“The Impossible Fact”

Palmstroem, old, an aimless rover,
Walking in the wrong direction
At a busy intersection
Is run over.

“How,” he says, his life restoring
And with pluck his death ignoring,
“Can an accident like this
Ever happen? What’s amiss?

“Did the state administration
Fail in motor transportation?
Did police ignore the need
For reducing driving speed?

“Isn’t there a prohibition
Barring motorized transmission
Of the living to the dead?
Was the driver right who sped … ?”

Tightly swathed in dampened tissues
He explores the legal issues,
And it soon is clear as air:
Cars were not permitted there!

And he comes to the conclusion:
His mishap was an illusion,
For, he reasons pointedly,
That which must not, can not be.

— Christian Morgenstern (translated by Max Knight)

| Poems

The Top Hat Illusion

https://archive.org/details/B-001-014-611/page/n69/mode/2up

A striking oddity from Matthew Luckiesh’s Visual Illusions, 1922. The height of this silk hat appears much greater than its width, but the two are the same.

“A pole or a tree is generally appraised as of greater length when it is standing than when it lies on the ground. This illusion may be demonstrated by placing a black dot an inch or so above another on a white paper. Now, at right angles to the original dot place another at a horizontal distance which appears equal to the vertical distance of the first dot above the original. On turning the paper through ninety degrees or by actual measurement, the extent of the illusion will become apparent.”

| Oddities · Science & Math

“The Worst of All Puns”

https://blog.le-miklos.eu/wp-content/HabeMortemPraeOcculis.jpg

At Nuremburg a wolf’s tooth was shown to travellers … on which an Abbé is represented lying dead in a meadow, with three lilies growing out of his posteriors. This is not only the worst pun that ever was carved upon a wolf’s tooth, but the worst that ever was or will be made. The Abbé is designed to express the Latin word Habe. He is lying dead in a meadow, … mort en pré; this is for mortem præ; and the three lilies in his posteriors are to be read oculis, … au cu lis. Thus, according to the annexed explanation, the whole pun, rebus, or hieroglyphic, is Habe mortem præ oculis.

— Robert Southey, Omniana, 1812

In other words, the French phrase Abbé mort en pré au cul lys (“Abbot died in a meadow with lilies in his rump”) sounds like the Latin phrase Habe mortem præ oculis (“Keep death before your eyes”). This joke appears to be referenced in Hieronymus Bosch’s 1504 triptych The Garden of Earthly Delights:

https://commons.wikimedia.org/wiki/File:The_Garden_of_earthly_delights.jpg

| Art · Humor · Language · Oddities

Cameo

Western Illinois University mathematician Iraj Kalantari published an unusual puzzle in Math Horizons in February 2019. A sphere B of radius 150 is centered at (150, 150, 0). A sphere M of radius 144, centered on the z-axis, lies entirely below the (x, y)-plane so that the volume of its intersection with B is 1/2. “Can we find a sphere S of radius 73 that has its center on the circle (x – 73)2 + (y – 73)2 = 1502 in the plane z = 73 so that the volume of B minus its intersections with M and S equals the volume of M minus its intersection with B plus the volume of S minus its intersection with B?”

The answer is no, because Vol(B – (M ∪ S)) = Vol((M – B) ∪ (S – B)) if and only if Vol(B) = Vol(M) + Vol(S), and that’s the case if and only if r_{B}^{3} = r_{M}^{3} + r_{S}^{3} , where rB, rM, and rS are the radii of the three spheres. “[A]nd because the radii are integers, this equality is impossible by Fermat’s last theorem!”

The placement of the spheres and the fact that the values differ by 1 are red herrings.

(Iraj Kalantari, “The Three Spheres,” Math Horizons 26:3 [February 2019]: 13, 25.)

02/28/2026 UPDATE: In my original statement of the problem I left out a vital phrase in Kalantari’s presentation: The sphere S should have its center on the circle (x – 73)2 + (y – 73)2 = 1502 in the plane z = 73.

I’d omitted the last phrase, a condition that guarantees that S lies above the xy plane and so does not intersect sphere M, which is required to deduce the equation involving the volumes. Many thanks to reader Francesco Veneziano for pointing this out.

| Science & Math

Devotion

A “prayer to the local deities” offered by Socrates in Plato’s Phaedrus:

Beloved Pan, and all ye other gods who haunt this place, give me beauty in the inward soul; and may the outward and inward man be at one. May I reckon the wise to be the wealthy, and may I have such a quantity of gold as a temperate man and he only can bear and carry.

“Anything more? The prayer, I think, is enough for me.”

| Society

Down Under

https://commons.wikimedia.org/wiki/File:La_rotondit%C3%A9_de_la_Terre,_Image_du_monde_(cropped).jpg

How is it with those who imagine that there are antipodes opposite to our footsteps? Do they say anything to the purpose? Or is there any one so senseless as to believe that there are men whose footsteps are higher than their heads? Or that the things which with us are in a recumbent position, with them hang in an inverted direction? That the crops and trees grow downwards? That the rains, and snow, and hail fall upwards to the earth? And does any one wonder that hanging gardens are mentioned among the seven wonders of the world, when philosophers make hanging fields, and seas, and cities, and mountains?

— Lactantius, Institutiones Divinae, 303

| History · Science & Math

Shapes of Things

https://commons.wikimedia.org/wiki/File:Sierra_Leone-Mappa.gif

In 2016, University of Buenos Aires computer science student Gonzalo Ciruelos worked out that the roundest country in the world is Sierra Leone, with a roundness index of 0.934 on a scale of 0 to 1.

He’d been inspired by David Barry, who’d found that the world’s most rectangular country is Egypt (0.955 on the same scale).

Metropolitan France is known as the Hexagon. I suppose each country has its claim to fame.

(Gonzalo Ciruelos, “What Is the Roundest Country?”, Math Horizons 26:3 [February 2019], 26-27.)

| Science & Math · Trivia