A Pi Diet

https://commons.wikimedia.org/wiki/File:Academ_rosette.svg Image: <a href="https://commons.wikimedia.org/wiki/File:Academ_rosette.svg">Wikimedia Commons</a>

Students beginning with the compass learn to draw this rosette, sometimes called the Flower of Life. If the arcs and the circle have the same radius, a, then the area of one petal is

\displaystyle X = a^{2}\left ( \frac{\pi }{3} - sin \frac{\pi }{3}\right ) = \left ( \frac{\pi }{3} - \frac{\sqrt{3}}{2}\right ) a^{2}

and the unshaded area of the circle is

\displaystyle Y = \pi a^{2} - 3X = \frac{3\sqrt{3}}{2}a^{2}.

Remarkably, though the area we sought is bounded entirely by the arcs of circles, the final expression is independent of π.

Related: When two cylinders of radius r meet at right angles, the volume of their intersection is 16r3/3 — again, no sign of π.

(J.V. Narlikar, “A Pi-Less Area,” Mathematical Gazette 65:431 [March 1981], 32-33.)

10/01/2024 UPDATE: Some deft rearranging shows that the unshaded area of the circle is just the area of an inscribed regular hexagon:

2024-09-28-a-pi-diet-2

So the absence of π isn’t that surprising. Thanks to readers Catalin Voinescu (who sent this diagram) and Gareth McCaughan for pointing this out.

| Science & Math

“Through the Looking-Glass”

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C.S. Kipping published this unusual problem in Chess Amateur in 1923. In each position, White is to mate in two moves.

Click for Answer
| Puzzles

Fish Story

https://commons.wikimedia.org/wiki/File:Blue_marlin.jpg

Then there is the other secret. There isn’t any symbolysm. The sea is the sea. The old man is an old man. The boy is a boy and the fish is a fish. The shark are all sharks no better and no worse. All the symbolism that people say is shit. What goes beyond is what you see beyond when you know. A writer should know too much.

— Ernest Hemingway, letter to Bernard Berenson, 1952

| Literature

A Belt Font

Suppose you have a collection of gears pinned to a wall (disks in the plane). When is it possible to wrap a conveyor belt around them so that the belt touches every gear, is taut, and does not touch itself? This problem was first posed by Manuel Abellanas in 2001. When all the gears are the same size, it appears that it’s always possible to find a suitable path for the belt, but the question remains open.

Erik Demaine, Martin Demaine, and Belén Palop have designed a font to illustrate the problem — each letter is a collection of equal-sized gears around which exactly one conveyor-belt wrapping outlines an English letter:

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Apart from its mathematical interest, the font makes for intriguing puzzles — when the belts are removed, the letters are surprisingly hard to discern. What does this say?

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Click for Answer
| Puzzles · Science & Math

The Sussman Anomaly

MIT computer scientist Gerald Sussman offered this example to show the importance of sophisticated planning algorithms in artificial intelligence. Suppose an agent is told to stack these three blocks into a tower, with A at the top and C at the bottom, moving one block at a time:

https://commons.wikimedia.org/wiki/File:Sussman-anomaly-1.svg
Image: Wikimedia Commons

It might proceed by separating the goal into two subgoals:

  1. Get A onto B.
  2. Get B onto C.

But this leads immediately to trouble. If the agent starts with subgoal 1, it will move C off of A and then put A onto B:

https://commons.wikimedia.org/wiki/File:Sussman-anomaly-2.svg
Image: Wikimedia Commons

But that’s a dead end. Because it can move only one block at a time, the agent can’t now undertake subgoal 2 without first undoing subgoal 1.

If the agent starts with subgoal 2, it will move B onto C, which is another dead end:

https://commons.wikimedia.org/wiki/File:Sussman-anomaly-3.svg
Image: Wikimedia Commons

Now we have a tower, but the blocks are in the wrong order. Again, we’ll have to undo one subgoal before we can undertake the other.

Modern algorithms can handle this challenge, but still it illustrates why planning is not a trivial undertaking. Sussman discussed it as part of his 1973 doctoral dissertation, A Computational Model of Skill Acquisition.

| Science & Math

Entre Nous

https://archive.org/details/strand-1897-v-14/page/690/mode/2up?view=theater

In 1896 the letter above arrived at the New York post office. As there was no Goat Street in New York, the office marked it misdirected and sent it on to Washington, where clerks eventually opened it, looking for further clues. They found this:

Dear Santa, — When I said my prayers last night I told God to tell you to bring me a hobby horse. I don’t want a hobby horse, really. A honestly live horse is what I want. Mamma told me not to ask for him, because I probably would make you mad, so you wouldn’t give me anything at all, and if I got him I wouldn’t have any place to keep him. A man I know will keep him, he says, if you get him for me. I thought you might like to know. Please don’t be mad. — Affectionately, John.

P.S. — A shetland would be enough.

P.S. — I’d rather have a hobby horse than nothing at all.

“I am very sorry to say that John did not get the horse,” wrote Mary K. Davis in the Strand. “Little boys who don’t do as their mothers tell them find little favour with Santa Claus.”

| Society

All the Uses of This World

https://commons.wikimedia.org/wiki/File:%C3%9Altima_escena_de_Hamlet,_por_Jos%C3%A9_Moreno_Carbonero.jpg

As a footnote to the above, I would like to say that I am getting very tired of literary authorities, on both the stage and the screen, who advise young writers to deal only with those subjects that happen to be familiar to them personally. It is quite true that this theory probably produced A Tree Grows in Brooklyn, but the chances are it would have ruled out Hamlet.

— Wolcott Gibbs, New Yorker, January 6, 1945

| Literature

In a Word

nimiety
n. superfluity

brachylogy
n. a condensed expression

scrimption
n. a very small amount or degree

perficient
adj. that accomplishes something; effectual

Leonard Bernstein conducts the Vienna Philharmonic without using his hands, 1984:

| Art · Language

The Sandwheel

https://commons.wikimedia.org/wiki/File:Sandwheel.gif

This is a variation on a perpetual motion machine proposed by the Indian mathematician Bhāskara II around 1150. Each of the wheel’s tilted spokes is filled with a quantity of sand. As the tubes descend on the right, the sand within them shifts outward, exerting greater torque in the clockwise direction and thus keeping the wheel turning forever.

Unfortunately the same design ensures that there’s always a greater quantity of sand on the left, so nothing happens.

| Science & Math