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The mathematics of Black Panther

In 2018, I watched the excellent Marvel film Black Panther, which has taken over a billion dollars at the box office worldwide! The film had a number of themes, including the question ‘what if an African country, named Wakanda, lead the world in technology?’  The film offered a cinematic picture of this, with an important emphasis on STEM subjects.

Now, one of the most interesting characters in the film is Shuri. Wikipedia describes her character like this:

Shuri, the princess of Wakanda, designs new technology for the country. She is ‘an innovative spirit with an innovative mind’ who ‘wants to take Wakanda to a new place’. Shuri is a good role model for young black girls as well as being one of the smartest persons in the world.

An illustration of Shuri, from the cover of a Black Panther comic. Image: Stanley Lau, fair use.

One of the technologies that Shuri designed was Black Panther’s suit. The suit is special because it can distribute the kinetic energy from an impact. The idea is that the kinetic energy will not be focused on one area, but move to another part of the suit where it can be absorbed. Okay, nice Hollywood science fiction stuff… or is it? Watching this scene took me back to my postgraduate days, when I was doing an MSc in Industrial Mathematical Modelling at Loughborough University. Here, I did a dissertation titled ‘Impact on an adhesive joint’. Continue reading

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David Blackwell and me

When I first came across the great Black mathematician and statistician, David Blackwell (1919-2010), circa 1975, I was actually informed that he was white. He was also then Irish. Or, so I was told by a triumphal fellow MSc economics and econometrics student at Southampton University, himself Irish, and now also a professor of economics.

The occasion of this initial meeting with Blackwell was our econometrics class’s introduction to the eponymous Rao-Blackwell theorem—a fundamental result in the theory of optimal statistical estimators. In simple terms, this theorem shows how to improve upon a rudimentary unbiased estimator of a statistical parameter, and indeed, get the best unbiased estimator of that parameter, when certain technical conditions are satisfied. I remember being struck by the beauty of this result. Perhaps it was my excitement about it that led my Irish colleague to try to deflate me by claiming his own racial and national part-ownership for the theorem by telling me that Blackwell was a white Irishman—Rao’s Indian extraction being self-evident. Maybe, more charitably, he was just engaging in supposedly characteristic Irish blarney, without malice. Regardless, I never bothered to check his claim—and, why should I have doubted a fellow student’s word about something as inconsequential as someone’s nationality, as I thought then?

So, for almost a decade afterwards, I happily persisted in the belief that Blackwell was indeed Irish and blithely assured others of this. I must have given much wry amusement to those who knew otherwise. It was not until the academic year 1984-85, which I spent as a joint fellow at CORE (Centre for Operations Research and Econometrics) and IRES (Institut de Recherches Économiques et Sociales) at Université Catholique de Louvain-la-Neuve, that I was finally disabused of my misinformation by another researcher. Continue reading

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Black Mathematician Month 2018

Happy October everyone and welcome to the second instalment of Black Mathematician Month! As you may have noticed, diversity in higher education has been in the news a lot recently and this is not without reason. We strongly believe that mathematics should be open to everyone, regardless of education, age or background. There has been very little progress in improving diversity in our field, and this is why we, as mathematicians and science communicators, think it is important to continue the discussion of the effects, as well as, what needs to be done for things to change. As a result we are working to actively promote the work of black mathematicians throughout the entirety of October alongside Black History Month in what we are calling Black Mathematician Month.
Continue reading

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Eight things you didn’t notice in Issue 07

On 19 October, Chalkdust issue 08 will be released (don’t forget to book your free ticket to our launch event). To help you to get as excited about the launch as we are, here are some of the things we hid in issue 07 that you may or may not have noticed.

1. No more editorial

Inspired by Katie Steckles’s article about the game No more women, we made the moves “no more vowels” and “no more consonants” in the editorial. With all the letters included, the editorial would’ve read:

I (no more vowels) expect this text is very hard to read, but maybe some of you will enjoy deciphering it. As Katie says in her article, we would classify banning vowels as a particularly mean move. But our next one is probably meaner…

We’re (no more consonants) really not expecting anyone to work out what this bit says, but good luck if you’re trying! Enjoy the magazine!

2. Scorpions

As usual, the scorpions that escaped from issue 03’s horoscope are still running around the magazine.


3. More scorpions

In issue 07, one scorpion did a particularly good job of hiding.


4. Crossnumber header

Crossnumber fans may have noticed that the square pattern in the header of the issue 07 crossnumber looked familiar, as the pattern was taken from the top right-hand corner of the issue 06 crossnumber’s grid.

5. Crossnumber grid

The black squares in the grid of the issue 07 crossnumber were arranged in a spiral. If you start from the outside and count inwards, the white squares in the spiral make a sequence you will be familiar with.

6. Captain Scarlet

One of the letters sent to Dirichlet was written by Paul Metcalfe from Winchester. This is the real name of Captain Scarlet.


7. Galois Knot Theory

The reviews page features a brief review of Galois Knot Theory, a book that was randomly generated by mathgen. Fittingly, the review was randomly generated using a Markov chain and the other reviews on the page.


8. Fields love

No, Fields medals are not actually heart shaped.


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Can you make something out of nothing?

The equation:

$$\emptyset + \{0\} = 1$$

looks a little bit like:

$$0 + 0 = 1.$$

According to Aristotle, nature could not create emptiness at all. Public Domain

Doesn’t it? The main point is: How to deal with zero. You can start with nothing and say: something is the opposite of zero.  On the other hand, one can say: zero is already something, I can count it: one element. But first things first. I would like to start with an old fear of people: the fear of emptiness. The Romans had no zero. For centuries physicists were afraid of the so-called horror vacui based on the idea of Aristotle that nature could not create any emptiness at all (Aristotle).  Even in art, there is the horror vacui as criticism of the art of the Victorian era (Mario Praz, 6/09/1986-23/03/1982).

Let us take one step back and start with sets. Why sets? Before we can handle this let us ask something else: How can a science be justified?  Religions work with dogmas. You need one example? Here it is: God is good and almighty. There is no further proof of this, for it has been established as a statement. Mathematics does not work with dogmas. You can start with something else: axioms.

What is an axiom? The word axiom comes from the Greek and means “perceived principle” and is in a theory, a science or an axiomatic system something that is not founded or deductively derived within this system. One amazing hint: Insufficiently strong systems, such as arithmetic, there must be statements that can neither be formally proven nor refuted (this is called the G\”odel incompleteness theorem).

Georg Cantor, a German mathematician that founded set theory. Public Domain

For a long time, mathematicians tried to design a clean axiomatic system. After many trials and errors, they decided to try it with quantities. So-called sets and finally we are in set theory. Set theory is a fundamental branch of mathematics that deals with the investigation of sets, i.e. summaries of objects. Set theory was founded by Georg Cantor and deals with representing (mathematical) objects as sets. By a set, we mean every summary M of certain well-different objects in our intuition or thinking (which are called the elements of M) into a whole. You need some examples?

$$ \mbox{Patrick Set} = \{A,B,C\}$$

This is the set of the letters A, B and C. The set is named after me. You can also write:

$$\mbox{Super Animal Set} = \{Dog,Cat,Duck\}$$

Another set with animals in it. And now an exciting question: What is an empty set?

The first symbol for an empty set comes from Andre Weil and is the letter of the Danish or Norwegian alphabet:

$$\emptyset.$$

Weil, a scientist from Strasbourg and one of the most important members of the Bourbaki group, a union of like-minded French mathematicians who undertook the task of formulating all mathematics in a new and relentlessly strict manner.

When dealing with sets of objects, there are two basic ways to make them. There is the predicative way, where we simply lay down verbally which elements contain a set or the constructive way in which new sets are made from given sets. Historically, the predicative path was first taken and brought to maturity in the nineteenth century but then something happened: the Russell paradox entered the mathematical stage.

The Russell paradox was discovered by Bertrand Russell in 1901. Public Domain

According to naive set theory, any definable collection is a set. Let A be the set of all sets that are not members of themselves. If A is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. It is like a barber who is allowed to cut all the hair of those who do not cut their own hair. Can the barber now cut his own hair or not?

So although mathematicians have operated implicitly with sets of objects for centuries, there was no real theoretical formalization until the nineteenth century. Frege (08/11/1848-26/07/1925) himself created the so-called predicate logic. So let us ask again: What is nothing in the set theory?

The return of zero

If you create a set of all even numbers you have the first set. All odd numbers are the second set but the cut of both sets is empty. George Boole said that the set is empty and wrote simply 0 (zero) for this. This is one possibility to create a nothing out of sets.

Ernst Zernelo. Public Domain

The Berliner mathematician Ernst Friedrich Ferdinand Zermelo (27/07/1871- 21/05/1953) studied mathematics, physics, and philosophy at the Universities of Berlin, Halle (Saale) and Freiburg. He tried to prove that sets have a so-called “Wohlordnung” (well-ordering).  A simple example of well-ordering is the normal arrangement of natural numbers. A totally ordered set is called well-ordered if every subset that is not empty has a smallest element. The set of integers with the usual order is not well ordered, because it does not have even the smallest element.

Why is this good for quantities? The so-called “Wohlordnungssatz” (the theorem of the well-ordering):
There is a well-ordering on every quantity. Apparently, every subset of a well-ordered set is well ordered. It is one idea to make sets like this beautiful and usable. It brings us closer to the constructive way.

With the help of Abraham Fraenkel (17/02/1891-15/10/1965), Zermelo published an axiomatic theory known as Zermelo-Fraenkel set theory. Zermelo (like Boole) used zero as the empty set sign, and with this convention, one can, for example, in Zermelo’s system construct a new set $\{0\}$. This symbol means an amount that contains only the empty set and it equals 1. In other words: $\{ \}$ means: let us count the number of elements in the set. A zero is one element. Therefore $\{0\} = 1$. It is another way of thinking about zero. Zero is something and at least one. It is not an emptiness at all. You can count it.

The idea behind sets is that you can create everything from your axiomatic system. Sets of sets or other theorems based on your created sets. You can then create a new set of two sets, such as $\{0, \{0\}\}$. Zermelo lends this trick to another mathematician named Dedekind.

Through such operations as inserting an element into a set, or through combinations of pairs, union, and cut of sets, as well as additional rules, one can always make new sets. Now we are here: that is the constructive approach to set theory.

The interesting fact is that we start from nothing the so-called “Urelement” (you can translate it with the primary element). You begin with a single object, with an empty set. In other variants of the set theory, an arbitrary set is taken and from this, the empty set and from this set everything will be created. New elements can then be identified with newly produced quantities. You start from any point and say what is true for “ n ” will be also true for “n + 1” and so on. In Zermelo’s logic, this is the axiom of infinity and you are able to create all natural numbers.

It looks strange: In the Zermelo-Fraenkel theory, there is no need to introduce individual elements. For example, one speaks not about the amount of all letters. Instead, one talks exclusively about quantities and quantities of quantities that are built up over the permitted operations. You start with the Urelement which is an empty set, so it contains no elements. If you count the number of elements in the set “a “or the empty element in a set you will get one (element). In the Zermelo world, we just need the empty set and axioms but we start from the pure empty. You can even think of the equation as:

$$\emptyset + \{0\} = 0 + a = 0 + 1 = 1.$$

Fantastic! You can write shortly:

$$\emptyset + \{0\} = 1.$$

In the set theory of Zermelo, you can find 10 axioms creating everything from empty or nothing. There is just one thing you have to decide. If you want to create all natural numbers you can include zero or not:

$$\mathbb{N} = {1,2,3,\dots}$$

or:

$$\mathbb{N} = {0,1,2,3,\dots}$$

Both conventions are used inconsistently. The older tradition does not count zero as a natural number. The zero became common in Europe only from the 13th century. Today one writes $$\mathbb{N}_{0}$$ for all natural numbers including zero and $$\mathbb{N}_{+}$$ for all natural numbers without zero (you may find $\mathbb{N}_{>0}$ or $\mathbb{N}^{+}$ as well).

John von Neumann (16/10/1831-12/02/1916) had the idea of a stepwise construction of the entire set universe with the help of ordinal numbers and the iteration of raizing a number to a given power. Written in empty sets the natural numbers look like this:

$$\mathbb{N} = \emptyset,\{ \emptyset \},\{ \emptyset,\{ \emptyset \} \}, \dots. $$

So Zermelo was not the only one with that idea. Once again: How to deal with zero? You can start with nothing and say: something is the opposite of zero. On the other hand, one can say: zero is already something, I can count it. It is not just about the different spelling: $\emptyset \, \mbox{or} \, \{0\}$.

The return to the void

Outer space has been seen for some as complete emptiness. Is it? Public Domain

Criticizing the financial world often means: how can you get something (for example, interest) from nothing? And nothing means securities that are based on other securities that are merely bets or only worthless loans. On the other hand, in philosophy, science, and mathematics we find a return to nothing or emptiness. Physicians found out that the vacuum is full of fluctuations from which even particles can arise. So particles from the nothing. In modern art, you can find a Renaissance of empty space. There is Yves Klein’s empty room of 1961 for example. There is a piece of music in which musicians think about the silence. The name is 4’33”, since it takes 4 minutes and 33 seconds in three musical movements by John Cage.

In philosophy, there is a long discussion about the nothing coming from pre-socratic ideas that something is the negation of nothing to modern thoughts like Bloch’s Not-Yet-Being (Noch-Nicht-Seienden).

So the nothing never really disappeared.

Or is it the return of empty and you can interpret this for our time? It might be the idea that something can come from nothing.  And finally, we can say (with a wink of the eye) that $0 + 0 = 1$ with a lot of curly braces and a slash going through of course.

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2018 Fields medal winners

Maths has always been about pushing the boundaries of knowledge, but the 21st century has also looked inward, forging a synthesis of previously disparate subjects. For example, Andrew Wiles’ proof of Fermat’s last Theorem was a triumph of algebraic geometry and number theory working side-by-side, and many ideas from string theory and quantum mechanics in physics have inspired new constructions in differential geometry. Nowhere was this philosophy more apparent than in the 2018 Fields Medal winners, several of which have drawn from many different fields in their most revolutionary works. In this article, we look at the maths that won the medals.

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Complex numbers and algebra

Imaginary and complex numbers. Perhaps they are incorrectly named, and Gauss even called for the ‘imaginary’ numbers to be renamed lateral numbers. However, most young mathematicians are not concerned with their naming but instead might be asking questions such as:

  • Do they even have a meaning?
  • Do they relate to any other area of mathematics?

These are justified questions. One might start with their discovery. Though complex numbers are commonly taught to students through considering the roots of the function $f(x)=x^2+1$, their origin is in fact from investigating cubic curves in the 16th century.

We start just before the 16th century. Cubic equations (equations of the form $ax^3+bx^2+cx+d=0$) had been reduced to simpler cubics in the form $x^3+px+q=0$. These cubics without an $x^2$ term were known as depressed cubics and this made solving cubic equations much easier.

Niccolò Fontana Tartaglia, the self-taught student that defeated Antonio Fiore. Public Domain

We now skip ahead to the 16th century. An Italian professor, Scipione del Ferro, at Bologna University had found a formula to solve the depressed cubics with $p$ positive and $q$ negative. On his death bed in 1526, he confided his proof and formula to his student Antonio Fiore. Fiore was at the university Bologna, where they regularly held mathematical competitions. Fiore now had the formula that mathematicians had been searching for and with the proof in hand, challenged a self-taught Niccolò Tartaglia. In an amazing feat, Tartaglia was able to derive the formula for the cubic before the competition and won against Fiore. The formula (without proof) was again passed on from a Tartaglia to Gerolamo Cardano. Cardano simply worked backwards and reconstructed the proof.

‘L’Algebra’, a central figure in the understanding of imaginary numbers. Public Domain

One problem was that part of the formula could have led to taking the root of a negative number and Cardano was the first to really consider it as a possibility. However, he never included this in any of his writing. On the other hand, Rafael Bombelli, in his writing ‘l’Algebra’ explicitly made note of this. It took time for complex numbers to be accepted but eventually, their usefulness outweighed the difficulty some mathematicians had in understanding them.

The cubic history of the complex numbers is fascinating. However, looking at quadratic functions might be easier. Let us reconsider the curve $f(x)=x^2+1$. According to an algebraic principle named the fundamental theorem of algebra, every polynomial with degree $n$ must have exactly $n$ roots. Therefore, since this curve has degree 2, it should have exactly two roots. An attempt to find the roots of $f(x)$ might look like this:

$$x^2+1=0 $$

$$x^2=-1 \, \, \, \therefore  \, \, \, \text{“No solutions”}$$

This doesn’t make sense though. The fundamental theorem of algebra states that there should be 2 roots, yet this answer suggests there are none. To correct the mistake, the phrase “No solutions” should read “No real solutions”. Taking the square root yields the result that $x=\pm \sqrt{-1}$. This produces exactly two roots for the function. $\sqrt{-1}$ was special and was given its own symbol, $\mathrm{i}$, and become the imaginary unit.

Creating beauty

Mathematicians’ belief in the existence of $\mathrm{i}$ as a plausible mathematical concept was vital for what is considered by some to be the most beautiful equations in all of mathematics. The first was discovered by Abraham De Moivre. We will slowly work towards it by first noticing that:

$$(\cos \theta+\mathrm{i}\sin \theta)^2 =\cos^2\theta – \sin^2\theta +(2\cos \theta \sin \theta)\mathrm{i}=\cos2\theta+\mathrm{i}\sin2\theta \, \, \text{(1)}.$$

Maybe we can start to see a pattern and we might even conjecture that:

$$(\cos\theta+\mathrm{i}\sin\theta)^n= \cos(n\theta)+\mathrm{i}\sin(n\theta).$$

Let $P(n)$ be the mathematical statement that equation (1) is true for all real values of $\theta$. We will prove this by induction on $n$ starting from the `base case’, $n=1$. We clearly have:

$$(\cos\theta+\mathrm{i}\sin\theta)^1=\cos1\theta+\mathrm{i}\sin1\theta$$

so $P(1)$ is true. Now, for the inductive step, we assume the `Inductive Hypothesis’ that $P(k)$ is true for some positive integer $k$. That means:

$$(\cos\theta+\mathrm{i}\sin\theta)^k=\cos k\theta+\mathrm{i}\sin k\theta.$$

We will use this to show that $P(k + 1)$ is correct.:

\begin{align*}
(\cos\theta+\mathrm{i}\sin\theta)^{k+1}&=(\cos\theta+\mathrm{i}\sin\theta)^k (\cos\theta+\mathrm{i}\sin\theta)\\
&=(\cos k\theta+\mathrm{i}\sin k\theta)(\cos\theta+\mathrm{i}\sin\theta)\\
&=\cos k\theta\cos\theta+\mathrm{i}\cos k\theta\sin\theta-\sin k\theta\sin\theta+\mathrm{i}\sin k\theta\cos\theta\\
&=\cos k\theta\cos\theta-\sin k\theta\sin\theta+\mathrm{i}(\sin k\theta\cos\theta+\cos k\theta\sin\theta)\\
&=\cos (k\theta+\theta)+\mathrm{i}\sin(k\theta+\theta)\\
&=\cos((k+1)\theta)+\mathrm{i}\sin((k+1)\theta)
\end{align*}

We conclude that, if $P(k)$ is true, then $P(k+1)$ is true. Therefore, since $P(1)$ is true $P(n)$ is true for all positive integers $n$ by the principle of mathematical induction.

The second formula I want to mention follows from the first and is perhaps even more aesthetically pleasing. Although discovered by Roger Coates, the equation is named Euler’s identity due to how much Euler was able to manipulate and further this identity. It is given as;

$$e^{i\theta} = \cos\theta+\mathrm{i}\sin\theta.$$

This proof we are going to give is one of three proofs that Euler gave to try and prove this beauty. To get to this mathematical gem, we start by considering the expansion of $e^x$:

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots.$$

Next, consider the case where the index is an imaginary number, say $\mathrm{i}x$:

\begin{align*}e^{\mathrm{i}x}&=1+\frac{\mathrm{i}x}{1!}+\frac{(\mathrm{i}x)^2}{2!}+\frac{(\mathrm{i}x)^3}{3!}+\frac{(\mathrm{i}x)^4}{4!}+\frac{(\mathrm{i}x)^5}{5!}+\frac{(\mathrm{i}x)^6}{6!}+\frac{(\mathrm{i}x)^7}{7!}+\frac{(\mathrm{i}x)^8}{8!}+\cdots \\
&=1+\mathrm{i}\frac{x}{1!}-\frac{x^2}{2!}-\mathrm{i}\frac{x^3}{3!}+\frac{x^4}{4!}+\mathrm{i}\frac{x^5}{5!}-\frac{x^6}{6!}-\mathrm{i}\frac{x^7}{7!}+\frac{x^8}{8!}+\cdots
\end{align*}

This can be separated into real parts and imaginary parts:

$$e^{\mathrm{i}x}=(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+ \cdots ) + \mathrm{i}(\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} +\cdots)$$

If you are familiar with the Maclaurin series expansion, you will quickly realise that the infinite sums in parenthesis are the expansions for $\cos x$ and $\sin x$ respectively. Ergo:

$$e^{\mathrm{i}x}=\cos x+\mathrm{i}\sin x$$

However, to show that $x$ here is an angle measured in radians, the formula is most commonly seen as:

$e^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$,

and therefore $e^{i\pi} = -1$. Wow! The elegance of Euler’s identity is that it yields more wonderful results. For example, we can continue exploring by taking $\theta$ to be the sum of two angles. Let these angles be A and B, where A and B are measured in radians. This produces the following identities which we used in the proof of De Moivre’s formula:

$$(\cos A+\mathrm{i}\sin A)(\cos B+\mathrm{i}\sin B) =e^{\mathrm{i}A} e^{\mathrm{i}B} = e^{\mathrm{i}(A+B)}=\cos (A+B)+\mathrm{i}\sin(A+B)$$

but also:

$$(\cos A+\mathrm{i}\sin A)(\cos B+\mathrm{i}\sin B) = \cos A\cos B-\sin A\sin B+\mathrm{i}(\cos A\sin B+\sin A\cos B).$$

Equating real and imaginary parts gives:

$$\cos A\cos B-\sin A\sin B= \cos(A+B)$$
$$\cos A\sin B+\sin A\cos B= \sin(A+B)$$

In the same way we could replace $\theta$ with $A-B$ to derive expressions for $\cos(A-B)$ and for $\sin(A-B)$. In addition, all the other trigonometric identities used in A-level mathematics can be derived just from these identities.

Many students might be aware of the geometric proof of these identities and might still be struggling to memorise these sometimes-confusing identities. However, I hope that after seeing this you might gain a deeper understanding of your learning, truly enjoying the hidden beauty of mathematics instead of rote learning mathematics. Perhaps the elegance of using Euler’s formula is that it highlights something quite interesting: there is a clear link between algebra and geometry, namely complex numbers. Furthermore, it provides evidence that mathematics is a collection of unified and collected ideas.

Why do they matter?

The discovery of complex numbers was extremely important in algebra. To fully appreciate this, we must consider our number system, something that many students are never explicitly taught. Initially, humans made use of natural numbers $\{1,2,3…\}$ to count naturally occurring objects- for example, 20 cows or 3 apples. Integers were then discovered to understand concepts such as debt. Integers extended the number of system and the natural numbers were a subset of the integers as shown by the image below. The next extension of the numbers system was the discovery of the rational numbers, which are defined as the ratio of two integers. The number system was yet again extended through the introduction of the real numbers, a set which contained the rational and irrational numbers. Mathematicians thought that the number system was complete, and that algebra was now understood. However, the discovery of the complex numbers exposed the incompleteness of the number system. In fact, the complex numbers were the final extension of the number system and are what is known as the algebraic closure of the number system. They completed traditional algebra.

More fun!

Let’s reconsider the function curve $f(x)=x^2+1$ we were working with at the beginning. When we attempted to find the roots of this function, we arrived at the equation $x^2=-1$. Now consider the case $z^n=\pm1$, where $n$ is a real number. The first equation, $z^n=1$, is a classic equation in complex analysis. Its solutions are named the roots of unity as they are the roots of one (unity). They are very interesting, but I was more intrigued with a variation of the roots of unity. Replacing 1 with $i$ to get:

$$z^n=\mathrm{i}.$$

I came across this idea whilst considering if I could take the square root of $\sqrt{-1}$. Since:

$$e^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta,$$

we could find a value of $\theta$ such that $\cos\theta=0$ and $\sin\theta=1$ to obtain an expression for $\mathrm{i}$. For example, $\theta=\pi/2$. Note: there are infinitely many values of $\theta$ that satisfy these requirements since $\sin\theta$ and $\cos\theta$ are periodic functions. However, working with values in the range $-\pi < \theta \leq \pi$ is easier. So we have $e^{\mathrm{i} \pi/2}=\cos \pi/2+\mathrm{i}\sin \pi/2$, or:

$$e^{\mathrm{i} \pi/2}=\mathrm{i}.$$

As a result, taking roots becomes simpler. For example:

$$\sqrt{\mathrm{i}}=e^{\mathrm{i}\pi/4}$$

$$e^{\mathrm{i} \pi/4}=\cos \frac{\pi}{4}+\mathrm{i}\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}+\mathrm{i}\frac{\sqrt{2}}{2}.$$

Think about the shape that all the roots of $\mathrm{i}$ would make on the complex plane. Try and plot these points on some graphing software such as Desmos. Could you explain why they form this shape by considering Euler’s identity and your knowledge of trigonometry.

We can continue exploring. What about taking the $\mathrm{i}$-th root of $\mathrm{i}$? Recall that $e^{\mathrm{i}\pi/2}=\mathrm{i}$, so maybe:

$$\sqrt[\mathrm{i}]{\mathrm{i}}= (e^{\mathrm{i} \pi/2})^{1/\mathrm{i}}$$

and

$$\sqrt[\mathrm{i}]{\mathrm{i}}=e^{\pi/2}.$$

Since $e$ and $\pi$ are real numbers, we can deduce that $\sqrt[\mathrm{i}]{\mathrm{i}}$ is a real number. This is a stunning result but there is a slight problem. In the same way that the principal value of $\sin^{-1}(1/2)$ is $\pi/6$ rads, $e^{\pi/2}$ can be considered to be the principal value of $\sqrt[\mathrm{i}]{\mathrm{i}}$. There are actually infinitely many values we could assign to the expression $\sqrt[i]{i}$.

In conclusion, complex numbers are something of beauty. They are just as real as real numbers, just as tangible and just as necessary. As I said before, complex numbers show that mathematics is not a collection of separated ideas but instead consists of linked and connected fields. In this case, we have seen the intrinsic link between complex analysis, geometry and algebra.

A challenge

But the fun doesn’t end there. As you might have seen, investigating complex numbers really is fascinating. Complex numbers can even be applied to situations that don’t seem to require them. We leave it is a challenge to the reader to use complex numbers to evaluate the following integral:

$$\int_{0}^{1}\frac{\sin(\log x)}{\log x}dx.$$

A solution will be uploaded to this blog in the near future.

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Folding nets into Johnson solids

Folding a truncated icosahedron.

Start with a collection of regular polygons — equilateral triangles, squares, pentagons, etc. Can this collection be glued together to make a solid? For instance, six squares can be fused to make a cube. Norman W. Johnson, a retired mathematician from Wheaton College (Massachusetts) who passed away in 2017, answered this question in the 1960’s: there are the five Platonic solids (eg cube, icosohedron), the thirteen Archimedean solids (eg the truncated icosahedron), the prisms and anti-prisms, and only 92 other solids that are now called the “Johnson solids”. (This fact was later proven by Victor Zalgaller.) All of these solids can be represented as nets, which are two-dimensional designs made up of the faces.

Truncated octahedron and its net.

Nets can be folded into three-dimensional solids called polyhedra. (See above for an example of a truncated octahedron.) Given a net of a polyhedron, Reuben Wattenhofer, an undergraduate student who worked with the author, recently created a method to automatically fold many nets into a three-dimensional polyhedron. In this article, we will explore these topics as well as a connection to Eugenia Cheng’s first book. Continue reading