# A Different View

Manet’s painting A Bar at the Folies-Bergère is sometimes criticized for its confused composition. The bottles to the barmaid’s right stand near the back of the bar, but in the reflection behind her they stand near the front. Her own image ought to stand behind her, not off to the right. And reflection of the man she’s addressing (in the position of the painter, or the viewer) ought also to be behind her — indeed, she herself should be blocking our view of it.

But in a dissertation at the University of New South Wales, art historian Malcolm Park found that the arrangement makes sense if certain assumptions are reconsidered. The barmaid is facing the viewer across the bar, with a mirror behind her. But she’s looking diagonally along the bar, not directly across it. (See the diagram here.)

The bottles in the background and the man she appears to be addressing are both in fact to the viewer’s left, beyond the edge of the frame and so visible only as reflections. And the barmaid’s own reflection appears to our right because, from our perspective, the mirror is not directly behind her — it’s “turned” somewhat, carrying her image over to one side.

(Malcolm Park, Ambiguity and the Engagement of Spatial Illusion Within the Surface of Manet’s Paintings, dissertation, College of Fine Arts, University of New South Wales, 2001.)

# All Together Now

In 1833, Heinrich Scherk conjectured that every prime of odd rank (accepting 1 as prime) can be composed by adding and subtracting all the smaller primes, each taken once. For instance, 13 is the 7th prime and 13 = 1 + 2 – 3 – 5 + 7 + 11.

In 1967 J.L. Brown Jr. proved that this is true.

# One Nation, Indivisible

The second professor of mathematics in the American colonies suggested reckoning coins, weights, and measures in base 8.

Arguing that ordinary arithmetic had already become “mysterious to Women and Youths and often troublesome to the best Artists,” the Rev. Hugh Jones of the College of William and Mary wrote that his proposal was “only to divide every integer in each species into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho’ all nations count universally by tens (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ten is uncapable.”

Successive powers of 8 would be called ers, ests, thousets, millets, and billets; cash, casher, and cashest would be used in counting money, ounce, ouncer, and ouncest in weighing, and yard, yarder, and yardest in measuring distance (so “352 yardest” would signify 3 × 82 + 5 × 8 + 2 yards).

Jones pressed this system zealously, arguing that “Arithmetic by Octaves seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic.” But he seems to have had no illusions about its prospects, acknowledging that “there seems no Probability that this will be soon, if ever, universally complied with.”

(H.R. Phalen, “Hugh Jones and Octave Computation,” American Mathematical Monthly 56:7 [August-September 1949), 461-465.)

# A Folded Cube

A simple and surprisingly effective illusion by Lisbon anamorphosis specialists Sonhos com Dimensão.

# The Pizza Theorem

If you’re sharing a pizza with another person, there’s no need to cut it into precisely equal slices. Make four cuts at equal angles through an arbitrary point and take alternate slices, and you’ll both get the same amount of pizza.

Larry Carter and Stan Wagon came up with this “proof without words”: Each piece in an odd-numbered sector corresponds to a congruent piece in an even-numbered sector, and vice versa.

Also: If a pizza has thickness a and radius z, then its volume is pi z z a.

(Larry Carter and Stan Wagon, “Proof Without Words: Fair Allocation of a Pizza,” Mathematics Magazine 67:4 [October 1994], 267-267.)

# Math Notes

In a 1752 letter to Euler, Christian Goldbach suggested that every odd integer is the sum of a prime and twice a square. (At the time, 1 was considered a prime number.)

Only two exceptions, 5777 and 5993, have ever been found.

# Illustration

A pleasing observation by W.V. Quine from 1988:

Fermat’s Last Theorem can be vividly stated in terms of sorting objects into a row of bins, some of which are red, some blue, and the rest unpainted. The theorem amounts to saying that when there are more than two objects, the following statement is never true:

Statement. The number of ways of sorting them that shun both colors is equal to the number of ways that shun neither.

He explains this, very concisely, here.

(W.V. Quine, “Fermat’s Last Theorem in Combinatorial Form,” American Mathematical Monthly 95:7 [September 1988], 636.)

# A New Perspective

I find myself more than half convinced by the oddly repellent hypothesis that the peculiarity of the time dimension is not … primitive but is wholly a resultant of those differences in the mere de facto run and order of the world’s filling. It is then conceivable, though doubtless physically impossible, that one four-dimensional area of the time part of the manifold be slewed around at right angles to the rest, so that the time order of that area, as composed by its interior lines of strain and structure, run parallel with a spatial order in its environment. It is conceivable, indeed, that a single whole human life should lie thwartwise of the manifold, with its belly plump in time, its birth at the east and its death in the west, and its conscious stream running alongside somebody’s garden path.

— Donald C. Williams, “The Myth of Passage,” Journal of Philosophy 48:15 (1951), 457-472

# A Near Miss

For a moment in the 1998 Simpsons episode “The Wizard of Evergreen Terrace,” it appears that Homer has found a solution to Fermat’s last theorem:

398712 + 436512 = 447212

If you check this on a calculator with a 10-digit display, it seems to work: Raise 3987 and 4365 each to the 12th power, take the 12th root of the sum, and you get 4472.

But that’s the fault of the display. The actual value for the third term is closer to 4472.000000007057617187512.

Simpsons writer David S. Cohen, who had studied physics at Harvard and contrived the ruse, told Simon Singh he was pleased at the consternation it caused online. “I feel great about it. It’s very easy working in television to not feel good about what you do on the grounds that you’re causing the collapse of society. So, when we get the opportunity to raise the level of discussion — particularly to glorify mathematics — it cancels out those days when I’ve been writing those bodily function jokes.”

(From Simon Singh, The Simpsons and Their Mathematical Secrets, 2013.)

# Vanishing Act

There is often peculiar humour about self-frustration. Consider, for example, a train of events which started outside the old Clarendon Laboratory, Oxford. I came across a dirty beaker full of water just when I happened to have a pistol in my hand. Almost without thinking I fired, and was surprised at the spectacular way in which the beaker disappeared. I had, of course, fired at beakers before; but they had merely broken, and not shattered into small fragments. Following Rutherford’s precept I repeated the experiment and obtained the same result: it was the presence of the water which caused the difference in behavior. Years later, after the War, I found myself having to lecture to a large elementary class at Aberdeen, teaching hydrostatics ab initio. Right at the beginning came the definitions — a gas having little resistance to change of volume but a liquid having great resistance. I thought that I would drive the definitions home by repeating for the class my experiments with the pistol, for one can look at them from the point of view of the beaker, thus suddenly challenged to accommodate not only the liquid that it held before the bullet entered it, but also the bullet. It cannot accommodate the extra volume with the speed demanded, and so it shatters.

— R.V. Jones, “Impotence and Achievement in Physics and Technology,” Nature 207:4993 (1965), 120-125

(When the Royal Engineers tried to use this trick to demolish a tall chimney, filling its base with 6 feet of water and firing an explosive charge into the water, “it succeeded so well that it failed completely”: The incompressible water flung the surrounding ring of bricks outward, leaving a foreshortened chimney suspended above in midair. This dropped down neatly onto the old foundation, upright and intact, “presenting the Sappers with an exquisite problem.”)