In Pascal’s triangle, above, the number in each cell is the sum of the two immediately above it.

If you “tilt” the triangle so that each row starts one column to the right of its predecessor, then the column totals produce the Fibonacci sequence:

pascal triangle fibonacci numbers

That’s from Thomas Koshy’s Triangular Arrays With Applications, 2011.

Bonus: Displace the rows still further and they’ll identify prime numbers.

Desperate Measures

From reader Jon Sweitzer-Lamme:

Pressed for materials during the siege of Vicksburg in 1863, the city’s Daily Citizen newspaper printed its last six issues on the back of wallpaper. Each of the issues for June 16, 18, 20, 27, 30, and July 2 was printed in four columns on a single sheet, as above; a reader who turned the sheet over would see this:

The last issue, on July 2, is still defiant:

The Yankee Generalissimo surnamed Grant has expressed his intention of dining in Vicksburg on the Fourth of July. … Ulysses must get into the city before he dines in it.

But two days later, when the city finally fell, Union troops added a final paragraph:

Two days bring about great changes, The banner of the Union floats over Vicksburg. Gen. Grant has ‘caught the rabbit:’ he has dined in Vicksburg, and he did bring his dinner with him. The ‘Citizen’ lives to see it. For the last time it appears on ‘Wall-paper.’ No more will it eulogize the luxury of mule-meat and fricassed kitten — urge Southern warriors to such diet never-more. This is the last wall-paper edition, and is, excepting this note, from the types as we found them. It will be valuable hereafter as a curiosity.

More at the Library of Congress. (Thanks, Jon.)


“All the glory of the world would be buried in oblivion, unless God had provided mortals the remedy of books.” — Richard de Bury

Podcast Episode 140: Ramanujan

In 1913, English mathematician G.H. Hardy received a package from an unknown accounting clerk in India, with nine pages of mathematical results that he found “scarcely possible to believe.” In this week’s episode of the Futility Closet podcast, we’ll follow the unlikely friendship that sprang up between Hardy and Srinivasa Ramanujan, whom Hardy called “the most romantic figure in the recent history of mathematics.”

We’ll also probe Carson McCullers’ heart and puzzle over a well-proportioned amputee.

See full show notes …

A Visitation

mead snow angel

The residents of Brattleboro, Vt., awoke to a surprise on New Year’s Day 1856: An angel of snow stood at the corner of Linden and North Main streets, holding a pen and notebook as if ready to record the events of the new year.

The artist turned out to be 21-year-old Larkin Goldsmith Mead Jr., the son of a local lawyer. He had worked through the night to fashion the eight-foot figure by lantern light, shaping it by hand from snow brought to him by two friends. He sculpted some parts separately in order to mold them more easily, attaching them later with wet snow, and he poured water over the finished creation to give it a smooth finish. The Vermont Phoenix marveled:

The inhabitants of the village discovered ‘The Snow Angel,’ in the prismatic glow of the morning sun’s reflection. The early risers and pedestrians about town were amazed, when they drew near, to see what appeared at a distance like a school-boy’s work turned to a statue of such exquisite contour and grace of form. … The passing school-boy was awed for once, as he viewed the result of adept handling of the elements with which he was so roughly familiar, and the thought of snowballing so beautiful an object could never have dwelt in his mind. It is related that the village simpleton was frightened and ran away, and one eccentric citizen, who rarely deigned to bow to his fellow men, or women either, lifted his hat in respect after he had gazed a moment upon Mead’s work.

The angel was featured in newspapers in Boston and New York, and it brought Mead commissions for more lasting works, including a statue of Ethan Allen and a wooden figure symbolizing agriculture for the Montpelier statehouse. His later projects include Lincoln’s tomb in Springfield, Illinois.

In 1886 he reproduced the angel in marble — it stands in the public library in Brattleboro. Across the street, commemorating the site of the original angel, stands a drinking fountain designed by his brother, architect William Mead — who immortalized the family name in his own way.


A desire path is a route made evident by foot traffic, often easier or more direct than a provided avenue:
Image: Wikimedia Commons

A holloway is a sunken lane formed by traffic or erosion — some in Europe date to the Iron Age:
Image: Wikimedia Commons

A snowy neckdown is a disused area of a roadway made evident by snowfall:,_2017-01-19_-_highlight.jpg
Image: Wikimedia Commons

In the absence of snow, some Australian engineers have dusted intersections with cake flour to reveal traffic patterns. Others study the oil stains left by traffic. Dan Burden, director of the Walkable and Livable Communities Institute, says, “I call something like that highway forensics.”

Image: Wikimedia Commons

Classical statues tend to lose their noses, and in the 19th century museums would commonly replace them with “restoration” noses, to preserve the appearance of the original sculpture.

In the 20th century some museums changed philosophies and “de-restored” their collections, thinking it better to present each piece in its authentic state.

This created a superfluity of noses, and some museums collect these into displays of their own. Charmingly, there’s even a word for this: A collection of noses is a Nasothek.

Above is the collection in the Ny Carlsberg Glyptotek museum in Copenhagen.

(Thanks, Carsten.)


You know the singularity has arrived when the robots start playing marimbas. Shimon, engineer Guy Hoffman’s robot musician, doesn’t play programmed music — it improvises in ensembles with human players, communicating with a “socially expressive head” and favoring musical ideas that are unlikely to be chosen by humans, so as to lead the performance in genuinely novel directions.

“The project, therefore, aims to combine human creativity, emotion, and aesthetic judgment with algorithmic computational capability of computers, allowing human and artificial players to cooperate and build off each other’s ideas,” notes the Georgia Tech Center for Music Technology, Shimon’s patron. “Unlike computer- and speaker-based interactive music systems, an embodied anthropomorphic robot can create familiar, acoustically rich, and visual interactions with humans.”

More at Georgia Tech.

Teaching by Example

“Two great duties, I think, we owe to posterity: one is progress, the other history. Only the former can we share in. The fruits of progress often apply to the generation which bears them. But the records of that progress come closer to being pure charity than any form of charity I know.” — Richard E. Byrd