Theorem 1. A crocodile is longer than it is wide.

Proof. A crocodile is long on the top and bottom, but it is green only on the top; therefore a crocodile is longer than it is green. A crocodile is green along both its length and width, but it is wide only along its width; hence a crocodile is greener than it is wide. Therefore a crocodile is longer than it is wide.

Theorem 2. Napoleon was a poor general.

Proof. Most men have an even number of arms. Napoleon was warned that Wellington would meet him at Waterloo. To be forewarned is to be forearmed. But four arms is a very odd number of arms for a man. The only number that is both even and odd is infinity. Therefore, Napoleon had an infinite number of arms in his battle against Wellington. A general who loses a battle despite having an infinite number of arms is very poor general.

Theorem 3. If 1/0 = ∞, then 1/∞ = 0.

Proof. Given:

Rotate both sides 90° counterclockwise:

Subtract 8 from both sides:

Now reverse the rotation:

Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.

(Thanks, Lee.)

I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.

For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.

I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.

The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.

German mathematician Erich Bessel-Hagen was often teased for his protruding ears.

In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.

That page makes no mention of Bessel-Hagen, but it does contain this figure:

Is that libel?

From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!

William D. Harvey offered this in Omni in 1980 — a mnemonic for spelling mnemonics:

Mnemonics neatly eliminate man’s only nemesis: insufficient cerebral storage.

We [Einstein and Ernst Straus] had finished the preparation of a paper and were looking for a paper clip. After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came upon a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When asked what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’

— Ernst Straus, “Memoir,” in A.P. French, ed., Einstein: A Centenary Volume, 1979

(Einstein said to an assistant at Princeton that this was the most characteristic anecdote that could be told of him.)