Four-for-All

https://commons.wikimedia.org/wiki/File:Zambezi_River_at_junction_of_Namibia,_Zambia,_Zimbabwe_%26_Botswana.jpg
Image: Wikimedia Commons

Here’s an odd bit of African geography: The finger of land in the upper left is Namibia, the region at the top is Zambia, Zimbabwe is at bottom right, and Botswana is at bottom left. Is the border between Zambia and Botswana long enough to permit a bridge to be built between the two? Or do the two peninsulas intrude far enough to make this impossible?

The answer isn’t clear. In 1970 Namibia had insisted that the four nations meet at a single point, meaning that the Kazungula Ferry linking Botswana and Zambia was illegal, as the border between them had no breadth. After an armed confrontation the ferry was sunk. Thirty-five years later Botswana and Zambia proposed building a bridge where the ferry had run. Is that geometrically permissible? The shaky consensus is that the two nations share a brief boundary of 150 meters between two “tripoints.” But the truth is as murky as the Zambezi itself.

(See Point of Interest. Thanks, Steve.)

The Simson Line

https://commons.wikimedia.org/wiki/File:Pedal_Line.svg

The three corners of any triangle ABC define a circle that surrounds it, called its circumcircle. And for any point P on this circle, the three points closest to P on lines AB, AC, and BC are collinear.

The converse is also true: Given a point P and three lines no two of which are parallel, if the closest points to P on each of the lines are collinear, then P lies on the circumcircle of the triangle formed by the lines.

This discovery is named for Robert Simson, though, as often happens, it was first published by someone else — William Wallace in 1797.

Mind’s Eye #3

mind's eye #3

Here’s this month’s piece from designer Von Glitschka — each month he composes an engraved artwork to accompany a Futility Closet quotation.

He’s provided a high-resolution PDF (2 MB) suitable to collect, print, and hang.

We’re releasing a new image each month, here and on social media — as with the wallpaper, I’ll put the links on the About page as well. Thanks for reading, and thanks again to Von!

Mind’s Eye #2

mind's eye #2

Here’s this month’s piece from designer Von Glitschka — each month he composes an engraved artwork to accompany a Futility Closet quotation. (Here’s last month’s composition.)

He’s provided a high-resolution PDF (2 MB) suitable to collect, print, and hang.

We’re releasing a new image each month, here and on social media — as with the wallpaper, I’ll put the links on the About page as well. Thanks for reading, and thanks again to Von!

Mind’s Eye #1

mind's eye #1

Another gift from designer Von Glitschka — he’s composing engraved artwork to accompany some of our quotations.

With each image he’s providing a high-resolution PDF (this one is 10 MB) suitable to collect, print, and hang.

We’re hoping to release a new image each month, here and on social media — as with the wallpaper, I’ll put the links on the About page as well. Thanks for reading, and thanks again to Von!

Wallpaper

futility closet wallpaper

Thanks again to the world-bestriding Von Glitschka for our new logo. At Von’s suggestion we’re calling the penguin Mr. Curious.

Von has also thoughtfully created wallpaper files for iPhone, iPad, and desktops, which are free for the downloading:

iPhone 4, iPhone 5, iPhone 6, iPhone 6 Plus

iPad Air, Mini, Pro

Desktop: 1920 × 1080, 2560 × 1440, 3410 × 1918

I’ll add these links to the About page as well.

As we get organized we’ll announce other offerings here and in the Patreon feeds, and Von will be creating custom promo graphics to download, print out, and collect. A million thanks again to him for all his contributions, and to you for your kind words about the redesign.

Redesign

We’ve upgraded our look today, to make the podcast and books easier to find and to make the site easier to use on mobile devices. Many thanks to Graph Paper Press for the new design and to the great Von Glitschka for our new logo (about which more in a future post).

Everything should work as it always has (including the RSS and email feeds), but hopefully things are easier to find now, especially for newcomers. If you spot any problems, please let me know.

This site is supported entirely by its readers. If you value Futility Closet and would like to contribute, either with a one-time donation or ongoing patronage, please see the support page. Thanks for reading!