We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here. The Operation … Continue reading The Power Tower Fractal
Tag: Analysis
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: $latex \displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,. &fg=000000$ The sequence of successive approximations to this function is $latex z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} … Continue reading The Imaginary Power Tower: Part II
The Imaginary Power Tower: Part I
The function defined by an `infinite tower' of exponents, $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ is called the Power Tower function. We consider the sequence of successive approximations to this function: $latex \displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,. &fg=000000$ As $latex {n\rightarrow\infty}&fg=000000$, the sequence $latex {\{y_n\}}&fg=000000$ converges for … Continue reading The Imaginary Power Tower: Part I
Vanishing Zigzags of Unbounded Length
We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity. Hopping Animals Let us compare a sequence of frog hops, cricket hops and flea hops. We assume each hop is a semi-circle so that the length is easily calculated. If the … Continue reading Vanishing Zigzags of Unbounded Length
Twin Peaks Entropy
Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks. Classical Entropy Entropy was introduced in classical thermodynamics about 150 years ago and, somewhat … Continue reading Twin Peaks Entropy
Squaring the Circular Functions
The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if $latex \displaystyle x = a\cos t \qquad\mbox{and}\qquad y = a\sin t … Continue reading Squaring the Circular Functions
A Few Wild Functions
Sine Function: $latex {\mathbf{y=\sin x}}&fg=000000$ The function $latex {y=\sin x}&fg=000000$ is beautifully behaved, oscillating regularly along the entire real line $latex {\mathbb{R}}&fg=000000$ (it is also well-behaved for complex $latex {x}&fg=000000$ but we won't consider that here). Chirp Function: $latex {\mathbf{y=\sin x^2}}&fg=000000$ Now $latex {y=\sin x^2}&fg=000000$ is also well-behaved: its oscillations become more rapid as $latex … Continue reading A Few Wild Functions
Which Way did the Bicycle Go?
``A bicycle, certainly, but not the bicycle," said Holmes. In Conan-Doyle's short story The Adventure of the Priory School Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal … Continue reading Which Way did the Bicycle Go?
Tap-tap-tap the Cosine Button
Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not … Continue reading Tap-tap-tap the Cosine Button
Café Mathematics in Lvov
For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at irishtimes.com). Zygmunt Janeszewski, who … Continue reading Café Mathematics in Lvov
The Birth of Functional Analysis
Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. … Continue reading The Birth of Functional Analysis
New Curves for Old: Inversion
Special Curves A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space $latex {\mathbb{R}^2}&fg=000000$ so that it generates a plane curve as it moves. … Continue reading New Curves for Old: Inversion
Curves with Singularities
Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it … Continue reading Curves with Singularities
Invention or Discovery?
Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture: Conjecture: Definitions are invented. Theorems are discovered. The goal is to prove … Continue reading Invention or Discovery?
Predator-Prey Models
Next week's post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here. The Lotka-Volterra Model Many ecological process can be modelled by simple systems of equations. An early example of this is the predator-prey model, developed independently by American mathematician Alfred Lotka … Continue reading Predator-Prey Models
The Unity of Mathematics
This week, That’s Maths in The Irish Times ( TM041 ) is about an ambitious program to unify mathematics. Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have … Continue reading The Unity of Mathematics
The Langlands Program
An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects. Substantial strides have been made in the … Continue reading The Langlands Program
The Prime Number Theorem
God may not play dice with the Universe, but something strange is going on with the prime numbers [Paul Erdös, paraphrasing Albert Einstein] The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural … Continue reading The Prime Number Theorem
A Mathematical Dynasty
The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers. In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More … Continue reading A Mathematical Dynasty
Sonya Kovalevskaya
A brilliant Russian mathematician, Sonya Kovalevskaya, is the topic of the That’s Maths column this week (click Irish Times: TM029 and search for "thatsmaths"). In the nineteenth century it was extremely difficult for a woman to achieve distinction in the academic sphere, and virtually impossible in the field of mathematics. But a few brilliant women managed … Continue reading Sonya Kovalevskaya
Ternary Variations
Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor's Ternary Set. In fact, the ternary set had been studied some ten years earlier … Continue reading Ternary Variations
The Lambert W-Function
Follow on twitter: @thatsmaths In a recent post ( The Power Tower ) we described a function defined by iterated exponentiation: $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ It would seem that when $latex {x>1}&fg=000000$ this must blow up. Surprisingly, it has finite values for a range of x>1. Below, we show that the power tower … Continue reading The Lambert W-Function
The Power Tower
Look at the function defined by an `infinite tower' of exponents: $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ It would seem that for x>1 this must blow up. But, amazingly, this is not so. In fact, the function has finite values for positive x up to $latex {x=\exp(1/e)\approx 1.445}&fg=000000$. We call this function the power tower … Continue reading The Power Tower
Sharing a Pint
Four friends, exhausted after a long hike, stagger into a pub to slake their thirst. But, pooling their funds, they have enough money for only one pint. Annie drinks first, until the surface of the beer is half way down the side (Fig. 1(A)). Then Barry drinks until the surface touches the bottom corner (Fig. … Continue reading Sharing a Pint
The Root of Infinity: It’s Surreal!
Can we make any sense of quantities like ``the square root of infinity"? Using the framework of surreal numbers, we can. In Part 1, we develop the background for constructing the surreals. In Part 2, the surreals are assembled and their amazing properties described. Part 1: Brunswick Schnitzel The number system has been built up … Continue reading The Root of Infinity: It’s Surreal!
The Popcorn Function
Continuity is not what it seems. In 1875, the German mathematician Carl Johannes Thomae defined a function P(x) with the following extraordinary property: P(x) is discontinuous if x is rational P(x) is continuous if x is irrational. A graph of this function on the interval (0,1) is shown below. The function has many names. We … Continue reading The Popcorn Function
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