Deluge

Dry Falls, in central Washington, has a stunningly dramatic history: At the end of the last glaciation, when ice dams to the east gave way, torrents of water roared through the landscape from flooded Montana, pouring over a 400-foot rock face at 65 mph in a waterfall five times as wide as Niagara and carrying 10 times the flow of all the world’s rivers combined.

The scale of the cataclysm is hard to imagine, so geologist Nick Zentner of Central Washington University commissioned these animations from Newlands & Company of Portland to help to convey its magnitude.

| Science & Math

Twice True

Each of these sums is valid in two ways, once when the words at taken at their face value and again when each letter is interpreted as a particular digit:

   THREE    79322           ONE       483       ZERO   4206      TRECE  69858
    NINE     6562          FIVE      7293        SEI    827      CINCO  57354
     TEN      726           TEN       138      SETTE  82112       OCHO   4504
FOURTEEN 40837226        ELEVEN    363938       OTTO   6116       ----   ----
 FIFTEEN  4547226      NINETEEN  82831338       NOVE   9652     QUINCE 127358
 -------  -------     FORTYFIVE 745107293       ----   ----       ONCE   4358
FIFTYONE 45471062     --------- ---------     TRENTA 102913
                      NINETYONE 828310483

All are from the Journal of Recreational Mathematics, collected by Leonard Gordon in “Doubly-True Alphametics,” Word Ways 27:1 [February 1994], 10-12. More alphametics.

| Language · Science & Math

Small World

tinyurl.com/FEmodel

Physicist Bruce Sherwood has created a 3-D computer model of a flat earth so that conspiracy theorists can examine the implications of their own theory.

For example, if Antarctica is a mountain range surrounding a flat disk, what should we expect to see in the sky? “Walk inside the model and look up,” Sherwood told philosopher Lee McIntyre. “If you’re standing at the North Pole, then Polaris should be directly overhead. Fair enough. But if you’re standing at the ‘edge of the Earth’ — and Polaris is only a few thousand miles overhead — shouldn’t you at best see it at an angle? But if you’re actually in Antarctica, you won’t be able to see it at all. Their model is inconsistent with physical observation. And they can see that for themselves.”

(Bruce Sherwood, “A Flat Earth?”, via Lee McIntyre, et al., How to Talk to a Science Denier, 2021.)

| Science & Math

In a Word

passeggiata
n. a leisurely walk

In the ancient world, distances were sometimes measured by pacing. Specialists known as bematists were employed for this purpose in both Egypt and Greece, and their accuracy could be startling: In his Naturalis Historia, Pliny the Elder notes that two bematists employed by Alexander the Great had measured the distance from Hecatompylos to Alexandria Areion on the Silk Road at 851 kilometers. The actual distance is 855 kilometers, a deviation of just 0.4 percent. In general, according to Pliny’s records, Alexander’s bematists showed a median deviation of just 2.8 percent from the true distances; a separate account by Strabo shows a median deviation of only 1.9 percent.

This accuracy suggests that the bematists may have been using an early odometer, such as one described by Heron of Alexandria, though the records don’t mention this.

12/30/2023 UPDATE: Reader Charlotte Fare has made a data visualization. (Thanks, Charlotte.)

| History · Language · Science & Math

Easy Street

The following was rather widely quoted a few years ago. It bothered one banker so much that he made a hasty trip to consult his neighbor, a college professor of mathematics. Assume we make a deposit of $50 in a bank.

    withdraw $20.00 leaving $30.00
now withdraw  15.00 leaving  15.00
now withdraw   9.00 leaving   6.00
now withdraw   6.00 leaving   0.00
             ------         ------
             $50.00         $51.00

We now present our figures to the bank, showing the discrepancy, and demand the extra dollar. Repeat ten thousand times, and retire for a while.

— Cecil B. Read, “Mathematical Fallacies,” School Science and Mathematics, June 1933

| Science & Math

On the Wing

https://en.wikipedia.org/wiki/File:Phoebe_snetsinger.jpg

Diagnosed with terminal melanoma at 50, Phoebe Snetsinger resolved to devote her remaining time to watching birds. Between 1981 and 1999, as her cancer went periodically into remission, she visited every continent several times over, traversing jungles, swamps, deserts, and mountains and surviving malaria, a boat accident, abduction in Ethiopia, and rape in Papua New Guinea. In 1995 she became the first person to see 8,000 species of bird, and in time she extended the list to 8,398, nearly 85 percent of the world’s known species. She died in 1999 when her van overturned during a birding trip in Madagascar. The last bird she’d observed was a red-shouldered vanga, a species that had been described as new to science only two years previously.

In her memoir, Birding on Borrowed Time, she wrote, “When I was given my death sentence by the doctors, one of my immediate reactions that I clearly remember was ‘Oh no — there are all those things I haven’t yet done, and now will never have a chance to do.’ … The preparation and primarily the birding itself, plus the record keeping afterwards, all enabled me to forget the threat to my life (or at least push it aside) and to immerse myself totally in what I was doing.”

| Death · Science & Math

Also-Ran

Arthur Conan Doyle tells us little about James Moriarty, the criminal mastermind in the Sherlock Holmes stories. But he does mention one intriguing accomplishment in The Valley of Fear:

Is he not the celebrated author of The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it?

Mathematicians Alain Goriely and Simon P. Norton have both pointed out that in 1887 King Oscar II of Sweden offered a bounty for the solution to the n-body problem in celestial mechanics. Doyle’s story was set in 1888, so it’s possible that Moriarty had intended his book as his entry in this contest.

If he did, he was disappointed — the prize went to Henri Poincaré.

| Literature · Science & Math

Fitting

From reader Snehal Shekatkar:

There exist exactly 17 numbers the sum of whose distinct prime factors is exactly 17:

17 = 17
52 = 2 × 2 × 13
88 = 2 × 2 × 2 × 11
99 = 3 × 3 × 11
147 = 3 × 7 × 7
175 = 5 × 5 × 7
210 = 2 × 3 × 5 × 7
224 = 2 × 2 × 2 × 2 × 2 × 7
250 = 2 × 5 × 5 × 5
252 = 2 × 2 × 3 × 3 × 7
300 = 2 × 2 × 3 × 5 × 5
320 = 2 × 2 × 2 × 2 ×2 × 2 × 5
360 = 2 × 2 × 2 × 3 × 3 × 5
384 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
405 = 3 × 3 × 3 × 3 × 5
432 = 2 × 2 × 2 × 2 × 3 × 3 × 3
486 = 2 × 3 × 3 × 3 × 3 × 3

(Thanks, Snehal.)

| Science & Math

Short Work

A simple proof that \sqrt{2} is irrational:

Assume that it’s rational. Then \sqrt{2} = p/q and p2 = 2q2. But in the latter equation, the left side must have an even number of prime factors and the right an odd number. That’s a contradiction, so our assumption must be wrong.

In the paper below, Manchester Polytechnic mathematician T.J. Randall credits this “marvellous” proof to Philip J. Davis and Reuben Hersh in their 1982 book The Mathematical Experience, but I can’t find it there.

(T.J. Randall, “67.45 \sqrt{2} Revisited,” Mathematical Gazette 67:442 [December 1983], 302-303.)

12/20/2023 UPDATE: Reader Hans Havermann finds the proof mentioned in Stuart Hollingdale’s Makers of Mathematics (1989), after the more familiar proof based on parity of p and q. Hollingdale writes that the alternative proof “can be traced back to the Classical period.”

| Science & Math

Changes

In 1967, psychiatrists Thomas Holmes and Richard Rahe asked 5,000 medical patients about recent difficult events in their lives. They found a positive correlation with illness. The sum of the “life change units” you’ve amassed in the past year, they said, gives a rough estimate of the effect on your health:

Death of a spouse: 100
Divorce: 73
Marital separation: 65
Imprisonment: 63
Death of a close family member: 63
Personal injury or illness: 53
Marriage: 50
Dismissal from work: 47
Marital reconciliation: 45
Retirement: 45
Change in health of family member: 44
Pregnancy: 40
Sexual difficulties: 39
Gain a new family member: 39
Business readjustment: 39
Change in financial state: 38
Death of a close friend: 37
Change to different line of work: 36
Change in frequency of arguments: 35
Major mortgage: 32
Foreclosure of mortgage or loan: 30
Change in responsibilities at work: 29
Child leaving home: 29
Trouble with in-laws: 29
Outstanding personal achievement: 28
Spouse starts or stops work: 26
Beginning or end of school: 26
Change in living conditions: 25
Revision of personal habits: 24
Trouble with boss: 23
Change in working hours or conditions: 20
Change in residence: 20
Change in schools: 20
Change in recreation: 19
Change in church activities: 19
Change in social activities: 18
Minor mortgage or loan: 17
Change in sleeping habits: 16
Change in number of family reunions: 15
Change in eating habits: 15
Vacation: 13
Major holiday: 12
Minor violation of law: 11

A score of 300+ means you’re at risk of illness.

A score of 150-299 means your risk of illness is moderate (but 30 percent lower than the foregoing group).

A score of less than 150 means your risk of stress-related illness is slight.

| Science & Math