If two triangles ABC and abc are oriented so that lines Aa, Bb, and Cc meet at a point, then the pairs of corresponding sides (AB and ab; BC and bc; and AC and ac) will meet in three collinear points.
The converse is also true: If the pairs of corresponding sides intersect in three collinear points, then the lines joining corresponding vertices will meet in a point.
Scotland’s 1904 antarctic expedition made a unique contribution to science:
A number of emperor penguins, which were here very numerous, were captured. … To test the effect of music on them, Piper Kerr played to one on his pipes, — we had no Orpheus to warble sweetly on a lute, — but neither rousing marches, lively reels, nor melancholy laments seemed to have any effect on these lethargic phlegmatic birds; there was no excitement, no sign of appreciation or disapproval, only sleepy indifference.
— Rudmose Brown et al., The Voyage of the “Scotia,” 1906
(This has produced a memorable Wikipedia image caption.)
In a joke issue of the Berichte der Deutschen Chemischen Gesellschaft in 1886, F.W. Findig offered an article on the constitution of benzene in which he finds that “zoology is capable of rendering the greatest service in clearing up the behavior of the carbon atom”:
Just as the carbon atom has 4 affinities, so the members of the family of four-handed animals possess four hands, with which they seize other objects and cling to them. If we now think of a group of six members of this family, e.g. Macacus cynocephalus, forming a ring by offering each other alternately one and two hands, we reach a complete analogy with Kekulé’s benzene-hexagon (Fig. 1).
Now, however, the aforesaid Macacus cynocephalus, besides its own four hands, possesses also a fifth gripping organ in the shape of a caudal appendix. By taking this into account, it becomes possible to link the 6 individuals of the ring together in another manner. In this way, one arrives at the following representation: (Fig. 2).
“It appears to me highly probable that a complete analogy exists between Macacus cynocephalus and the carbon atom,” Findig wrote. “In this case, each C-atom also possesses a caudal appendix, which, however, cannot be included among the normal affinities, although it takes part in the linking. Immediately this appendix, which I call the ‘caudal residual affinity’, comes into play, a second form of Kekulé’s hexagon is produced; this, being obviously different from the first, must behave differently.”
(From John Read, Humour and Humanism in Chemistry, 1947.)
“A Kiss and Its Consequences,” English carte de visite, 1910.
In 1965, Caltech computer scientist Donald Knuth privately circulated a theorem that, “under special circumstances, 1 + 1 = 3”:
Proof. Consider the appearance of John Martin Knuth, who exhibits the following characteristics: Weight 8 lb. 10 oz. (3912.23419125 grams) (3) Height 21.5 inches (0.5461 meters) (4) Voice loud (60 decibels) (5) Hair dark brown (Munsell 5.0Y2.0/11.8) (6) Q.E.D.
He conjectured that the stronger result 1 + 1 = 4 might also be true, and that further research on the problem was contemplated. “I wish to thank my wife Jill, who worked continuously on this project for nine months. We also thank Dr. James Caillouette, who helped to deliver the final result.”
(From Donald E. Knuth, Selected Papers on Fun & Games, 2011.)
Write out the positive integers in a row and underline every fifth number. Now ignore the underlined numbers and record the partial sums of the other numbers in a second row, placing each sum directly beneath the last entry that it contains.
Now, in this second row, underline and ignore every fourth number, and record the partial sums in a third row. Keep this up and the entries in the fifth row will turn out to be the perfect fifth powers 15, 25, 35, 45, 55 …
If we’d started by ignoring every fourth number in the original row, we’d have ended up with perfect fourth powers. In fact,
For every positive integer k > 1, if every kth number is ignored in row 1, every (k – 1)th number in row 2, and, in general, every (k + 1 – i)th number in row i, then the kth row of partial sums will turn out to be just the perfect kth powers 1k, 2k, 3k …
This was discovered in 1951 by Alfred Moessner, a giant of recreational mathematics who published many such curiosa in Scripta Mathematica between 1932 and 1957.
(Ross Honsberger, More Mathematical Morsels, 1991.)
“‘Tis further from London to Highgate than from Highgate to London.” — James Howell, Proverbs, 1659
In his 1991 Dictionary of Scientific Quotations, Alan L. Mackay calls this “an example of a non-commutative metric.” Highgate is at the top of a hill.
Draw a triangle and color its vertices red, green, and blue. Then divide it into as many smaller triangles as you like (the smaller triangles must meet edge to edge and vertex to vertex). Now color the vertices of these smaller triangles using the same three colors. You can do this however you like, with one proviso: The vertices that lie on a side of the large triangle must take the color of either of its ends (so, for instance, the point at the bottom center of the triangle above must be colored either green or blue, not red).
No matter how this is done, there will always exist a small triangle with vertices of three colors. In fact, there will always be an odd number of such triangles.
“Every honest researcher I know admits he’s just a professional amateur. He’s doing whatever he’s doing for the first time. That makes him an amateur. He has enough sense to know that he’s going to have a lot of trouble, so that makes him a professional.” — Charles F. Kettering
German scientist Otto von Guericke conducted a memorable experiment on May 8, 1654: He connected two hemispheres, sealed their rims together, and drew out the air between them using a pump of his own devising. The resulting vacuum was so strong that 30 horses could not pull them apart.
At the time the experiment was seen as a strike against Aristotle’s dictum that nature abhors a vacuum. It’s repeated today as a dramatic demonstration of the power of atmospheric pressure.
How many circles can be drawn through an arbitrary point on a torus? Surprisingly, there are four. Two are obvious: One is parallel to the equatorial plane of the torus, and another is perpendicular to that.
The other two are produced by cutting the torus obliquely at a special angle. They’re named after French astronomer Yvon Villarceau, who first described them in 1848.
