**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

## One Comment