the father of all beamer talks

By lieven

continued fractions

the father of all beamer talks
the Manin-Marcolli cave
devilish symmetries

Who was the first mathematician to give a slide show talk? I don’t have the definite answer to this question, but would like to offer a strong candidate : Hermann Minkowski gave the talk “Zur Geometrie der Zahlen” (On the geometry of numbers) before the third ICM in 1904 in Heidelberg and even the title page of his paper in the proceedings indicates that he did present his talk using slides (Mit Projektionsbildern auf einer Doppeltafel)

Seven of these eight slides would be hard to improve using LaTeX

What concerns us today is the worst of all slides, the seventh, where Minkowski tries to depict his famous questionmark function ?(x) , sometimes also called the devil’s staircase

The devil’s staircase is a fractal curve and can be viewed as a mirror (taking a point on the horizontal axis to the point on the vertical axis through the function value) having magical simplifying properties : - it takes rational numbers to dyadic numbers, that is those of the form $n.2^{-m}$ with $n,m \in \mathbb{Z}$ . - it takes quadratic irrational numbers to rational numbers. So, iterating this mirror-procedure, the devil’s staircase is a device solving the main problem of Greek Mathematics : which lengths can be constructed using ruler and compass? These constructible numbers are precisely those real numbers which become after a finite number of devil-mirrors a dyadic number. The proofs of these facts are not very difficult but they involve a piece of long-forgotten mathematical technology : continued fractions. By repeted approximations using the floor-function (the largest natural number less than or equal to the real number), every positive real number can be written as

$a = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \frac{1}{\dots}}}}$

with all a_i natural numbers. So, let us just denote from now on this continued fraction of a by the expression

$a = \langle a_0;a_1,a_2,a_3,\dots \rangle$

Clearly, a is a rational number if (and also if but this requires a small argument using the Euclidian algorithm) the above description has a tail of zeroes at the end and (slightly more difficult) $a$ is a real quadratic irrational number (that is, an element of a quadratic extension field $\mathbb{Q}\sqrt{n}$ ) if and only if the continued fraction-expression has a periodic tail. There is a lot more to say about continued-fraction expressions and I’ll do that in another ‘virtual-course-post’ (those prepended with a (c): sign). For the impatient let me just say that two real numbers will lie in the same $GL_2(\mathbb{Z})$ -orbit (under the action via Moebius-transformations) if and only if their continued fraction expressions have the same tails eventually (which has applications in noncommutative geometry as in the work of Manin and Marcolli but maybe I’ll come to this in the (c): posts).

Right, now we can define the mysterious devil-stair function ?(x) . If a is in the real interval [0,1] and if $a \in \mathbb{Q}$ then $a = \langle 0;a_1,a_2,\dots,a_n,0,0,\dots \rangle$ and we define $?(a) = 2 \sum_{k=1}^{n} (-1)^k 2^{-(a_1+a_2+\dots+a_k)}$ and if a is irrational with continued fraction expression $a = \langle 0;a_1,a_2,a_3,\dots \rangle$ , then

$?(a) = 2 \sum_{k=1}^{\infty} (-1)^{k+1} 2^{-(a_1+a_2+\dots+a_k)}$

A perhaps easier description is that with the above continued-fraction expression, the binary expansion of ?(a) has the following form

$?(a) = 0,0 \dots 01 \dots 1 0 \dots 0 1 \dots 1 0 \dots 0 1 \dots 1 0 \dots$

where the first batch of zeroes after the comma has length a_1-1 , the first batch of ones has length a_2 the next batch of zeroes length a_3 and so on.

It is a pleasant exercise to verify that this function does indeed have the properties we claimed before. A recent incarnation of the question mark function is in Conway’s game of contorted fractions. A typical position consists of a finite number of boxed real numbers, for example the position might be

$\boxed{\pi} + \boxed{\sqrt{2}} + \boxed{1728} + \boxed{-\frac{1}{3}}$

The Rules of the game are : (1) Both players L and R take turns modifying just one of the numbers such that the denominator becomes strictly smaller (irrational numbers are supposed to have $\infty$ as their ‘denominator’). And if the boxed number is already an integer, then its absolute value must decrease.
(2) Left must always decrease the value of the boxed number, Right must always increase it. (3) The first player unable to move looses the game. To decide who wins a particular game, one needs to compute the value of a position $\boxed{x}$ according to the rules of combinatorial game theory (see for example the marvelous series of four books Winning Ways for your Mathematical Plays. It turns out that this CG-value is no other than $?(x)$ … And, Conway has a much improved depiction of the devil-staircase in his book On Numbers And Games

Next in series

Conway, games, geometry, latex, Manin, Marcolli, noncommutative

1 comment
Posted in modular
Written on Fri, 02 March 2007 at 10:12 am
Tags: Conway, games, geometry, latex, Manin, Marcolli, noncommutative
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One Response to “the father of all beamer talks”

devilish symmetries | neverendingbooks Says:
January 3rd, 2008 at 10:32 am
[...] another post we introduced Minkowski’s question-mark function, aka the devil’s straircase and [...]