Today in Incoherent Numerology

Or, “Oh, Wikipedia, How I Love Thee. Let me count the ways: one, two, phi…”

From Wikipedia’s page on Duchamp’s Nude Descending a Staircase, No. 2 (today’s version):

It has been noted disquisitively [link] that the number 1001 of Duchamp’s entry at the 1912 Indépendants catalogue also happens to represent an integer based number of the Golden ratio base, related to the golden section, something of much interest to the Duchamps and others of the Puteaux Group. Representing integers as golden ratio base numbers, one obtains the final result 1000.1001φ. This, of course, was by chance—and it is not known whether Duchamp was familiar enough with the mathematics of the golden ratio to have made such a connection—as it was by chance too the relation to Arabic Manuscript of The Thousand and One Nights dating back to the 1300s.

Euhhhhh, non.

As best I can tell, all this is saying is that the catalogue number of Duchamp’s painting contains only 0s and 1s.

The idea behind the “golden ratio base” is that we can write integer numbers in terms of the golden ratio, $\phi$, if we add up different powers of $\phi$. For example, anything to the zeroth power is 1, so $1 = \phi^0$. Less obviously, we can say from the definition of $\phi$ that

$\frac{1}{\phi} = \phi – 1.$

Squaring both sides of this equation,

$\frac{1}{\phi^2} = (\phi – 1)^2 = \phi^2 – 2\phi + 1.$

So,

$\frac{1}{\phi^2} + \phi = \phi^2 – \phi + 1 = \phi(\phi – 1) + 1.$

Referring back to our first equation,

$\frac{1}{\phi^2} + \phi = \phi\left(\frac{1}{\phi}\right) + 1,$

which means that

$\frac{1}{\phi^2} + \phi = 2.$

Another way of writing this would be to say

$2 = \phi^{-2} + \phi^1.$

With more cleverness, we can write any positive integer as a sum of powers of $\phi$:

$N = \phi^{k_1} + \phi^{k_2} + \cdots + \phi^{k_n},$

where the numbers $k_1$ through $k_n$ are distinct integers. Notice that we don’t have any coefficients in front of the terms—or, to say it more carefully, the coefficient of any term in the sum is either zero or one. So, “1001” could be a representation of a number in the golden-ratio base, if we read it as

$1001_\phi = 1\cdot\phi^3 + 0\cdot\phi^2 + 0\cdot\phi^1 + 1\cdot\phi^0.$

In the same way, “1000.1001” can stand for a number in base $\phi$. It’s the number we normally write as 5. It is not the “final result” of “representing integers as golden ratio base numbers.”

I tried making sense of the disquisition to which Wikipedia credits this observation. The stuff about writing numbers in the golden-ratio base isn’t even there. What we do get is that the number 1001 is

le nombre figuré pentagonal en relation avec le mythique « nombre d’or » que l’on retrouve dans toute forme pentagonale et dans l’étoile à cinq branches.

[the pentagonal number in relation with the mythic “golden number” which one finds in all pentagonal forms and in the five-pointed star.]

It’s true: 1001 is a pentagonal number (so are 1, 2, 5, 7, 12, 15, 22, 26, 35, …). The sense of the argument appears to be, “1001 is a pentagonal number [true], and because pentagon therefore GOLDEN RATIO!” The golden ratio occurs in a regular pentagon, as the ratio of the diagonal length to the side length. That doesn’t make the free word association of “pentagon” and “mystical golden number” a valid argument.

But hey, when you feel the need for uninhibited babble slicked over with a superficial veneer of pseudoscholarship, there’s no better place to find it than an encyclopaedia article, right?

“Painters who definitely did make use of GR include Paul Serusier, Juan Gris, and Giro Severini, all in the early 19th century, and Salvador Dali in the 20th, but all four seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason. Also, the Cubists did organize an exhibition called “Section d’Or” in Paris in 1912, but the name was just that; none of the art shown involved the Golden Ratio.”
Keith Devlin

EDIT TO ADD (12 August 2014): I might as well include the proof that 1000.1001φ is 5. We figured out above that

$2 = \phi^{-2} + \phi^1.$

So, we square this:

$4 = (\phi^{-2} + \phi^1)^2.$

Using the binomial theorem,

$4 = \phi^{-4} + 2 \phi^{-1} + \phi^2.$

We want all the coefficients to be 0 or 1, so we split up the middle term:

$4 = \phi^{-4} + \phi^{-1} + \phi^{-1} + \phi^2.$

Next, we use the basic fact we know about the golden ratio.

$4 = \phi^{-4} + \phi^{-1} + \phi – 1 + \phi^2.$

To find an expression for the integer five, we add one to our expression for the integer four:

$5 = \phi^{-4} + \phi^{-1} + \phi + \phi^2.$

We can shorten this by the following move:

$5 = \phi^{-4} + \phi^{-1} + \phi^2(\phi^{-1} + 1).$

Again using our basic fact about the golden ratio, we recognize the expression in parentheses:

$5 = \phi^{-4} + \phi^{-1} + \phi^2 \cdot \phi.$

Therefore,

$5 = \phi^{-4} + \phi^{-1} + \phi^3.$

Q.E.D.

UPDATE (14 November 2015): The article was “fixed” in July, with the comment, “2 equals 10.01 in the golden ratio base, previous sentence made no sense” (true). Unfortunately, this just means that the article attributes to its source a statement which is not actually there. The claim is now,

It has been noted disquisitively[4] that the number 1001 of Duchamp’s entry at the 1912 Indépendants catalogue also happens to represent in the form 10.01 the integer 2 in the Golden ratio base, related to the golden section, something of much interest to the Duchamps and others of the Puteaux Group.

This looks more respectable: the mathematics is not complete bafflegab, and “10.01” is now related to the title of the piece. But the new and improved claim isn’t anywhere in reference [4]!

It’s not quite classic citogenesis, but it’s similar.