The Creativity Code. Marcus du Sautoy
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in Ancient Greece coincides with the development of one of the very first algorithms. In Euclid’s Elements, alongside the proof that there are infinitely many prime numbers, we find a recipe that, if followed step by step, solves the following problem: given two numbers, find the largest number that divides them both.
It may help to put the problem in more visual terms. Imagine that the floor of your kitchen is 36 feet long by 15 feet wide. You want to know the largest square tile that will enable you to cover the entire floor without cutting any tiles. So what should you do? Here is the 2000-year-old algorithm that solves the problem:
Suppose you have two numbers, M and N (and suppose N is smaller than M). Start by dividing M by N and call the remainder N1. If N1 is zero, then N is the largest number that divides them both. If N1 is not zero, then divide N by N1 and call the remainder N2. If N2 is zero, then N1 is the largest number that divides M and N. If N2 is not zero, then do the same thing again. Divide N1 by N2 and call the remainder N3. These remainders are getting smaller and smaller and are whole numbers, so at some point one must hit zero. When it does, the algorithm guarantees that the previous remainder is the largest number that divides both M and N. This number is known as the highest common factor or greatest common divisor.
Now let’s return to our challenge of tiling the kitchen floor. First we find the largest square tile that will fit inside the original shape. Then we look for the largest square tile that will fit inside the remaining part – and so on, until you hit a square tile that finally covers the remaining space evenly. This is the largest square tile that will enable you to cover the entire floor without cutting any tiles.
If M = 36 and N = 15, then dividing N into M gives you a remainder of N1 = 6. Dividing N1 into N we get a remainder of N2 = 3. But now dividing N1 by N2 we get no remainder at all, so we know that 3 is the largest number that can divide both 36 and 15.
You see that there are lots of ‘if …, then …’ clauses in this process. That is typical of an algorithm and is what makes algorithms so perfect for coding and computers. Euclid’s ancient recipe gets to the heart of four key characteristics any algorithm should ideally possess:
1. It should consist of a precisely stated and unambiguous set of instructions.
2. The procedure should always finish, regardless of the numbers you insert. (It should not enter an infinite loop!)
3. It should give the answer for any values input into the algorithm.
4. Ideally it should be fast.
In the case of Euclid’s algorithm, there is no ambiguity at any stage. Because the remainder grows smaller at every step, after a finite number of steps it must hit zero, at which point the algorithm stops and spits out the answer. The bigger the numbers, the longer the algorithm will take, but it’s still proportionally fast. (The number of steps is five times the number of digits in the smaller of the two numbers, for those who are curious.)
If one of the oldest algorithms is 2000 years old then why does it owe its name to a ninth-century Persian mathematician? Muhammad Al-Khwarizmi was one of the first directors of the great House of Wisdom in Baghdad and was responsible for many of the translations of the Ancient Greek mathematical texts into Arabic. ‘Algorithm’ is the Latin interpretation of his name. Although all the instructions for Euclid’s algorithm are there in the Elements, the language that Euclid used was very clumsy. The Ancient Greeks thought very geometrically, so numbers were lengths of lines and proofs consisted of pictures – a bit like our example with tiling the kitchen floor. But pictures are not a rigorous way to do mathematics. For that you need the language of algebra, where a letter can stand for any number. This was the invention of Al-Khwarizmi.
To be able to articulate clearly how an algorithm works you need a language that allows you to talk about a number without specifying what that number is. We already saw it at work in explaining how Euclid’s algorithm worked. We gave names to the numbers that we were trying to analyse: N and M. These variables can represent any number. The power of this new linguistic take on mathematics meant that it allowed mathematicians to understand the grammar that underlies the way that numbers work. You didn’t have to talk about particular examples where the method worked. This new language of algebra provided a way to explain the patterns that lie behind the behaviour of numbers. A bit like a code for running a program, it shows why it would work whatever numbers you chose, the third criterion in our conditions for a good algorithm.
Algorithms have become the currency of our era because they are perfect fodder for computers. An algorithm exploits the pattern underlying the way we solve a problem to guide us to a solution. The computer doesn’t need to think. It just follows the instructions encoded in the algorithm again and again, and, as if by magic, out pops the answer you were looking for.
Desert island algorithm
One of the most extraordinary algorithms of the modern age is the one that helps millions of us navigate the internet every day. If I were cast away on a desert island and could only take one algorithm with me, I’d probably choose the one that drives Google. (Not that it would be much use, as I’d be unlikely to have an internet connection.)
In the early days of the internet (we’re talking the early 1990s) there was a directory that listed all of the existing websites. In 1994 there were only 3000 of them. The internet was small enough for you to pretty easily thumb through and find what you were looking for. Things have changed quite a bit since then. When I started writing this paragraph there were 1,267,084,131 websites live on the internet. A few sentences later that number has gone up to 1,267,085,440. (You can check the current status here: http://www.internetlivestats.com/.)
How does Google figure out exactly which one of the billion websites to recommend? Mary Ashwood, an 86-year-old granny from Wigan, was careful to send her requests with a courteous ‘please’ and ‘thank you’, perhaps imagining an industrious group of interns on the other end sifting through the endless requests. When her grandson Ben opened her laptop and found ‘Please translate these roman numerals mcmxcviii thank you’, he couldn’t resist tweeting the world about his nan’s misconception. He got a shock when someone at Google replied with the following tweet:
Dearest Ben’s Nan.
Hope you’re well.
In a world of billions of Searches, yours made us smile.
Oh, and it’s 1998.
Thank YOU
Ben’s Nan brought out the human in Google on this occasion, but there is no way any company could respond personally to the million searches Google receives every fifteen seconds. So if it isn’t magic Google elves scouring the internet, how does Google succeed in so spectacularly locating the answers you want?
It all comes down to the power and beauty of the algorithm Larry Page and Sergey Brin cooked up in their dorm rooms at Stanford in 1996. They originally wanted to call their new algorithm ‘Backrub’, but eventually settled instead on ‘Google’, inspired by the mathematical number for one followed by 100 zeros, which is known as a googol. Their mission was to find a way to rank pages on the internet to help navigate this growing database, so a huge number seemed like a cool name.
It isn’t that there weren’t other algorithms out there being used to do the same thing, but these were pretty simple in their conception. If you wanted to find out more about ‘the polite granny and Google’, existing algorithms would have identified all of the pages with these words and listed them in order, putting the websites with the