“An Elevation Puzzle”

This does my head in — it’s a puzzle from the October 1958 issue of Eureka, the journal of the Cambridge University Mathematical Society:

“Below are shown the front elevation and plan of a mathematical figure. What is the side elevation?”

https://archive.org/details/eureka-21/page/7/mode/2up?view=theater — Image: Eureka

The terms (I believe) refer to multiview orthographic projection, the illustration technique used in architectural drawings: The front elevation is the view looking squarely at the “front” of the object, and the plan view looks down from above. What is the side view?

	Select Click for Answer
Here’s the answer given in the journal: Image: Eureka No explanation is given. I had imagined something like a curved disk, but that wouldn’t give a circular plan view. The nearest similar puzzle I know is the “architect’s puzzle,” in which you’re asked to imagine a single object that can cast three different shadows: a circle, a square, and a triangle. That’s possible, but it’s a distinctly different task, and it’s hard to see how to adapt that object into the one we’re seeking here. 09/27/2023 UPDATE: I’ve received a ton of mail about this puzzle — a million thanks to everyone who’s contributed; I’m constantly impressed by your intelligence, imagination, and resourcefulness. The consensus is that the solution was published upside down! And the puzzle as presented is a bit ambiguous. Reader Catalin Voinescu explains: Note there are no additional lines to the semicircle that is the front elevation: that’s all there is to see from that direction. The plan view could be filled or not, and we can’t tell. So the figure must be the intersection of the bottom half of a horizontal cylinder surface (like a half-round gutter) and either a vertical cylinder surface, like a gutter downpipe, or a solid vertical cylinder. If you think of what a T joint of two cylinders of the same diameter looks like from the side, the joints appear as straight lines that bisect the angle between the cylinders. (They must be straight lines bisecting the angle for reasons of symmetry.) So the given answer is one of two possibilities. When the plan view is hollow, the figure is a closed curve in three dimensions: draw an ellipse with a long diameter $\displaystyle \sqrt{2}$ times the short diameter, and bend the paper at 90 degrees along the short diameter. The top view is a circle, the side view a semicircle, the other side view is the angle made by the paper. However, if the plan view was solid, the figure would be a curved surface: what you need to cut out of the half-round gutter to get the vertical cylinder to go through it. Viewed from the side, it would look like a triangle — like the given solution, but with a line at the bottom. That second solution looks like this: Here’s an interactive plot of the first solution, and here’s the second. Click and drag these figures to examine them. In both cases the side elevation is an inverted V, so the published solution is upside down (and I’m not crazy!). My thanks to Drake Thomas and to Catalin for their work on all this, and thanks again to everyone else who wrote in about it. I’ve actually gone through the subsequent issues of the journal to see whether an erratum notice was ever published, and I don’t find one. I wonder how much mail this generated in 1958!

Click for Answer

Here’s the answer given in the journal:

https://archive.org/details/eureka-21/page/29/mode/2up?view=theater — Image: Eureka

No explanation is given. I had imagined something like a curved disk, but that wouldn’t give a circular plan view.

The nearest similar puzzle I know is the “architect’s puzzle,” in which you’re asked to imagine a single object that can cast three different shadows: a circle, a square, and a triangle. That’s possible, but it’s a distinctly different task, and it’s hard to see how to adapt that object into the one we’re seeking here.

09/27/2023 UPDATE: I’ve received a ton of mail about this puzzle — a million thanks to everyone who’s contributed; I’m constantly impressed by your intelligence, imagination, and resourcefulness. The consensus is that the solution was published upside down! And the puzzle as presented is a bit ambiguous. Reader Catalin Voinescu explains:

Note there are no additional lines to the semicircle that is the front elevation: that’s all there is to see from that direction. The plan view could be filled or not, and we can’t tell. So the figure must be the intersection of the bottom half of a horizontal cylinder surface (like a half-round gutter) and either a vertical cylinder surface, like a gutter downpipe, or a solid vertical cylinder. If you think of what a T joint of two cylinders of the same diameter looks like from the side, the joints appear as straight lines that bisect the angle between the cylinders. (They must be straight lines bisecting the angle for reasons of symmetry.)

So the given answer is one of two possibilities.

When the plan view is hollow, the figure is a closed curve in three dimensions: draw an ellipse with a long diameter $\displaystyle \sqrt{2}$ times the short diameter, and bend the paper at 90 degrees along the short diameter. The top view is a circle, the side view a semicircle, the other side view is the angle made by the paper.

However, if the plan view was solid, the figure would be a curved surface: what you need to cut out of the half-round gutter to get the vertical cylinder to go through it. Viewed from the side, it would look like a triangle — like the given solution, but with a line at the bottom.

That second solution looks like this:

elevation puzzle cylinder solution

Here’s an interactive plot of the first solution, and here’s the second. Click and drag these figures to examine them. In both cases the side elevation is an inverted V, so the published solution is upside down (and I’m not crazy!).

My thanks to Drake Thomas and to Catalin for their work on all this, and thanks again to everyone else who wrote in about it. I’ve actually gone through the subsequent issues of the journal to see whether an erratum notice was ever published, and I don’t find one. I wonder how much mail this generated in 1958!

September 26, 2023September 27, 2023 | Puzzles

“Auld Saws Speak Truth”

Scottish proverbs:

The shortest road’s the nearest.
The less wit a man has the less he kens the want o’t.
The gude dog doesna aye get the best bane.
Sorrow an’ ill weather come unca’d.
Silence and thought hurt nae man.
Self praise is nae honour.
Every bird thinks its ain nest best.
He that keeks through a keyhole may see what will vex him.
There’s naething mair precious than time.
He that lends money to a friend has a double loss.
Haste and anger hinder gude counsel.
Friends ‘gree best at a distance.
Weel done, soon done.
Be a friend to yoursel’ and ithers will.
A wise man wavers, a fool is fixed.
The langest day has an end.

And “There’s naething sae gude on this side o’ time but it micht hae been better.”

(From Colin S.K. Walker, Scottish Proverbs, 2000.)

September 25, 2023September 24, 2023 | Literature

Nice Try

In 2021 Denmark’s Kunsten Museum of Modern Art engaged artist Jens Haaning to recreate two of his earlier works, An Average Austrian Year Income and An Average Danish Annual Income. Those pieces had presented framed piles of kroner and euro bills to represent the sums indicated in their titles, so the museum gave Haaning 532,549 kroner for the purpose.

Instead, he delivered two framed blank canvases. The title of this new work, he said, was Take the Money and Run.

“This is only a piece of art if I don’t return the money,” he told the New York Times. “I believe that I have created a good and relevant piece of artwork, which could be hung on the wall.”

It didn’t work out that way — on Sept. 18, a Copenhagen court ruled that he must return the money.

Haaning acknowledged that he hadn’t fulfilled the commission, but “I completed something else,” he argued. “You’re asked to show a 10- and a 12-year-old work, and suddenly you have a better concept.”

(Thanks, Sharon.)

September 25, 2023September 24, 2023 | Art

“Elessdé”

In a certain fair island, for commerce renown’d,
Whose fleets sailed in every sea,
A set of fanatics, men say, there was found,
Who set up an island and worship around,
And called it by name Elessdé.

Many heads had the monster, and tails not a few,
Of divers rare metals was he;
And temples they built him right goodly to view,
Where oft they would meet, and, like idolists true,
Pay their vows to the great Elessdé.

Moreover, at times would their frenzy attain
(‘Twas nought less) to so high a degree,
That his soul-blinded votaries did not complain,
But e’en laid down their lives his false favour to gain–
So great was thy power, Elessdé.

As for morals, this somewhat unscrupulous race
Were lax enough, ‘twixt you and me;
Men would poison their friends with professional grace,
And of the fell deed leave behind ne’er a trace,
For the sake of the fiend, Elessdé.

Then forgery flourished, and rampant and rife
Was each form of diablerie;
While the midnight assassin, with mallet and knife,
Would steal on his victim and rob him of life,
And all for thy love, Elessdé.

There were giants of crime on the earth in that day,
The like of which we may not see:
Although, peradventure, some sceptic will say
There be those even now who acknowledge the sway
Of the god of the world — £ s. d.

(That is, pounds, shillings, and pence, the pre-decimal currencies of the British Isles.)

From William T. Dobson, Poetical Ingenuities and Eccentricities, 1882.

September 24, 2023September 23, 2023 | Poems

Roll Call

More than 500 of the commanders in the American Civil War were West Point graduates, and all but a handful of these studied under the same military theorist, Dennis Hart Mahan, whose course “Engineering and the Science of War” applied the lesson of Napoleon’s campaigns — that the key to success was rapid maneuver to concentrate one’s forces at a critical point of weakness in the enemy position. (Of Grant’s encirclement of Vicksburg, Mahan said, “European warfare can produce nothing equal to it since the 1st Napoleon.”)

In addition to Grant, Mahan’s pupils included William Tecumseh Sherman, George Meade, George B. McClellan, Henry Halleck, George Henry Thomas, John Sedgwick, Ambrose Burnside, Joseph Hooker, and Philip Sheridan among the Union forces and Robert E. Lee, Jefferson Davis, Stonewall Jackson, James Longstreet, Albert Sidney Johnston, P.G.T. Beauregard, John Bell Hood, Richard S. Ewell, J.E.B. Stewart, and William J. Hardee in the Confederacy. Historian James Robertson called him “the guru of the generals.”

(James Robertson, The Untold Civil War, 2011.)

September 23, 2023September 22, 2023 | History

Robinsonades

The “Swiss family Robinson” is not named Robinson. The title of Johann David Wyss’s 1812 novel shows the enormous influence of Daniel Defoe’s Robinson Crusoe even a century after its publication; the 18th century was filled with “Robinsonades” in German, Dutch, French, Danish, Swiss, Swedish, and Italian:

Teutsche Robinson, 1722
Americanische Robinson, 1724
Nordische Robinson, 1741
Hollandsche Robinson, 1743
Dänische Robinson, 1750
Walchersche Robinson, 1752
Maldivschen Philosophen Robine, 1753
Oude en Jongen Robinson, 1753
Isländische Robinson, 1755
Hartz-Robinson, 1755
Robinson vom Berge Libonon, 1755
Haagsche Robinson, 1758
Robertson [sic] aux terres australes, 1766
Steyerische Robinson, 1791
Böhmische Robinson, 1796

Wyss’s marooned Swiss family is nameless.

(Gary Dexter, Why Not Catch-21?, 2007.)

September 22, 2023September 21, 2023 | Literature

Brave New World

A Canadian notice to new telephone users, 1896:

To Listen: Place the telephone fairly against the ear, with an upward motion, so that the lower extremity or lobe of the ear is gathered in, into the cavity of the telephone; in this position it will be found to fit snugly and comfortably — the lobe of the ear acting as a cushion and at the same time closing out all ulterior sounds, thus enabling the voice to be heard with clearness and precision.

One California instruction read, “Speak directly into the mouthpiece keeping mustache out of the opening.”

With no social conventions to follow, users had to be taught propriety. AT&T promoted a “Telephone Pledge” that read, “I believe in the Golden Rule and will try to be Courteous and Considerate over the Telephone as if Face to Face.” The winner of a 1910 Bell essay contest wrote, “Would you rush into an office or up to the door of a residence and blurt out ‘Hello! Hello! Who am I talking to?’ No, one should open conversations with phrases such as ‘Mr. Wood, of Curtis and Sons, wishes to talk with Mr. White …’ without any unnecessary and undignified ‘Hello’s.'”

In America Calling (1992), Claude S. Fischer notes, “Companies cut off service to abusers and obtained legislation that fined or even jailed profane customers.”

September 22, 2023September 21, 2023 | Society · Technology

In a Word

metrophobia
n. an irrational fear of poetry

September 21, 2023September 20, 2023 | Language

Never Mind

In a 1961 contribution to Analog, Maurice Price explained that, while a new engineer might arrive to a clean desk, a “cycle of confusion” quickly begins as papers pile up so deeply that the engineer can do little more than fish old technical journals from the pile and read them (this is known as “keeping up with the state of the art”). Eventually the engineer resolves to clean the desk, but that only returns the system to the start of the cycle. The key equation is

C = K₁ exp (K₂t).

“In this equation K₁ is the constant of confusion and K₂ is the coefficient of chaos. These may vary from desk to desk and from engineer to engineer, but the general form of the curve is not altered. Note that the amount of work to be done does not affect the curve at all. This is because the amount of work expands to overflow the available desk area.”

The only way to break the cycle is to promote the engineer, but that only restarts the cycle at a new desk. And productivity and concentration actually increase with clutter. So the best solution is for the engineer to “bring his desk to the state of stagnation as soon as possible. From then on, the desk should be ignored completely. All work must be carried out by telephone.”

(Maurice Price, “An Introduction to the Calculus of Desk-Clearing,” Analog Science Fact & Fiction, British edition, 1961, 68-72.)

September 21, 2023September 20, 2023 | Humor · Science & Math

Peak Performance

This is fascinating if it’s true: Ten municipalities meet at the summit of Mount Etna, producing a “decipoint” and one of the most complex arrangements of political boundaries outside Antarctica.

I say “if it’s true” because, for such a striking fact, it’s surprisingly hard to confirm. Many sources point to a blog post at Condé Nast Traveler by Jeopardy champion Ken Jennings, but that cites no source.

On the other hand, no one else seems to doubt it, and this map by Patrick McGranaghan won the American Geographical Society’s monthly map contest in November 2017. Maybe I’m too skeptical?

09/21/2023 UPDATE: Wow, it seems to be true. The official website of the Parco dell’Etna includes a map (PDF) showing the distribution of comuni metropolitani around the peak. And the website of the Italian National Institute of Statistics offers data files on statistical districts that can be opened using the desktop version of Google Earth. Many thanks to readers Rob Miller and Ross Ogilvie for looking into this.

September 20, 2023September 22, 2023 | Oddities · Trivia