## Archive for March, 2016

### The Imaginary Power Tower: Part II

This is a continuation of last week’s post: LINK

The complex power tower is defined by an infinite tower’ of exponents:

$\displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,.$

The sequence of successive approximations to this function is

$z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} \qquad \dots$

If the sequence ${\{z_n(z)\}}$ converges it is easy to solve numerically for a given ${z }$.

In Part I we described an attempt to fit a logarithmic spiral to the sequence ${\{z_n(i)\}}$. While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large ${n}$.

### The Imaginary Power Tower: Part I

The function defined by an infinite tower’ of exponents,

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

is called the Power Tower function. We consider the sequence of successive approximations to this function:

$\displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,.$

As ${n\rightarrow\infty}$, the sequence ${\{y_n\}}$ converges for ${e^{-e}. This result was first proved by Euler. For an earlier post on the power tower, click here.

### Computus: Dating the Resurrection

Whatever the weather, St Patrick’s Day occurs on the same date every year. In contrast, Easter springs back and forth in an apparently chaotic manner. The date on which the Resurrection is celebrated is determined by a complicated convolution of astronomy, mathematics and theology, an algorithm or recipe that fixes the date in accordance with the motions of the Sun and Moon [TM087, or search for “thatsmaths” at irishtimes.com].

Iona Abbey, the last Celtic monastery to hold out against  Easter reform [Image Wikimedia Commons]

### Peano Music

The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel’s incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the focus of Gödel’s Theorems.

The Peano Axioms in symbolic form.

### The Abel Prize – The Nobel Prize for Mathematics

There is no Nobel Prize for mathematics, but there is a close equivalent: The prestigious Abel Medal is awarded every year for outstanding work in mathematics [TM086, or search for “thatsmaths” at irishtimes.com]. This years winner, or winners, will be announced soon.

When Alfred Nobel’s will appeared, the absence of any provision for a prize in mathematics gave rise to rumours of discord between Nobel and Gösta Mittag-Leffler, the leading Swedish mathematician of the day. They are without foundation, the truth being that Nobel had little interest in the subject, and probably didn’t appreciate the practical benefits of advanced mathematics.