# Getting Around

Jakob von Uexküll used to say: ‘When a dog runs, the dog moves its legs; when a sea urchin runs, the legs move the sea urchin.’ This assertion was based on the following experiment reported by von Uexküll. A sea urchin was broken in half and the inner sides of both halves of the shell were scraped using sandpaper. The whole of the ambulacral system as well as the nervous system was thus completely removed. Then the two halves were joined together again by means of a spring clasp. The spines of the sea urchin still worked in coordination with one another. In this special case, the riderless horse of von Holst’s parable does indeed exist; the sea urchin’s reaction of fleeing from a star fish still functioned. And in this sense, von Uexkull’s description of a sea urchin being a ‘reflex republic’ is justified, provided one keeps in mind that the ‘reflex’ no longer plays the all-important role ascribed to it during von Uexküll’s time.

— Konrad Lorenz, The Foundations of Ethology, 1982

(Lorenz described this direct mutual influence among the peripheral organs as “a panic spreading among the spines.”)

# Letters and Numbers

ONE + TWO – THREE – FOUR + FIVE = 1

That’s true if we replace each word either with the number it denotes or with the quantity of its letters: Either way we’re left with 1. Another:

ONE + TWO – THREE – FOUR + FIVE – SIX + SEVEN + EIGHT + NINE – TEN + ELEVEN + TWELVE – THIRTEEN – FOURTEEN = 5

These are the only two such sequences using 20 or fewer consecutive number names, found Leonard Gordon, although other sequences of plus and minus signs are possible.

In a separate but related project, Gordon assigned the number names ONE through FIFTEEN, ONE through NINETEEN, and ONE through TWENTY to either side of an equals sign so that the denoted equation is mathematically correct and each equation “balances,” with the same number of letters on each side:

ONE + FOUR + SEVEN + TEN + ELEVEN + THIRTEEN + FOURTEEN = TWO + THREE + FIVE + SIX + EIGHT + NINE + TWELVE + FIFTEEN

ONE + THREE + FIVE + SEVEN + NINE + SIXTEEN + SEVENTEEN + EIGHTEEN + NINETEEN = TWO + FOUR + SIX + EIGHT + TEN + ELEVEN + TWELVE + THIRTEEN + FOURTEEN + FIFTEEN

ONE + THREE + SIX + NINE + TEN + TWELVE + THIRTEEN + FIFTEEN + SEVENTEEN + NINETEEN = TWO + FOUR + FIVE + SEVEN + EIGHT + ELEVEN + FOURTEEN + SIXTEEN + EIGHTEEN + TWENTY

(“Self-Referential Sums Revisited,” in “Kickshaws,” Word Ways 28:3 [August 1995], 170-180.)

# Relative

During an eclipse in 1919, Sir Arthur Eddington confirmed Albert Einstein’s prediction of the gravitational bending of light rays, upholding the general theory of relativity. That Christmas, Einstein wrote to his friend Heinrich Zangger in Zurich:

“With fame I become more and more stupid, which, of course, is a very common phenomenon. There is far too great a disproportion between what one is and what others think one is, or at least what they say they think one is. But one has to take it all with good humor.”

(From Helen Dukas and Banesh Hoffmann, eds., Albert Einstein, the Human Side: New Glimpses From His Archives, 1979.)

# The Seconds Pendulum

An interesting historical fact from these MIT notes: Christiaan Huygens proposed defining the meter conveniently as the length of a pendulum that produces a period of 2 seconds. A pendulum’s period is

$\displaystyle T = 2\pi \sqrt{\frac{l}{g}},$

so, using Huygens’ standard of T = 2s for 1 meter,

$\displaystyle g = \frac{4\pi ^{2}\times 1\ \textup{meter}}{4s^{2}} = \pi ^{2}ms^{-2}.$

“So, if Huygens’s standard were used today, then g would be π2 by definition.”

# An Elevated Perspective

Consider a triangle ABC and three other triangles (ABD1, BCD2, and ACD3) that share common sides with it, and assume that the sides adjacent to any vertex of ABC are equal, as shown. The altitudes of the three outer triangles, passing through D1, D2, and D3 and orthogonal to the sides of ABC, meet in a point.

This can be made intuitive by imagining the figure in three dimensions. Fold each of the outer triangles “up,” out of the page. Their outer vertices will meet at the apex of a tetrahedron. Now if we imagine looking straight down at that apex and folding the sides down again, each of those vertices will follow the line of an altitude (from our perspective) on the way back to its original position, because each follows an arc that’s orthogonal to the horizontal plane and to one of the sides of ABC. The result is the original figure.

(Alexander Shen, “Three-Dimensional Solutions for Two-Dimensional Problems,” Mathematical Intelligencer 19:3 [June 1997], 44-47.)

# “Appendicitis”

The symptoms of a typical attack
A clearly ordered sequence seldom lack;
The first complaint is epigastric pain
Then vomiting will follow in its train,
After a while the first sharp pain recedes
And in its place right iliac pain succeeds,
With local tenderness which thus supplies
The evidence of where the trouble lies.
Then only — and to this I pray be wise —
Then only will the temperature rise,
And as a rule the fever is but slight,
Hundred and one or some such moderate height.
‘Tis only then you get leucocytosis
Which if you like will clinch the diagnosis,
Though in my own experience I confess
I find this necessary less and less.

From Zachary Cope, The Diagnosis of the Acute Abdomen in Rhyme, 1947.

# More Loops

Further to my March post “A Lucrative Loop,” reader Snehal Shekatkar of S.P. Pune University notes a similar discovery of iterates leading to strange cycles among natural numbers.

Here is a simple example. Take a natural number and factorize it (12 = 2 * 2 * 3), then add all the prime factors (2 + 2 + 3 = 7). If the answer is prime, add 1 and then factorize again (7 + 1 = 8 = 2 * 2 * 2) and repeat (2 + 2 + 2 = 6). Eventually ALL the natural numbers greater than 4 eventually get trapped in cycle (5 -> 6 -> 5). Instead of adding 1 after hitting a prime, if you add some other natural number A, then depending upon A, numbers may get trapped in a different cycle. For example, for A = 19, they eventually get trapped in cycle (5 -> 24 -> 9 -> 6 -> 5).

For some values of A, several cycles exist. For example, when A = 3, some numbers get trapped in cycle (5 -> 8 -> 6 -> 5) while others get trapped in the cycle (7 -> 10 -> 7).

(Made with Tian An Wong of Michigan University.) (Thanks, Snehal.)

# Podcast Episode 341: An Overlooked Bacteriologist

In the 1890s, Waldemar Haffkine worked valiantly to develop vaccines against both cholera and bubonic plague. Then an unjust accusation derailed his career. In this week’s episode of the Futility Closet podcast we’ll describe Haffkine’s momentous work in India, which has been largely overlooked by history.

We’ll also consider some museum cats and puzzle over an endlessly energetic vehicle.

See full show notes …

# Cutting Cake

In the College Mathematics Journal in 2001, Rick Mabry published this “proof without words” that

$\displaystyle \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \cdots = \frac{1}{2}.$

He gives a charming explanation here.

(Rick Mabry, “Mathematics Without Words,” College Mathematics Journal 32:1 [January 2001], 19.)

# First Things First

It is a traditional axiom of medicine that health is the absence of disease. What is a disease? Anything that is inconsistent with health. If the axiom has any content, a better answer can be given. The most fundamental problem in the philosophy of medicine is, I think, to break the circle with a substantive analysis of either health or disease.

— Christopher Boorse, “Health as a Theoretical Concept,” Philosophy of Science 44:4 (1977), 542-573