“The first author would like to acknowledge and thank Jesus Christ, through whom all things were made, for the encouragement, inspiration, and occasional hint that were necessary to complete this article. The second author, however, specifically disclaims this acknowledgement.”

— Michael I. Hartley and Dimitri Leemans, “Quotients of a Universal Locally Projective Polytope of Type {5, 3, 5},” Mathematische Zeitschrift 247:4 (2004), 663-674

First Light

I think if there’s one thing that you could truly say it the most beautiful sight you can possibly see as a human, it is watching sunrise over the Earth, because imagine, you’re looking at blackness out the window, black Earth, black space, and then as the Sun comes up, the atmosphere acts as a prism, and it splits the light into the component colors. It splits the white light of the Sun into the component colors, so you get this rainbow effect, and it starts with this deep indigo eyelash, just defining the horizon, and then as the Sun rises higher, you get these reds and oranges and blues in this rainbow. … You never got tired of looking at those.

— Astronaut Mike Mullane, quoted in Ariel Waldman, What’s It Like in Space?, 2016

Looking Up

Astronomer Clyde Tombaugh assembled his first telescope from spare parts on his family’s Kansas farm — the crankshaft of a 1910 Buick, a cream-separator base, and mechanical components from a straw spreader. He used this to make sketches of Jupiter and Mars that so impressed the astronomers at Lowell Observatory that they gave him a job there.

Years later, after he had made his name by discovering Pluto, the Smithsonian Institution asked if it could exhibit this early instrument. He told them he was still using it — he was making observations from his backyard near Las Cruces, N.M., until shortly before his death in 1997.

“Its mirror was hand-ground and tested in a storm cellar,” wrote Peter Manly in Unusual Telescopes in 1991. “It’s not the most elegant-looking optical instrument I’ve ever used, but it is one of the better planetary telescopes around. … Because of its role in the history of astronomy, I would classify this as one of the more important telescopes in the world.”

Settling Accounts

In 1880 art collector Charles Ephrussi commissioned Manet to paint A Bundle of Asparagus for 800 francs.

When Manet delivered the painting, Ephrussi gave him 1,000 francs.

So later that year Manet delivered the small painting below with a note: “There was one [sprig] missing from your bundle.”

(Thanks, Jon.)


The following story is true. There was a little boy, and his father said, ‘Do try to be like other people. Don’t frown.’ And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions.

Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle’s dictum about the 30 millions be still true, yet other people do not like it. So, if you are different, you had better hide it, and pretend to be solemn and wooden-headed. Until you make your fortune. For most wooden-headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden-headed. So be rigorous; that will cover a multitude of sins. And do not frown.

— Oliver Heaviside, “Electromagnetic Theory,” in The Electrician, Feb. 23, 1900

(When asked the population of England, Thomas Carlyle had said, “Thirty million, mostly fools.”)

Black and White

gold chess problem

Henry Dudeney in Strand, June 1911: “It would be difficult to find a prettier little chess problem in three moves, produced from such limited material as a rook and a pawn, than the one given this month, by Dr. S. Gold. The novice will probably find the task of discovering the key move quite perplexing. White plays and checkmates in three moves.”

Click for Answer

Court Order

From Chapter 12 of Ken Follett’s novel The Pillars of the Earth:

‘My stepfather, the builder, taught me how to perform certain operations in geometry: how to divide a line exactly in half, how to draw a right angle, and how to draw one square inside another so that the smaller is half the area of the larger.’

‘What is the purpose of such skills?’ Josef interrupted.

‘Those operations are essential in planning buildings,’ Jack replied pleasantly, pretending not to notice Josef’s tone. ‘Take a look at this courtyard. The area of the covered arcades around the edges is exactly the same as the open area in the middle. Most small courtyards are built like that, including the cloisters of monasteries. It’s because these proportions are most pleasing. If the middle is bigger, it looks like a marketplace, and if it’s smaller, it just looks as if there’s a hole in the roof. But to get it exactly right, the builder has to be able to draw the open part in the middle so that it’s precisely half the area of the whole thing.’

How is this done? Inscribe a diamond within a square and then rotate it 45 degrees:

court order

A Silver Lining

The opening of England’s Liverpool and Manchester Railway in 1830 took a direful turn when William Huskisson, a member of Parliament for Liverpool, approached the Duke of Wellington’s railway carriage. Huskisson became so engrossed in their conversation that he failed to notice an oncoming train, and when he realized his danger and tried to climb into Wellington’s carriage, the door swung outward and deposited him in its path. His leg was badly mangled.

“Immediately after the accident, he was placed on the ‘Northumbrian’ — another of Stephenson’s engines — and raced to Liverpool at the then unprecedented speed of 36 m.p.h., with Stephenson himself as driver,” writes Ernest Frank Carter in Unusual Locomotives. “It was the news of this accident, and the speed of the engine, which was one of the causes of the immediate adoption and rapid spread of railways over the world. Thus was the death of the first person to be involved in a railway accident turned to some good account.”

Langley’s Adventitious Angles
Image: Wikimedia Commons

Edward Mann Langley, founder of the Mathematical Gazette, posed this problem in its pages in 1922:

ABC is an isosceles triangle. B = C = 80 degrees. CF at 30 degrees to AC cuts AB in F. BE at 20 degrees to AB cuts AC in E. Prove angle BEF = 30 degrees.

(Langley’s description makes no mention of D; perhaps this is at the intersection of BE and CF.)

A number of solutions appeared. One, offered by J.W. Mercer in 1923, proposes drawing BG at 20 degrees to BC, cutting CA in G. Now angle GBF is 60 degrees, and angles BGC and BCG are both 80 degrees, so BC = BG. Also, angles BCF and BFC are both 50 degrees, so BF = BG and triangle BFG is equilateral. But angles GBE and BEG are both 40 degrees, so BG = GE = GF. And angle FGE is 40 degrees, so GEF is 70 degrees and BEF is 30 degrees.