Yin into Yang; and Yang into Yin…

I never realized how closely related the two halves of the yin-yang symbol are:

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Installing the Windows 10 Preview in a Virtual Machine

Do you want to try out Windows 10 for yourself, but are worried that it may be interfere with your current setup? Have no fear because this video shows how you can easily install the full preview in a VirtualBox virtual machine. That way it is fully sandboxed and your computer is protected from potential bugs. What’s more, none of your current settings will be modified, and you can even install the preview on a Mac or a Linux PC, too.

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My new book series will be out in January

I have a new series of books, 20 Lessons to Successful Web Development, coming out in January. So far there are four books in the series, covering PHP, JavaScript, HTML & HTML5 and CSS & CSS3. Each book contains 20 lessons, which you should be able to complete in under an hour each, including the accompanying free online video tutorial that comes with each lesson. In total each book has over four hours of video and dozens of tested and working, easy-to use examples.

You can pre-order your copies at all good bookstores and online retailers. Here’s a link to the books on Amazon. The examples can all be downloaded for free at 20lessons.com.

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Store it as fat!


Science is funny at Beatrice the Biologist

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Thanks Scotland

Whichever way you voted I want to thank all the Scottish people for benefiting the union as a whole. The consequence of your referendum is that the English, Welsh and Northern Irish will now also get a much bigger say in how they are governed at a local level, as the UK reinvents itself into a modern federal democracy, similar to the USA.

Now we have to address the other big question that’s quickly coming our way – what to do about Europe…

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So, what’s it going to be, Scotland?

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The Future of the UK, Scotland, New Britain, and The European Union

No more Scotland in the UK?So, the latest poll shows that support for Scottish independence from the UK is rapidly growing, and has now passed the 50% required to win in the upcoming referendum – and increasing by about 5% a week. If this rate continues, an overwhelming majority of around 60% of Scots will vote for separation from the hundreds of years old union.

First of all, as a born and bred English southerner, I wish the Scottish people success and prosperity with whatever decision they take. In fact it must be very exciting to contemplate the possibility of such a massive change, and the reclaiming of so many powers from Westminster. Fail or prosper, the future success of the Scottish nation will rest on their own endeavours and their own choices, and that must be a very appealing and satisfying prospect.

I think if I were Scottish I would choose independence, for the very same reasons that most British voters recently voted for the UK Independence Party in the European elections and who, in next year’s general election, are likely to return this party into a position of holding the balance of power, forcing the Liberal Democratic party (currently in a ruling alliance with the Conservatives) into fourth place

If the UK didn’t have a first past the post system, UKIP might even win the general election outright, with over 50% of the vote. However, our system favours incumbents, who change boundaries to remain in power, and probably 65% of the vote would be necessary for a minority party to obtain a landslide. Still, not unachievable – but probably not this time around.

We are fed up with far away bureaucrats ruling our lives

What the British as a whole are saying they want is more local control over what affects our lives. The Scottish want London to keep its fingers out of Scottish business, while the British as a whole want Brussels to do the very same thing. It seems that while business people love the idea of ‘ever closer union’, because it helps them to become ever richer, the general populace don’t, because these unions results in more rules and fewer freedoms, and we have all simply had enough of petty bureaucracy costing us time and money. We don’t like being pushed around by people we see as having no business interfering in our lives.

I do wonder whether Scotland working together with an Independent UK, withdrawn from the EU would be stronger than the two separated, but I believe the Scots actually want to remain in the EU after independence, so that would not have been an option. However if, as appears likely, Scotland votes to leave the UK, then it means the pro-European Scots will not be able to vote in the 2017 European referendum for UK withdrawal from the EU, and that will give a huge boost to UKIP.

Speaking of which, when (or maybe it’s still if)  the English and Welsh see how the Scottish manage to gain their independence, they will want British independence from the EU more than ever. We will think that if Scotland can do it, so can the rest of the former UK. So, for me, the indications clearly are that the UK as we currently know it will be no more by next year, and by 2018, the remaining member countries of the UK will be independent themselves, from the rest of Europe.

After separation: The question of names

And then, what will be our respective names? Well, obviously Scotland will be Scotland, that’s an easy one. But the UK? It will no-longer be as united, so United Kingdom is surely now out of the question. How about Great Britain then? Well, it won’t be as great as it was with Scotland out of the union, so that won’t be fitting either.

Also I seriously doubt any contraction of Wales and England will work (Wengland, Walesland, Engales, etc), so I see no choice remaining, other than that the UK will become simply ‘Britain’. Many of us refer to ourselves a ‘Brits’ anyway, so I think we’ll easily get used to our new country name, which will be logical and fitting.

The future of Scotland and Britain (and the EU)

The UK, or Britain as it may be known, has a long series of past connections with former countries it once occupied. But its former colonies have all been given their independence and obtained self governance. And now there remains a very strong connection in the form of the Commonwealth. A voluntary organisation of 53 independent countries with a population of about 2.2 billion people (about a third of the world). In contrast to this the USA has about 315 million people, and the EU around 500 million.

The Commonwealth does many things. It just recently marked a very successful Commonwealth games (in Glasgow, Scotland), and the Charter of the Commonwealth is every bit as important as the US Constitution, or the EU’s Charter of Human Rights, but there is no desire for uniting the Commonwealth countries into any larger entity. The aim is simply to work together to benefit the lives of all Commonwealth members.

And that’s where I see the future of new Britain (hey, maybe New Britain could be our new name), to strengthen Commonwealth links, build up commercial and trading relationships, and prosper as the biggest partnership of its kind in the world.

The European Union doesn’t have such a relationship, but the European countries do have each other. Great, so let them unite. Europe will continue absorbing satellite countries until it fully unites politically into one large country, like the USA and its 50 states. Interestingly, Scotland will very likely end up a part of the expanded EU, handing its hard won powers back, but this time to Brussels.

Britain, however, doesn’t need Europe when we have the Commonwealth. Especially when there is no requirement to unite or follow orders issued by a central bureaucracy, just the commitment to work closely on world issues for the benefit of all. We will all retain full local control over our own laws and the vastly different ways of life of our different peoples, and still enjoy the benefits of cross-border collaboration and trade. The best of both worlds.

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Has the popularity of PHP (the most-used web server language) now peaked?

PHP and JavaScriptAfter a number of years of consolidation, in which users flocked from Perl to PHP because it was so much faster, easier to use, and could handle more connections at a time (although Perl nowadays is extremely fast and sleek, but I think it’s had its day), there’s now another similar change looming on the horizon.

This time the instigator is the growth of HTML5, in which you need to master JavaScript in order to achieve the more fun and useful features, such as geolocation and accessing the canvas. This has caused an incredible (and mostly unforeseen) rise in the popularity of JavaScript, which over the last five years has shot far ahead of all other web development languages, yes beating even Java and Ruby in popularity.

And while PHP has shown slow growth over the same period (presumably at the expense of Ruby and Python, which are both declining in popularity), I think it has probably peaked and is even starting to decline. Yet, just a few years ago most people would have thought PHP was going to dominate the web, and never would have thought that a language primarily developed to run in a client environment could become a challenger to PHP.

But that’s what seems to be happening with the likes of Node.js, which uses the Google Chrome JavaScript run-time to run on a web server (in place of a language such as PHP), providing highly scalable and ultra fast applications. Even if you haven’t used Node.js but think about the concept for a minute, you’ll begin to understand the product’s success. You see, why learn two languages for server/client communications, when you need to learn only one – namely JavaScript?

So where do I see PHP in five years time? Well, it will still be an important technology, simply because it has the momentum of being used by something like 60% of all websites. But I believe that JavaScript is going to migrate fairly and squarely to the server environment and, whether or not Node.js remains at the forefront, I believe JavaScript will become so widely used that it will continue to increase its already impressive lead in popularity over the other competing languages, for a considerable number of years to come.

So what should you do as a developer (or would-be developer)? Well, right now you absolutely need to know one or more of the main server languages, and for me the best choice would be PHP. But you should also become expert in using JavaScript too. Not only will learning it pay dividends in the browser, in the future you’ll be able to draw on the same skill set for your server programming too. Fortunately both these languages (and other technologies you should know) are covered in depth in my book Learning PHP, MySQL, JavaScript, CSS & HTML5.

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What if… well, you decide…

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Primes, Pseudo Primes, And Yet More Primes!

Prime numbersIt’s possible to tell whether or not a number is prime in a number of ways. You can crunch through all possible factors of a number until you have a definitive result (which can take an exceedingly long time for very large numbers), or you can use a formula to return a very high level of probability about whether a number is prime. Recently I’ve been looking into both these methods.

In this post:

  • How to create huge arrays of primes quickly using JavaScript
  • Graphically displaying and animating prime numbers
  • How to test for the probability of primality (Fermat’s Little Theorem)
  • Numbers that resist these tests (Pseudo Primes and Carmichael Numbers)
  • How to overcome this resistance

Finding Primes With JavaScript

At its simplest, to find all prime numbers between 1 and n in a language such as JavaScript, you can use code like this:

function getprimes(n)
{
  var p = []
  for (var j = 1 ; j < n ; ++j)
    if (isprime(j)) p.push(j)
  return p
}

function isprime(n)
{
  for (var j = 2 ; j <= Math.sqrt(n) ; ++j)
    if (n % j == 0) return false
  return true
}

The first function builds an array of prime numbers, which it returns in the array p[] by calling the second function to test all values of n for primality, up to the square root of n. The square root of n is chosen because all factors come in pairs and we only need to check the lowest of each pair. So (for example) the number 64 has the pair of primes 8 and 8, so you can clearly see that, therefore, the square root of a number is the largest that the smaller of a pair of primes can be.

It is possible to improve on code such as this because we know that all composite (non-prime) numbers can be created with a pair of prime factors. Therefore the previous code can be updated to only test for prime factors, like this (with the modifications shown in blue):

function getprimes(n)
{
  var p = []
  for (var j = 1 ; j < n ; ++j)
    if (isprime(p, j)) p.push(j)
  return p
}

function isprime(p, n)
{
  for (var j = 1 ; p[j] <= Math.sqrt(n) ; ++j)
    if (n % p[j] == 0) return false
  return true
}

The second function is also given access to the array p[] by adding it as an additional argument, so that it can look up previously determined prime numbers for use in its primality tests.

Now if, for example, the number being tested is 36, instead of checking possible factors 2, 3, 4, 5, and 6 (the square root of 36), only the values, 2,3, and 5 will be tested since they are the only primes numbers equal to or less than 6. This has saved two tests (4 and 6). And the bigger the number, the more tests you save until you can ignore about 5 in every 6 or so.

Indeed, you can also improve the main loop by ignoring all even numbers plus those that are divisible by 5, like this (with updates in blue):

function getprimes(n)
{
  var p = [1, 2]
  for (var j = 3 ; j < n ; j += 2)
    if (j == 5 || j % 5 != 0 && isprime(p, j)) p.push(j)
  return p
}

function isprime(p, n)
  {
    for (var j = 1 ; p[j] <= Math.sqrt(n) ; ++j)
      if (n % p[j] == 0) return false
    return true
  }
}

In this version of the function, the p[] array is now pre-populated with the values 1 and 2 since they work slightly differently to the others, so they are hand-inserted. Then the loop starts at the number 3, with the (other unusual) number 5 accepted, but not any other multiples of 5. And now the code is fairly efficient, except that the square root function is not optimized for this particular use, and so a faster version of the inner function would be to remove the expression p[j] <= Math.sqrt(n), replacing it with p[j] * p[j] <= n.

You can now call this function with the following command (for example) to return the prime numbers up to 100, which will then be displayed in the current browser document:

document.write(getprimes(100))

The result of this command will be as follows, and I have tested it on an i7 PC with values of up to 10 million (which are calculated in about a second), and 100 million (which takes around 9 seconds):

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The PrimeDancer Web App

My new web app, PrimeDancer.com uses the preceding code to generate the primes it displays and animates. It’s a vastly improved and speeded up version of my earlier prime spiral animator, which now uses simple sliders to adjust the speed, direction, and zoom. It also lets you click in the animation to detect all the numbers in a sequence of a spiral arm, or use the sliders on the left to find them. When running the program looks like this, and I think you’ll find it rather fun to play with:

Prime Dancer

I find it fascinating (and quite relaxing) to watch all the possible patterns resulting from the distances between potential plot points on the spiral being slightly changed with each frame. What happens is that, for example, all the even numbers often reveal themselves as spiral gaps, but then so do multiples of three, four, five, and all the way up, as and when they are lined up above above each other in each turn of the spiral.

What’s more you can see multiple gaps at a time causing ‘interference’ patterns, and a tremendous amount of complexity reveals itself. One of my first theories was that where a spiral arms was particularly flush with pixels, it might be possible to follow that arm out like a rich vein of coal. Indeed when I did this it was the case, but only for a while. The further out you go the more the ratio of primes to composites returns back to the norm. Still, as you’ll see, there are other ways of finding seams of primes.

So this app is probably now more of a toy than a maths experiment that will yield new information, but I think it’s unique, and primes have never been displayed in quite such an interesting and dynamic manner before.

Continuing The Search For Prime Numbers

But why all this tinkering in the first place, you might ask? Well, things like Internet security rest on the employment of incredibly large prime numbers, so any new discoveries in this area could lead either to new, more powerful encryption systems, the cracking of current techniques, or even both. So it’s a fun (and potentially profitable) area to study.

The trouble is that (fast as it is), this type of code will never be good enough to identify enormous prime numbers quickly enough for improving (or hindering) ecommerce and private communications (even when written in compiled C or hand-coded assembly language).

A Better Way to Test For Primality

Thankfully, though, there’s a better way (in fact there are several of them), the first of which was invented (or perhaps discovered) all the way back in the 1600s by the mathematician Fermat (yes, he of the famous Last Theorem). What is less known is that Fermat also came up with what is called his Little Theorem, in which the primality of a number can be predicted with a very high rate of probability by selecting what is termed a coprime number, and then  raising this coprime to the power of the number to be tested. The coprime number can be any integer less than the one being tested, and need not be prime.

So, for example, to test the number 7, you could choose the coprime base value of 4 and raise it to the power of 7 (which is 4 × 4 × 4 × 4 × 4 × 4 × 4). If you can then subtract the coprime from this result such that the final result is exactly divisible by the number being tested, then the number could well be prime. It looks like this: (47 – 4) / 7. Or like this in JavaScript:

if ((Math.pow(4, 7) - 4) % 7 == 0)
  document.write('Possibly prime')

In this code, the % instruction divides by 7 and returns only the remainder (modulus) of the operation. So if the result is zero there was no remainder and so the the test is passed. If, however, there is a remainder then the number is definitely composite.

The Problem With Pseudo Primes

Interestingly, when applying Fermat’s little theorem, a few very strange numbers crop up which look very much like prime numbers to it, but they aren’t – and so they are given the title pseudo primes.

Some examples of Fermat’s pseudo primes include the numbers 341, 561, 645, 1105, 1387, and 1629. Actually there’s an infinite number of them, but you can see that they are fairly uncommon.

However, by changing the coprime used in the theorem to a different base number, and re-running the test, you can almost always weed out these pseudo primes to narrow down your certainty.

The Resistant Carmichael Numbers

But you can only narrow down to an extent, because a subset of these pseudo primes (known as the Carmichael numbers after the mathematician who discovered the first one), resist the Fermat Little Theorem test in every base you choose:

In number theory, a Carmichael number is a composite number
n which satisfies the modular arithmetic congruence relation:
b n ≡ b (mod n) for all integers 1 < b < nWikipedia

This set of numbers includes 561, 1105, and 1721, and has an even rarer occurrence (only about 1 in 2.5 million). Nevertheless, these numbers are also infinite and are a considerable nuisance to cryptographers, who have now developed some very advanced mathematics to deal with this issue.

Before moving on to looking more deeply at these pseudo primes, though, let’s take a moment to write a program to simply map all prime and composite numbers onto a spiral, in a simpler version of my PrimeDancer App, like this:

<canvas id='canvas' width='630' height='630'></canvas>
<script>
  var E = document.getElementById('canvas'),
      C = E.getContext('2d'),
      P = getprimes(1024),
      j, x, y

  for (j = 1 ; j < 1024 ; ++j)
  {
    x = 315 + Math.sin(j) * j * .3
    y = 315 + Math.cos(j) * j * .3

    if (isprime(P, j)) C.fillStyle = '#00f'
    else               C.fillStyle = '#faa'

    C.fillRect(x, y, 2, 2)
  }
</script>

This code uses a 630 × 630 pixel HTML5 canvas, onto which the first 1024 numbers are plotted by simply calculating the sine and cosine of of each number to rotate it a distance according to its size. Each consecutive number is then plotted slightly further out from the center of the canvas in either a salmony color for composites, or in blue for prime numbers.

This code obviously includes some HTML and the getprimes() function from earlier, but these are not shown. The result of running this program, which you can test for yourself here, is shown below:

Number Spiral

The Chinese Hypothesis

Now let’s have some fun and slightly modify this code to enforce a variation of Fermat’s Little Theorem (known as the Chinese Hypotheses) on all these plot points, like this (with the changes highlighted in blue):

x = 315 + Math.sin((Math.pow(2, j) % j) * j * .3
y = 315 + Math.cos((Math.pow(2, j) % j) * j * .3

This time when you run the program the various plot point rotations are first vastly increased in size, and then restrained back again, causing spokes of point clustering around the circle. The result looks like this, which (for want of another word) I’ll call a Chinese Spiral:

Chinese Spiral

Holy cow Batman. See how those primes just lined right up for us? And, interestingly, how many of the composites tried to also get in on the act along a number of other arms too (and many didn’t, which is also interesting). What happened is that the primes got in an orderly line because the expression used returned a modulus of 2 for every one.

Unfortunately, though, the Chinese Hypothesis also suffers from the pseudo prime problem. Let me illustrate this by running all the numbers from 1 to 1024 through the Chinese Hypothesis

for (j = 0 ; j < 1024 ; ++j)
  if (Math.pow(2, j) % j == 2)
    if (!isprime(P, j))
      document.write(j + ' is pseudo prime<br>')

When you run this code the result is as follows:

341 is pseudo prime
561 is pseudo prime
645 is pseudo prime

Aha – we have found Fermat’s first three pseudo primes, and also the first Carmichael number (561). OK, so let’s tighten up the code a little and use the trick Fermat described, of testing using one or more additional bases, like this (with the change highlighted in blue):

for (j = 0 ; j < 1024 ; ++j)
  if (Math.pow(2, j) % j == 2 && Math.pow(2, j) % j == 2)
    if (!isprime(P, j))
      document.write(j + ' is pseudo prime<br>')

Now when you run this code, no pseudo primes are returned – all prime numbers up to 1024 are successfully recognized, as confirmed by the isprime() test. I’d like to take this example further, but the limit of JavaScript’s Math library is reached at powers of 1024. So that will have to wait until I’ve written (or borrowed) a library for dealing with exceptionally large numbers.

A Possible New Angle?

But what I find interesting is that I thought the Carmichael numbers were supposed to be resistant to all base coprimes supplied to Fermat’s Little Theorem. But it seems that by using the Chinese Hypothesis variant with a different base, you can weed out (at least some of) the Carmichael numbers. Any mathematicians care to investigate this – it could possibly be significant?

Anyway, that’s enough messing around with primes for the moment – I have to get back to recording the videos for my new CSS book due for publication by McGraw-Hill in January. But I’d welcome distractions in the meantime if you have any ideas or thoughts to share in the comments :)

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