LET’S TALK ABOUT MATHS
LET'S TALK ABOUT MATHS – PART ONE
VIDEO TRANSCRIPT: LET’S TALK ABOUT MATHS! YouTube Talk 5a © Nigel Peace, 2023
Yes, this is Maths, folks – but not as you know it. I spent more than three decades as a mathematics teacher and I truly love the subject. But I can pretty much guarantee that what we’re going to talk about was never mentioned when you were at school. And that’s despite it being not only pretty simple stuff, it’s of absolutely vital importance in understanding the world, including ourselves. This is sacred Maths…
First off, here’s a question for you. What does the Great Pyramid of Giza have in common with the United Nations Secretariat? Need a clue? It’s the same thing that Leonardo da Vinci has in common with Claude Debussy and your head. I hope that helps.
Before we answer that, I should mention a few things that are not sacred Maths. Numerology, for one. There’s an awful lot written and talked about on numerology and most of it is, I’m afraid, awful. Turn the letters of your name into numbers (a equals one, b equals two, etc) then add the numbers of your birthdate then add all digits together, unless you’ve got eleven, twenty-two or thirty-three, until you get one digit. This will define your entire character and life purpose.
It seems there are only twelve types of character and life purposes. And this is assuming you’re not a Chaldean, who doesn’t recognise the number nine. Or you have a different alphabet or calendar. Extra layers have even been thought up to give us a Life Path and Soul Urge numbers.
I’m a six, so I’m compassionate and I channel positive energy to everyone. Hmm, not sure many of my students would agree with that. There are a few other flaws in the system, too. For one thing, people change their names or are known by family and friends by different names or have important middle names. Pablo Picasso had fourteen names. Actually, that’s nothing. A certain German typographer had twenty-eight and his surname had five hundred and ninety-six letters. I’m guessing Hubert Blaine, for short, wasn’t a big fan of numerology.
Then we have so-called angel numbers… Do you keep seeing 11:11 or 555? Well, you’re embarking on a new path or going through some changes in your life. You probably knew that anyway, didn’t you? But at least an angel has confirmed it.
A belief that you keep seeing 11:11 is an example of Yellow Elephant Syndrome. What’s that? Well, whatever you do… do not think about yellow elephants… Oh no, there they are. This is an example of self-affirmation, of simply seeing what we expect or want to see and ignoring all other similar sights. It’s a Y.E.S.
These beliefs about numbers may be an interesting way to pass the time, but at the end of the day they’re not really helpful on our spiritual journey, are they? They’re not going to affect the way we live or guide our responses to significant events. And let’s remember that numbers are, after all, basically just a human invention to help us count things.
That doesn’t stop some conspiracy theories about the mystical nature certain numbers. A famous example arose in the 1960s, when the American author and artist William Burroughs wrote that he’d met a ferry operator in Tangiers named Captain Clark who boasted that he hadn’t had an accident in twenty-three years. That night, the ferry sank and Clark died. The same night, an Eastern Airlines ’plane piloted by a different Captain Clark crashed; it was flight number 23. Burroughs called such events ‘The 23 Enigma’, ascribing mystical properties to the number, and now it even has its own Wikipedia page.
If true – and let’s not forget the story started over a few drinks in a rough bar – it’s surely no more than an interesting coincidence. A similar example was reported by C G Jung, who described an evening outing when the number of his tram ticket was the same as his theatre ticket. Then later that evening he received a call when the same digits were mentioned as a telephone number. Yet these events didn’t mean anything. Just coincidence.
Those who love numbers are often not worried at all about a lack of meaningfulness. And arithmetic is full of peculiarities. There are Mystery Numbers, which are the sum of two other numbers whose digits are reversed. For example, 121 = 29 + 92. There are even Perfect Numbers, which are the sum of their own factors excluding themselves. 6 = 1 + 2 + 3. There is nothing either mysterious or mystical about these things. They’re just oddities.
So far, admittedly, I’ve been rather negative about mathematics so let’s turn the tables and admit that there may be something a bit mysterious about mandalas. The name means ‘circle’ in Sanskrit and indeed most mandalas are circular geometric designs, sometimes within a square. Originally, they symbolised the universe or paradise or the spiritual journey, from the outer edge towards the centre. They are usually painted but can even be made, with astonishing skill and reverence, with coloured sand. An example of this was the Chenrezig mandala created to celebrate the Dalai Lama’s visit to the UK in 2008.
Mandalas have been used for millennia in Hinduism, Buddhism and Shinto as an aid to meditation. They embody deep meanings in the symbols used and their geometric forms.
Lately, mandalas have gained fresh popularity in New Age culture. And any number of spiritually-minded artists have turned out their own versions, with circles and spirals and nice colours in a more or less complex pattern. Few of them, however, bear any comparison to the beauty and significance of the ancient religious creations. They’re just pictures.
Ah, sorry, I’m being negative again. But look, the truth is that even the most extraordinary Buddhist or Hindu mandala is still a purely human creation. Many people call this “sacred geometry”. Yet there is nothing at all inherently sacred about geometry, especially when it’s used to represent one particular human belief.
Surely the word ‘sacred’ should be reserved for something beyond human invention, beyond our thinking minds, something numinous and ineffable? And yes, mathematics can indeed help us with this…
Don’t worry, folks, this is nothing like the Maths you did at school! Actually, I love the subject and I taught it for more than three decades. But what I always wanted to know was whether there really is such a thing as ‘Sacred Maths’…
Let’s start, in Part One of this talk, with what it definitely isn’t – despite the beliefs of many good and spiritual people…
Are you a fan of numerology, turning the letters of your name into numbers and adding them up to discover your life purpose? Hubert Blaine wasn’t, his real surname had 596 letters. Do you keep seeing 11:11? It seems the angels are telling you that you’re going through some life changes – but then you probably knew that already.
There are Mystery Numbers and Perfect Numbers and 23 even has its own Wikipedia page. It’s all very entertaining, but does any of it actually help us to navigate the challenges of the spiritual path?
What about mandalas, then, the beautiful designs created by ancient religions as aids for meditation? Well, that’s the problem. They were created.
LET'S TALK ABOUT MATHS – PART TWO
VIDEO TRANSCRIPT: LET’S TALK ABOUT MATHS! YouTube Talk 5b © Nigel Peace, 2023
Mathematics was invented by the Sumerians some four and a half thousand years ago. The Chinese introduced negative numbers two thousand years later and then the Mayans included zero.
People being what they are, the first applications of Maths were for commerce and taxation, followed by astronomy and calendars, since knowing the rhythms of the year was vital for agriculture. But it would be the fifteenth century before new discoveries and ways of thinking began to come at pace. The visionary work of Galileo, Newton and Leibniz revolutionised our beliefs and laid the foundations for modern science and technology.
Can mathematics ever be called ‘sacred’? The answer lies in the fact that it’s virtually impossible to measure or calculate anything exactly. And some of the most interesting and important numbers – take a deep breath – don’t even have a fixed value. What? That’s weird. And literally irrational.
Grab a pencil and ruler and draw a right-angled triangle with, say, base five centimetres and height five centimetres. It’s easy to measure the length of the long, sloping side, isn’t it? But Pythagoras tells us that this length is five times the square root of two, which is not an exact number. As a decimal, it’s 1.41… continuing to infinity with no pattern. The roots of prime numbers turn up frequently in mathematical problems.
Actually, the Babylonians, ancient Egyptians and Greeks knew about this long before Pythagoras. But they could never quite put their fingers on it because they used fractions, not decimals, for numbers that aren’t whole. So the square root of two is about seven divided by five.
The most well-known of these irrational numbers is pi, the ratio of a circle’s circumference to its diameter. It’s about 3.14… or twenty-two divided by seven. Its decimal value has now been calculated to over thirty-one trillion places, without any pattern. This special number is even celebrated worldwide on March the 14th (3.14 in American date format) by, naturally, eating pies. Is all this, er, pointless? Not at all, irrational numbers are used to encrypt our financial transactions online as well as to create national security codes.
Now, it turns out that pizza and ice cream are not the only good things to come out of Italy. In 1202, Leonardo Bonacci, also known as Fibonacci, introduced the world to the Golden Ratio, named by the Greek letter phi. True, it had been known to Indian mathematicians fourteen hundred years earlier, and the Babylonians, ancient Greeks and Egyptians also realised that it had something to do with the geometry of nature. It is built into the design of the Great Pyramid of Giza, presumably to make the structure in harmony with the Earth.
Fibonacci, though, showed us how fundamental it is for understanding how things grows. And that includes us. Here’s a simple example (you may like to get your pencil and paper out again):
There are two kinds of reproductive bees, drones and queens. Drones have only one parent, a queen, whereas queens need two parents, a drone and another queen. Let’s count the ancestors of one drone. He has one parent, who herself has two parents. So our friend has three great-grandparents, two queens and a drone, then five great-great-grandparents, three queens and two drones… you get the idea.
The family tree reads: 1, 1, 2, 3, 5, 8, 13, 21, 34… now called the Fibonacci Sequence. It’s a mathematical model of natural growth, because each number is the sum of the two before it. And dividing one number by the one before it gets us closer and closer to the value of phi, about 1.62. But of course the sequence is infinite, making phi an irrational number.
Fibonacci numbers are found throughout nature, for example the head of a sunflower having twenty-one and thirty-four intertwining spirals of seeds. The branches of plants and trees grow in this sequence. The main trunk produces a branch, which creates two growth points. One of the new stems then branches into two, while the other one lies dormant, and so on. The eye, fins and tail of a dolphin divide its body in the Golden Ratio.
Grab your ruler, a calculator and a mirror. Measure the length of your head, excluding hair, and divide this by its width, excluding ears. Well, no-one is perfect but on average the result will be 1.62. The same proportions are found throughout the rest of our bodies. It’s natural growth.
Have you ever wondered why the pentagram has been revered throughout history as a magical symbol? Divide the length of any of its lines by the next shorter length and we get phi, exactly. So this shape represents our life-force, the Spirit. Starfish always knew this, of course.
The DNA molecule measures 34 by 21 angstroms for each cycle of its double helix spiral. And speaking of spirals, rotating lengths of Fibonacci numbers around a point creates the logarithmic spiral, discovered by the German artist Albrecht Dürer. This is found on snails, pine cones and nautilus shells, the nerves of our eyes’ corneas and the spiral arms of galaxies in the Milky Way.
Still not convinced of its important? The Fibonacci Sequence is used to create Internet search algorithms.
But it’s not really the numbers that are important here, it’s the geometric proportions of things. Phi represents natural beauty and the spirit of life itself, and it was those Italians again who made us realise it. In the fifteenth century, the mathematician and monk, Luca Pacioli, told his friends about it. Among his friends were Pietro Perugino, Piero della Francesca and Leonardo da Vinci.
The Renaissance was born. These artists began to use phi in the design of their work, canvases being divided into rectangles with Fibonacci lengths, such as eight units by thirteen. The result was that paintings now had natural proportions, enhancing their beauty. The most famous examples, of course, are the Mona Lisa and the Baptism of Christ.
The same idea has often been used in architecture, with Fibonacci rectangles seen from the Parthenon in Greece to the United Nations Secretariat building in New York. In music, the structure of the formal boundaries in Debussy’s haunting symphony, La Mer, also follows the Fibonacci Sequence.
I have shamelessly used the same thinking in the structure of my novel, The Unbroken, a love story and an adventure through mysterious worlds of nature. Each chapter has a Fibonacci number and every section of text has an exact Fibonacci number of words, in thousands. So you’re bound to think this book is beautiful!
Mathematics is a wonderful subject, invented to help us understand the world. But it also contributes to our spiritual path. Bonacci, da Vinci and Dürer have shown us that Maths can be genuinely sacred.
For something to be called ‘sacred’, surely it should be beyond human invention, something numinous and ineffable. And it turns out that one of those weird numbers is at the very heart of nature herself, in the growth patterns and geometric proportions of all living things on Earth. It’s called Phi, or the Golden Ratio, and it’s been used by architects and artists throughout history to introduce natural beauty into their work.
The spirit (and spirits) of nature are central characters in my novel The Unbroken. So for the first time ever in print, the very structure of the book, its chapters and word count, reflect the Sacred Maths of Phi.
I loved to unsettle my students by telling them about weird and important numbers that don’t have any fixed value. Yes, that’s irrational. Yet some of these are the very building blocks of life itself. Including you!
Hint: for Part Two of this talk you may like to get yourself a ruler and a calculator…