Sylver Coinage

This curious game was invented by Princeton mathematician John H. Conway. Two players take turns naming positive integers, but an integer is off limits if it’s the sum of nonnegative multiples of integers that have already been named. Once 1 is named, everything is off limits (because any positive integer is a sum of 1s), so that ends the game; the player who is forced to name 1 is the loser. An example gives the idea:

  • I start by naming 5. From now on neither of us can name 5, 10, 15, 20, …
  • You name 4. Now neither of us can name a number built of 5s and 4s, that is, 4, 5, 8, 9, 10, or any number greater than 11.
  • I name 11. This reduces the list of available numbers to 1, 2, 3, 6, and 7.
  • You name 6. Now we’re down to 1, 2, 3, and 7.
  • I name 7. Only 1, 2, and 3 are still available to name.

Now we’ll use our next turns to name 2 and 3, and you’ll be forced to name 1, losing the game.

Though the game is very easy to understand, it’s still full of mysteries. For example, R.L. Hutchings has shown that playing a prime number as the first move guarantees that a winning strategy exists, but no one has figured out how to find the strategy. And no one knows whether there are any winning opening moves at all that aren’t prime.

More on the Sylver Coinage page.

| Science & Math

Perspective

https://commons.wikimedia.org/wiki/File:Van_Eyck_-_Arnolfini_Portrait.jpg

There’s an interesting detail in Jan van Eyck’s Arnolfini Portrait of 1434. The painting depicts Italian merchant Giovanni di Nicolao Arnolfini and his wife in their home in Bruges. In the background is a convex mirror over which is inscribed the legend “Johannes de eyck fuit hic 1434” (“Jan van Eyck was here 1434”). And curiously, reflected in the mirror, where we’d expect to see the painter and his easel, are two figures:

https://commons.wikimedia.org/wiki/File:The_Arnolfini_Portrait,_d%C3%A9tail_(2).jpg

In 1934 art historian Erwin Panofsky offered a unique explanation for this: The painting is an “artistic marriage certificate.” Neither Arnolfini nor his new wife had any relatives at Bruges at the time of their marriage, and the custom at the time was to record two witnesses to the wedding. “So that we can understand the original idea of a picture which was a memorial portrait and a document at the same time, and in which a well-known gentleman-painter signed his name both as artist and as witness.”

Amused by the reflection, Hungarian graphic designer István Orosz set up his own arrangement in his 1997 etching Johannes de Eyck fuit hic, below. “Here I attempt to show the world behind the door with the help of two mirrors.”

http://art.ayay.co.uk/art/optical_illusion/istvan_orosz/door-and-mirror/

For a thousand bucks you can buy your own Van Eyck mirror and investigate for yourself.

(Erwin Panofsky, “Jan van Eyck’s Arnolfini Portrait,” Burlington Magazine for Connoisseurs 64:372 [March 1934], 117-119.)

| Art

His and Hers

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

When Triberg, Germany, unveiled a new parking garage in 2012, it included 12 parking spaces designated for women’s use, as has been required by German law since the 1990s. The spaces are somewhat larger than normal and located near the mall entrance, to reduce the possibility of sexual assault.

Due to the Triberg mall’s shape, it also has one space on each level that requires a complex parking maneuver. Mayor Gallus Strobel decided to designate these for men … which brought on a backlash from people who felt that it insults women’s parking skills.

“It’s a joke,” Strobel told ABC. “Everyone in Triberg thinks it is a joke. We looked at the two parking spaces and we said, ‘They could be dangerous for your car,’ so at the same time, we decided to make them for men, and then give 12 others for women.”

Is this sexist? Considerate? Unnecessary? Germany isn’t the only nation that uses women’s parking spaces — they’ve also been introduced in Austria, Switzerland, and China, where the spokesman for one northern mall told AFP, “The fact that our women’s parking spaces are wider is simply due to practical reasons and shouldn’t imply that women are worse at driving than men.”

For his part, Strobel seems to have his eye on the bottom line — after Triberg controversy made the news, the town’s tourist traffic increased from 250,000 to 400,000. “Women can come here and prove me wrong,” he told the Times of India, “and while they’re at it, they can see the town’s attractions.”

| Society

Turnabout

Here’s an especially vivid example of the illusion created by Dick Termes’ six-point perspective.

If you can convince yourself that the front half of this sphere is transparent, and that the image is painted on the interior of the back half, you’ll find that you’re inside the cage, turning to your left, while the birds are outside the cage, looking in at you. (To get started, I find it helps to focus on an edge of the sphere, rather than the center.)

There are many more examples on Termes’ YouTube channel.

| Art · Oddities · Science & Math

The Bear Gates

https://www.flickr.com/photos/tomblackie/138196120/in/photolist-a72jJ3-6rfpnR-o7aY4s-6rfpDM-2GDkxv-ddhnr-7fiLNR-ddhTJ-5BTgdv-axp9MK-ddbTs-ddihk-wzPed2-7jXJPG
Image: Flickr

These gates, at the main entrance to the grounds of Scotland’s Traquair House, were installed in 1738.

After a visit by Bonnie Prince Charlie in 1745, earl Charles Stuart vowed they would never open again until a Stuart king returned to the throne.

They never have.

| History · Oddities

“Mother Nature in Tears”

http://mikenolanwildlifeimages.blogspot.com/2009/09/mother-nature-in-tears.html

Photographer Michael S. Nolan took this photo in 2009 at the Austfonna ice cap in Norway’s Svalbard archipelago. He writes:

When I took the image early in the morning on July 16, 2009 from the bow of the National Geographic Explorer I was struck by the unmistakable likeness of the face of a woman crying. In fact once my mind locked onto the face it was hard to see any other pattern in the ice cap. I was moved to photograph this particular waterfall several different ways with a couple of different lenses. It was one of the best examples of a human likeness I have ever witnessed in nature.

“The icescape changes every year I visit,” he told the Telegraph. “Every summer the route has less ice as the polar cap retreats.”

(Via Brad Honeycutt, The Art of Deception, 2014.)

| Oddities

The Last Detail

A puzzle by R.P. Cross:

Find the last digit in the evaluation of \sum_{n = 1}^{100}n!

Click for Answer
| Puzzles · Science & Math

Podcast Episode 164: Vigil on the Ice

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

In 1930, British explorer Augustine Courtauld volunteered to spend the winter alone on the Greenland ice cap, manning a remote weather station. As the snow gradually buried his hut and his supplies steadily dwindled, his relief party failed to arrive. In this week’s episode of the Futility Closet podcast we’ll follow Courtauld’s increasingly desperate vigil on the ice.

We’ll also retreat toward George III and puzzle over some unexpected evidence.

See full show notes …

| History · Podcast · Science & Math

A Good Start

opalka 1 million

In 1965, Polish artist Roman Opałka hung a 196 × 135 cm canvas in his Warsaw studio. In the top left corner he painted a tiny numeral 1, then a 2, and so on until he had filled the canvas with numbers. Then he put up a new canvas and continued where he had left off. He called these images “details”; all of them had the same size and the same title, 1965 / 1 – ∞.

He vowed to spend the rest of his life on the project. “All my work is a single thing,” he said, “the description from number one to infinity. A single thing, a single life. … The problem is that we are, and are about not to be.”

At the start he painted white numbers on a black background, but in 1972 he began gradually to lighten the black with each detail, saying that his goal was “to get up to the white on white and still be alive.” He expected that this would happen when he reached 7777777 … but at the time of his death, in 2011, he’d got only as far as 5607249.

| Art · Science & Math

Mens Agitat Molem

http://www.jeremywood.net/artworks/traverse_me.html

In 2010 Jeremy Wood walked around the campus of the University of Warwick with a GPS device to “draw” a map at 1:1 scale. Altogether he covered 238 miles in 17 days.

“He stayed in the Maths Houses on Gibbet Hill so the line through Tocil Wood to the Mead Gallery is exceptionally dark since it was walked so many times,” the university reports. “As he worked his way across the fields towards Kenilworth he began to ‘draw’ images associated with the University, from its crest, to a mortar board, to a globe in homage to the many ‘international’ centres that he encountered in his journeys. Reported to security several times for walking in ‘a suspicious manner’ around Claycroft and Lakeside residences, he soon disappeared from view, walking the countryside that surrounds the University but which is far removed from central campus.”

“I responded to the structure of each location and avoided walking along roads and paths when possible,” Wood writes. “Security was called on me twice on separate occasions and I lost count of how many times I happened to trigger an automatic sliding door.” More at his website and at GPS Drawing.

Somewhat related: Mathematician Jerry Farrell invented a two-player coin-pushing game played on a map of Butler University, his institution. Rebecca Wahl analyzed it in Barry Cipra’s Tribute to a Mathemagician (2005), and Aviezri Fraenkel of Israel’s Weizmann Institute of Science revisited it the following year (PDF).

| Art · Science & Math · Technology