Moondance

What is the shape of the moon’s path around the sun? The moon orbits the earth, and the earth orbits the sun, so many of us imagine it looks something like the image on the left, a looping motion in which the moon periodically slides “backward” during its progress around the larger body.

But it’s not! The shape is closer to a 13-gon with rounded corners; there are no loops. Helmer Aslaksen, a mathematician at the National University of Singapore, writes, “I like to visualize this as follows. Imagine you’re driving on a circular race track. You overtake a car on the right, and immediately slow down and go into the left lane. When the other car passes you, you speed up and overtake on the right again. You will then be making circles around the other car, but when seen from above, both of you are driving forward all the time and your path will be convex.”

More at his page.

(Thanks, Drake.)

March 8, 2019 | Science & Math

For Pi Day

From the prolifically interesting Catriona Shearer: The red line is perpendicular to the bases of the three semicircles. What’s the total area shaded in yellow?

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At first it appears there’s not enough information, but there is! No matter the relative sizes of the two smaller semicircles, the area of the shaded region is π. There are any number of ways of proving this; here’s a pretty “proof without words” that relies on the fact that the Pythagorean theorem works with similar figures of any shape (here, semicircles). (Thanks to Bill Bowser for the illustration.)

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March 2, 2019March 3, 2019 | Puzzles · Science & Math

A Thorough Anagram

This is incredible. In 2005, mathematician Mike Keith took a 717-word section from the essay on Mount Fuji in Lafcadio Hearn’s 1898 Exotics and Retrospective and anagrammed it into nine 81-word poems, each inspired by an image from Hokusai’s famous series of landscape woodcuts, the Views of Mount Fuji.

That’s not the most impressive part. Each anagrammed poem can be arranged into a 9 × 9 square, with one word in each cell. Stacking the nine grids produces a 9 × 9 × 9 cube. Make two of these cubes, and then:

In Cube “D” (for Divisibility), assign each cell the number “1” if the sum of the letter values in the corresponding word (using A=1, B=2, C=3 etc.) is exactly divisible by 9, or “0” if it is not.
In Cube “L” (for Length), assign each cell the number “1” if its word has exactly nine letters, or “0” if it does not.

Replace each “1” cell with solid wood and each “0” cell with transparent glass. Now suspend the two cubes in a room and shine beams of light from the top and right onto Cube D and from the front and right onto Cube L:

mike keith anagram cubes

The shadows they cast form reasonable renderings of four Japanese kanji characters relevant to the anagram:

The red shadow is the symbol for fire.
The green shadow is the symbol for mountain.
Put together, these make the compound Kanji symbol (“fire-mountain”) for volcano.

The white shadow is the symbol for wealth, pronounced FU
The blue shadow is the symbol for samurai, pronounced JI
Put together, these make the compound word Fuji, the name of the mountain.

See Keith’s other anagrams, including a 211,000-word recasting of Moby-Dick.

March 1, 2019 | Language · Science & Math

A Pretty Puzzle

I don’t know who came up with this; I found it on r/mathpuzzles. What’s the area of the red region?

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Draw a line from each vertex to the interior point where the four regions meet. This divides each region into two triangles. On each side of the large figure (north, south, east, and west), two of these triangles are neighbors and have the same area (because they have the same base and height) (here’s a figure). The four triangles that make up the northeast and southwest regions have the same area as the four triangles that make up the northwest and southeast regions. This means that each pair of diagonally opposite regions makes up half the total area of the figure. So the missing area is 28 cm². 02/24/2019 Reader Bill Bowser sent this diagram (thanks, Bill):

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February 21, 2019February 24, 2019 | Puzzles · Science & Math

Topology

https://commons.wikimedia.org/wiki/File:Bridge_29_Macclesfield_Canal.jpg — Image: Wikimedia Commons

Like the Tehachapi Loop, this is a beautiful solution to a nonverbal problem. When the towpath switches to the other side of a canal, how can you move your horse across the water without having to unhitch it from the boat it’s towing?

The answer is a roving bridge (this one is on the Macclesfield Canal in Cheshire). With two ramps, one a spiral, the horse passes through 360 degrees in crossing the canal, and the tow line never has to be unfastened.

February 20, 2019 | Science & Math · Technology

A Bite

From the Royal Society of Chemistry’s Chemistry World blog: In 1955, when impish graduate student A.T. Wilson published a paper with his humorless but brilliant supervisor, Melvin Calvin, Wilson made a wager with a department secretary that he could sneak a picture of a man fishing into one of the paper’s diagrams. He won the wager — can you find the fisherman?

February 20, 2019 | Science & Math

“Algebraic Theory of French Letters”

From Lee Sallows:

sallows french letters

(Thanks, Lee!)

February 16, 2019 | Language · Science & Math

Plutchik’s Wheel of Emotions

In 1958 psychologist Robert Plutchik suggested that there are eight primary emotions: joy, sadness, anger, fear, trust, disgust, surprise, and anticipation. Each of the eight exists because it serves an adaptive role that gives it survival value — for example, fear inspires the fight-or-flight response.

He arranged them on a wheel to show their relationships, with similar emotions close together and opposites 180 degrees apart. Like colors, emotions can vary in intensity (joy might vary from serenity to ecstasy), and they can mix to form secondary emotions (submission is fear combined with trust, and awe is fear combined with surprise).

When all these combinations are included, the system catalogs 56 emotions at 1 intensity level. And in his final “structural model” of emotions, the petals are folded up in a third dimension to form a cone.

February 15, 2019 | Science & Math

Dictum

Hunters of the 19th century defended their practice in part because it was the only way to identify species that would otherwise remain unknown.

They distilled this into an adage: “What’s hit is history, what’s missed is mystery.”

February 14, 2019 | Science & Math · Society

Art Appreciation

https://commons.wikimedia.org/wiki/File:African_Hooded_Vulture_(Necrosyrtes_monachus),_Kruger_National_Park.jpg — Image: Wikimedia Commons

In 1833, to show that vultures found their prey by sight rather than smell, naturalist John Bachman made “a coarse painting representing a sheep skinned and cut open”:

This proved very amusing — no sooner was this picture placed on the ground than the Vultures observed it, alighted near, walked over it, and some of them commenced tugging at the painting. They seemed much disappointed and surprised, and after having satisfied their curiosity, flew away. This experiment was repeated more than fifty times, with the same result.

He confirmed the result by setting the painting within two feet of a heap of camouflaged offal in his garden. “They came as usual, walked around it, but in no instance evinced the slightest symptoms of their having scented the offal which was so near them.” He concluded that, while vultures may have a sense of smell, they don’t use it to find food.

See Vulture Picnic.

February 8, 2019 | Science & Math