All the Way Down

https://commons.wikimedia.org/wiki/File:Geometric_series_14_square2.svg

The infinite series 1/4 + 1/16 + 1/64 + 1/256 + … was one of the first to be summed in the history of mathematics; Archimedes had found by 200 BC that it totals 1/3. There are two neat visual demonstrations that make this fact immediately apparent. In the unit square above, the largest black square has area 1/4, the next-largest black square has area 1/16, and so on. Regions of black, white, and gray make up equal areas in the total figure, so the black squares, taken together, must have area 1/3.

The same argument can be made using triangles (below). If the area of the largest triangle is 1, then the largest black triangle has area 1/4, the next-largest 1/16, and so on. Areas of black, white, and gray make up equal parts of the total figure, so the black regions must total 1/3.

https://commons.wikimedia.org/wiki/File:Geometric_series_triangle.svg
Image: Wikimedia Commons
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Misc

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

  • Lady Godiva’s horse was named Aethenoth.
  • UGHA in BROUGHAM is silent.
  • 7 × 58 × 73 × 28 = 7587328
  • APHELIOTROPISMS is an anagram of OMPHALOTRIPSIES.
  • “The French for London is Paris.” — Ionesco

“No general proposition is worth a damn.” — Oliver Wendell Holmes Jr. (a general proposition)

| Language · Quotations · Science & Math · Trivia

Midy’s Theorem

The decimal expansion of 1/7 is

0.142857142857 …

Interestingly, if you split the repeating decimal period in half and add the two complements, you get a string of 9s:

142 + 857 = 999

It turns out this is true for every fraction with a prime denominator and a repeating decimal period of even length:

1/11 = 0.090909 …
0 + 9 = 9

1/13 = 076923 …
076 + 923 = 999

1/17 = 0.0588235294117647 …
05882352 + 94117647 = 99999999

1/19 = 0.052631578947368421 …
052631578 + 947368421 = 999999999

It was discovered by French mathematician E. Midy in 1836.

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Suggestion

As to your method of work, I have a single bit of advice, which I give with the earnest conviction of its paramount influence in any success which may have attended my efforts in life — Take no thought for the morrow. Live neither in the past nor in the future, but let each day’s work absorb your entire energies, and satisfy your widest ambition. That was a singular but very wise answer which Cromwell gave to Bellevire — ‘No one rises so high as he who knows not whither he is going,’ and there is much truth in it. The student who is worrying about his future, anxious over the examinations, doubting his fitness for the profession, is certain not to do so well as the man who cares for nothing but the matter in hand, and who knows not whither he is going!

— William Osler, advice to students, McGill College, 1899

| Science & Math · Society

A Late Valentine?

In 1818, English zoologist William Elford Leach named nine new genera of parasitic isopods: Anilocra, Canolira, Cirolana, Conilera, Lironeca, Nelocira, Nerocila, Olencira, and Rocinela.

Each of these is an anagram of the name Caroline (or its Latinized form Carolina). But Leach was not married and had no known relationship with any woman of that name.

The genera stand as a “tantalizing puzzle for posterity.” More here.

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Lahaina Noon

Twice a year, objects Hawaii lose their shadows as the sun passes directly overhead.

A “zero shadow day” occurs biannually between the Tropics of Cancer and Capricorn, arriving at each location when the sun’s declination equals its latitude.

| Oddities · Science & Math

Number Theory

What’s the funniest number? Yale physicist Emily Pottebaum proposed the Perceived Specificity Hypothesis, which states that “for nonnegative integers < 100, the funniness of a number increases with its apparent precision." She surveyed 68 acquaintances and found that:

  • Among integers divisible by 10, 0 is funniest.
  • Odd numbers are consistently funnier than even.
  • “Furthermore, the most oddly specific numbers — odd numbers with a degree of specificity of 2 — are the most funny, according to the data presented here.”

The degree of specificity characterizes the distance between an integer and the nearest multiple of 5:

https://arxiv.org/abs/2503.24175
Image: arxiv.org

So 3, 7, 13, 17, etc. were judged to be funniest.

“I acknowledge my Ph.D. advisor, who I shall not name out of respect for her academic integrity, for her exasperation upon learning about this study. I thank her for putting up with my antics and plead that she continue to do so until I graduate.”

(E.G. Pottebaum, “What Is the Funniest Number? An Investigation of Numerical Humor,” arXiv preprint, arXiv:2503.24175 [2025].)

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