Any pair of points define an infinity of ellipses and an infinity of hyperbolas.
The ellipses do not touch one another, nor do the hyperbolas.
But every ellipse meets every hyperbola at a right angle.
Science & Math
The Black Hole
If Satan plays miniature golf, this is his favorite hole. A ball struck at A, in any direction, will never find the hole at B — even if it bounces forever.
The idea arose in the 1950s, when Ernst Straus wondered whether a room lined with mirrors would always be illuminated completely by a single match.
Straus’ question went unanswered until 1995, when George Tokarsky found a 26-sided room with a “dark” spot; two years later D. Castro offered the 24-sided improvement above. If a candle is placed at A, and you’re standing at B, you won’t see its reflection anywhere around you — even though you’re surrounded by mirrors.
The Franklin Square
In a 1769 letter, Ben Franklin describes a magic square he devised in his youth. The magic total of 260 can be reached by adding the numbers in each row or column, as in a normal magic square. But “bent rows” (shaded) produce the same total, even when “wrapped across” the border of the table. This works in all four directions.
Further: Half of each row or column sums to half of 260, as does any 2×2 subsquare. And the four corners and the four center squares sum to 260. (Alas, the main diagonals don’t, so this doesn’t strictly qualify as a magic square by the modern definition.)
Interestingly, no one knows how Franklin created the square. Many methods have been devised, but none apparently as quick as his, which he claimed could generate them “as fast as he could write.”
The Perfect Family
Take any Platonic solid, join the centers of its faces, and, charmingly, you get another Platonic solid. The cube and the octahedron produce one another, as do the dodecahedron and the icosahedron, and the tetrahedron produces another tetrahedron.
Bonus factoid: If you inscribe a dodecahedron and an icosahedron in the same sphere, the dodecahedron will occupy more of the sphere’s volume. It has fewer faces than the icosahedron, but its faces are more nearly circular, so it fits the sphere more snugly.
See The Pup Tent Problem.
Mathematicians’ Graves
Archimedes wanted no other epitaph than a sphere inscribed within a cylinder — he had determined the sphere’s relative volume and considered this his greatest achievement.
Henry Perigal’s tomb in Essex displays his graphic proof of the Pythagorean theorem (left).
Gauss wanted to be buried under a heptadecagon, which he’d shown can be constructed with compass and straightedge. (The stonemason demurred, fearing he’d produce only a circle.)
And Jakob Bernoulli opted for a logarithmic spiral and the words Eadem mutata resurgo—the motto means “I shall arise the same though changed.”
Benardete’s String Paradox
Let us take a piece of string. In the first half minute we shall form an equilateral triangle with the string; in the next quarter minute we shall employ the string to form a square; in the next eighth minute we shall form a regular pentagon; etc. ad infinitum. At the end of the minute what figure or shape will our piece of string be found to have assumed? Surely it can only be a circle. And yet how intelligible is that process? Each and every one of the polygons in our infinite series contains only a finite number of sides. There is thus a serious conceptual gap separating the circle, as in the limiting case, from each and every polygon in the infinite series.
— Jose Amado Benardete, Infinity: An Essay in Metaphysics, 1964
Math Notes
Turn each of these palindromes “inside out” and their sum remains the same:
13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097
Remarkably, this holds true even if you square or cube them:
130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972
130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973
From Albert Beiler, Recreations in the Theory of Numbers, 1964.
Misc
- Douglas Adams claimed that the funniest three-digit number is 359.
- Romeo has more lines than Juliet, Iago than Othello, and Portia than Shylock.
- Friday the 13th occurs at least once a year.
- “By nature, men love newfangledness.” — Chaucer
- John was the only apostle to die a natural death.
The Paradox of Dives and Lazarus
The Gospel of Luke contains a parable about a rich man and a beggar. Both men die, and the rich man is consigned to hell while the beggar is received into the bosom of Abraham. The rich man pleads for mercy, but Abraham tells him that in his lifetime he received good things and the beggar evil things: “now he is comforted and thou art tormented.” The rich man then begs that his brothers be warned of what lies in store for them, but Abraham rejects this plea as well, saying, “If they hear not Moses and the prophets, neither will they be persuaded though one rose from the dead.”
Now, writes E.V. Milner:
Suppose … that this last request of Dives had been granted; suppose, in fact, that some means were found to convince the living, whether rich men or beggars, that ‘justice would be done’ in a future life, then, it seems to me, an interesting paradox would emerge. For if I knew that the unhappiness which I suffer in this world would be recompensed by eternal bliss in the next world, then I should be happy in this world. But being happy in this world I should fail to qualify, so to speak, for happiness in the next world. Therefore, if there were such a recompense awaiting me, its existence would seem to entail that I should at least be not wholly convinced of its existence.
“Put epigrammatically, it would appear that the proposition ‘Justice will be done’ can only be true for one who believes it to be false. For one who believes it to be true justice is being done already.”
Countdown
Write the numbers 82 to 1 in descending order and string them together:
8281807978777675747372717069686766656463626160595857565554535251504948474645444-3424140393837363534333231302928272625242322212019181716151413121110987654321
The resulting 155-digit number is prime.