-based math educator and author of a number of books, including Common Core Math for Parents for Dummies, tells Mental Floss. Our mathematical thinking includes not just algebra or geometry, but trigonometry, calculus, probability, statistics, and any of the at least 60 types [PDF] of math out there. Here are five examples.

1. COOKING // ALGEBRA

Of all the maths, algebra seems to draw the most ire, with some people even writing entire books on why college students shouldn't have to endure it because, they claim, it holds the students back from graduating. But if you cook, you're likely doing algebra. When preparing a meal, you often have to think proportionally, and "reasoning with proportions is one of the cornerstones of algebraic thinking," Droujkova tells Mental Floss.

You're also thinking algebraically whenever you're adjusting a recipe, whether for a larger crowd or because you have to substitute or reduce ingredients. Say, for example, you want to make pancakes, but you only have two eggs left and the recipe calls for three. How much flour should you use when the original recipe calls for one cup? Since one cup is 8 ounces, you can figure this out using the following algebra equation: n/8 : 2/3.

algebraic equation illustrates adjustment of a recipe
Lucy Quintanilla

However, when thinking proportionally, you can just reason that since you have one-third less eggs, you should just use one-third less flour.

You're also doing that proportional thinking when you consider the cooking times of the various courses of your meal and plan accordingly so all the elements of your dinner are ready at the same time. For example, it will usually take three times as long to cook rice as it will a flattened chicken breast, so starting the rice first makes sense.

"People do mathematics in their own way," Droujkova says, "even if they cannot do it in a very formalized way."

2. LISTENING TO MUSIC // PATTERN THEORY AND SYMMETRY

woman enjoys listening to music in headphones
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The making of music involves many different types of math, from algebra and geometry to group theory and pattern theory and beyond, and a number of mathematicians (including Pythagoras and Galileo) and musicians have connected the two disciplines (Stravinsky claimed that music is "something like mathematical thinking").

But simply listening to music can make you think mathematically too. When you recognize a piece of music, you are identifying a pattern of sound. Patterns are a fundamental part of math; the branch known as pattern theory is applied to everything from statistics to machine learning.

Danielson, who teaches kids about patterns in his math classes, says figuring out the structure of a pattern is vital for understanding math at higher levels, so music is a great gateway: "If you're thinking about how two songs have similar beats, or time signatures, or you're creating harmonies, you're working on the structure of a pattern and doing some really important mathematical thinking along the way."

So maybe you weren't doing math on paper if you were debating with your friends about whether Tom Petty was right to sue Sam Smith in 2015 over "Stay With Me" sounding a lot like "I Won't Back Down," but you were still thinking mathematically when you compared the songs. And that earworm you can't get out of your head? It follows a pattern: intro, verse, chorus, bridge, end.

When you recognize these kinds of patterns, you're also recognizing symmetry (which in a pop song tends to involve the chorus and the hook, because both repeat). Symmetry [PDF] is the focus of group theory, but it's also key to geometry, algebra, and many other maths.

3. KNITTING AND CROCHETING // GEOMETRIC THINKING

six steps of crocheting a hyperbolic plane
Cheryl, Flickr // CC BY-SA 2.0

Droujkova, an avid crocheter, she says she is often intrigued by the very mathematical discussions fellow crafters have online about the best patterns for their projects, even if they will often insist they are awful at math or uninterested in it. And yet, such crafts cannot be done without geometric thinking: When you knit or crochet a hat, you're creating a half sphere, which follows a geometric formula.

Droujkova isn't the only math lover who has made the connection between geometry and crocheting. Cornell mathematician Daina Taimina found crocheting to be the perfect way to illustrate the geometry of a hyperbolic plane, or a surface that has a constant negative curvature, like a lettuce leaf. Hyperbolic geometry is also used in navigation apps, and explains why flat maps distort the size of landforms, making Greenland, for example, look far larger on most maps than it actually is.

4. PLAYING POOL // TRIGONOMETRY

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If you play billiards, pool, or snooker, it's very likely that you are using trigonometric reasoning. Sinking a ball into a pocket by using another ball involves understanding not just how to measure angles by sight but triangulation, which is the cornerstone of trigonometry. (Triangulation is a surprisingly accurate way to measure distance. Long before powered flight was possible, surveyors used triangulation to measure the heights of mountains from their bases and were off by only a matter of feet.)

In a 2010 paper [PDF], Louisiana mathematician Rick Mabry studied the trigonometry (and basic calculus) of pool, focusing on the straight-in shot. In a bar in Shreveport, Louisiana, he scribbled equations on napkins for each shot, and he calculated the most difficult straight-in shot of all. Most experienced pool players would say it’s one where the target ball is halfway between the pocket and the cue ball. But that, according to Mabry’s equations, turned out not to be true. The hardest shot of all had a surprising feature: The distance from the cue ball to the pocket was exactly 1.618 times the distance from the target ball to the pocket. That number is the golden ratio, which is found everywhere in nature—and, apparently, on pool tables.

Do you need to consider the golden ratio when deciding where to place the cue ball? Nope, unless you want to prove a point, or set someone else up to lose. You're doing the trig automatically. The pool sharks at the bar must have known this, because someone threw away Mabry's math napkins.

5. RE-TILING THE BATHROOM // CALCULUS

tiled bathroom with shower stall
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Many students don't get to calculus in high school, or even in college, but a cornerstone of that branch of math is optimization—or figuring out how to get the most precise use of a space or chunk of time.

Consider a home improvement project where you're confronted with tiling around something whose shape doesn't fit a geometric formula like a circle or rectangle, such as the asymmetric base of a toilet or freestanding sink. This is where the fundamental theorem of calculus—which can be used to calculate the precise area of an irregular object—comes in handy. When thinking about how those tiles will best fit around the curve of that sink or toilet, and how much of each tile needs to be cut off or added, you're employing the kind of reasoning done in a Riemann sum.

Riemann sums (named after a 19th-century German mathematician) are crucial to explaining integration in calculus, as tangible introductions to the more precise fundamental theorem. A graph of a Riemann sum shows how the area of a curve can be found by building rectangles along the x, or horizontal axis, first up to the curve, and then over it, and then averaging the distance between the over- and underlap to get a more precise measurement. 

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