Complex Prime Patterns

Have you ever caught yourself wondering what an old German banknote, Jerusalem's Dome of the Rock, and the Ultimate Fighting Championship have in common?

Of course you have. Silly question.

The purple chap with the mutton chops on the 10 Mark note is Carl Friedrich Gauss, an extraordinarily prolific mathematician of the early 19th Century. His inevitable Wikipedia page contains an enviable list of items named in his honour; including a bell-shaped curve (the bedrock of Statistics), an extinct Antarctic volcano, two laws of magnetism and the hilariously-named associated process, Degaussing.

Dog being thoroughly Degaussed

Not included in this Wikilist are Gaussian Integers or Gaussian Primes despite the former having its own Wikipage. Many people who study post-16 mathematics will have come across Gaussian Integers without actually referring to them by this name; they form an integral part (pun intended) of the field of complex numbers.

For those who are a bit rusty on this topic or who have never encountered it before, here's a quick and easy explanation. [For those who can already tell their real parts from their imaginary parts, skip forward to the picture of umbrellas.]

Here is the real number line:

The number zero is in the middle. All the positive numbers lie to the right of it. All the negative numbers lie to the left of it. It extends forever in both directions and contains no gaps. Every real number lies somewhere on this line.

Sometimes, however, an uncountably infinite set of numbers is simply not enough.

We can extend the idea of a number by adding a second number line, at right angles, like this:

The vertical line is called the imaginary number line. It is constructed using a unit called i. The imaginary line contains precisely the same quantity of numbers as the real number line. This new number line also gives a home to all the square roots of negative numbers: There is no real number that we can multiply by itself to give -1 but if we let i be a number with this property, then we can infer that we get -4 when we square 2i and -9 when we square 3i and so on.

The real strength in drawing two number lines in this way lies in the quadrants in between. We could give a label to every point on the page. For example, the number marked with a red cross below could be called 3+2i. Similarly, the blue cross represents the complex number 1-4i.

Complex-confident readers rejoin here.

Any number in the form of the sum of a whole real number and a whole imaginary number is known as a Gaussian Integer. The collection of all the Gaussian Integers form the vertices of a square lattice that covers the entire complex plane.

In my very first matheminutes post, I discussed the importance of a collection of real integers called the primes. These were the ones that had no interesting factors; they couldn't be divided into products of smaller numbers. These primes can generate all the integers greater than 1 when they are multiplied by each other.

It shouldn't really be a surprise that this idea might extend to Gaussian Integers. After all, if we multiply two together, we always get a third. Take the two numbers indicated by crosses above:

This means that the Gaussian Integer 14-5i can be split into two factors. It just so happens that neither of these two factors, 2+3i and 1-4i can be split any further. They are Gaussian Primes.

If the Gaussian Primes are anything like their one-dimensional counter-parts, then they won't have any sort of regular distribution in the complex plane. They don't disappoint:

In the above diagram, each cell represents a Gaussian Integer. They range from -25 to 25 (left to right) and -25i to 25i (bottom to top). The five black cells in the middle represent the numbers 0, 1, i, -1, and -i.

Counting to the right along the middle row, you can see that the positive real numbers which are Gaussian Primes are 3, 7, 11, 19, 23. This throws up the curious fact that some prime numbers are not primes in the complex sense. For example, 5 has no real integer factors but it is nevertheless the product of 1+2i and 1-2i.

You will undoubtedly have noticed that there is a certain symmetry to the above pattern. Adding a few neatly placed lines shows that it splits into eight identical (given suitable reflections) segments.

This might remind you of buildings such as Jerusalem's Dome of the Rock...

...or arenas such as the Ultimate Fighting Championship ring...

...or stock images such as these umbrellas again.

Incidentally, in researching the symmetry of the anything-goes-punch-up arena, I stumbled across an insightful article entitled, Why UFC Ring is Called "The Octagon"? which you might like to browse just in case some clarification is required.

We find similar symmetry when we investigate collections of Gaussian Integers which have the same number of Gaussian Prime factors. Below are maps of the numbers with (from left to right) two, three, and four complex factors.

I find these patterns slightly beguiling; their juxtaposition of randomness and symmetry is kaleidoscopic. If you're wondering how I produced them then I'm happy to reveal that I spent several hours typing 2601 numbers into a Microsoft Excel spreadsheet in order that I could use the conditional formatting option to pick out those cells I wanted.

Using this method, it's also straightforward to shade all the squares according to their primeness. In this diagram, primes are red and, of the other Gaussian Integers, the ones with fewest factors are darker.

Alternatively, we could of course make those with many factors the darkest:

So there you have it. The idea of a prime extends beyond that of the positive integers. Mathematics is full of structured sets of objects called rings. Those rings that contain prime elements, in the sense that we're familiar with, are called Unique Factorisation Domains. Often abbreviated to UFD, they lie just one letter away from the common abbreviation of Ultimate Fighting Championship.

Coincidence?

from matheminutes http://ift.tt/2krfkrC

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