Beyond Newton · Reality

Squaring The Circle.

Beyond Newton, Part 12.

the details of the jam jar’s design. There will have been a very great deal of humming and haa’ing over the details, and calculations involving ‘π’ the Greek letter commonly used to express this ratio will have been used continually
From the side, from the weight, there is no way to tell the accuracy to which the dimensions were calculated.
It doesn’t matter if you’re only buying on price…
… or because it’s a statement.

Note: this is about mathematics. It is not intended to tax you in the way school demanded of you.

We’re taught at school about the ratio of the diameter of a circle to its circumference. Roll a plate on its side and having marked a point on its perimeter, the length it travels by the time it reaches that point again is around 3.14; actually it’s a fraction more. But with a dinner plate of 30cms diameter the difference would be a little over half a millimetre. Not enough for any serious schoolchild to worry about. “A little over three” would suffice here.

The difference between the rough estimation of two decimal points – that is to say, 3.14, and the accuracy of the reality isn’t enough for most of us to worry about in our daily lives. If we’re buying preserved food means looking at the price and the weight, not the details of the jam jar’s design that will imply careful and exact measurement. There will have been a very great deal of humming and haa’ing over the details, and calculations involving ‘π’ the Greek letter commonly used to express this ratio will have been used continually.

Mind you, jam jars are more a matter of how much glass is used along with a standardized metal or plastic lid. Weight, earthiness by any other name, overcomes all in our day and age. This we can calculate with the accuracy of the perimeter of a square: which is precisely four times the length of one side.

They really did have to be precise: perfectly circular, the accuracy of which is eye-popping. Well beyond the scope of the vernier measuring device – and my old one measured down to 1/50th of a millimetre.
The simplicity of the form, its ubiquitous nature belie the unimaginable tolerances that are hidden from the eye.

The circular plastic water beakers that you see popping out of the tube at the side of the water cooler are a different matter altogether. I met a chap who used to make the moulds for them, and a mould would take him some three to four weeks to make. They really did have to be precise: perfectly circular, the accuracy of which is eye-popping. Well beyond the scope of the vernier measuring device – and my old one measured down to 1/50th of a millimetre.

Then there is the spacing of the ribs that keep their gossamer thin walls from collapsing under their own weight. These have to be calculated with tolerances that exact the exacting mind of an engineer, and not that of the casual sipper of cooled water. There is the size and quality of the rib itself, allowing the beaker to be reasonably firm in the hand… and also to make allowance for wall thickness. All pointing to the all important fact that they don’t stick together.

Because if they did stick together, they’d not fall out of the tube when you pull the little lever, would they? And that would never do. The effect of gravity on such a tiny sliver of matter means they really do have to be perfect.

This isn’t the laser interferometry used to make the helicopter rotor blades that my tutor at college helped design. They used to get the measurements right, down to a few nanometres. But that is real strain for the brain.

The ratio of the diameter to the circumference is rarely noticable beyond two decimal places.
Put a car into a skid and you’ll wear enough off the tyre’s surface that would condemn a rotor blade to the scrap-heap.
That’s the first line of figures,by the way. There are four more lines… talk about the unnecessary!

It saved money. But money in this respect is weight against the original spark of thought that led to the development of laser interferometry…

But then, a few nanometres is the difference in the circumference of your car’s tyre when you’ve driven as far as the traffic lights.

Probably less.

That’s modern engineering for you: there are tolerances that to the ordinary eye are meaningless, even ridiculous. Yet in our contorted, upside down world of economics, saving a few pennies is worth a lot of thought. When you make millions of little plastic beakers, saving half a penny on each package soon mounts up… It’s different for helicopter rotor blades; but even here, costs override every other consideration. In this world, aesthetics are but a raiment. That doesn’t mean they’re any less effective, but that is for another post to explore.

Do We Need This?

There are things we need to know, and things we don’t. Mathematicians used logarithms to design camera lenses, and used logs with twenty decimal places. Newton liked calculating these things, in his own idiosyncratic and rather obsessive-compulsive way. He’d go through the process again and again to arrive at another figure. Perhaps it was the different figure that charmed him?

Newton liked calculating these things, in his own idiosyncratic and rather obsessive-compulsive way.
Obsessive compulsive disorder? Calculations made in Newton’s own notebook show how obsessive he really was about an accuracy he couldn’t imagine – leave alone see.
What else was there to entertain Newton’s mind in a world where anything new would be a danger?

Who knows? He was such a recluse. What is true is that he did enjoy repeating the process time and time and time and time again… rather than meet with friends and learn something new from them.

In our daily lives there is just so much we need in terms of accuracy. Give or take five minutes on the hour when it comes to lunch: five minutes early, ten minutes late. Enough to rile the boss, not enough to anger him. We all need wheels that are round, we don’t need them machined to the tolerances of a helicopter rotor blade. Put a car into a skid and you’ll wear enough off the tyre’s surface that would condemn a rotor blade to the scrap-heap.

Tolerance is expensive, especially when you cross a certain boundary. At least when you’re talking about things that are made by hand.

I remember a lathe turner telling me about a trick to finish a piece of steel using some very fine sand paper. I remember another telling me how he could always tell when somebody had used very fine sandpaper to achieve a certain finish. I guess it depends on how well tuned your eyes are to the quality of machining!

Automated machines have changed all that! They’ve increased the tolerances we’re used to, mainly because it’s not so expensive as it used to be. You’d still be pushed to see it, though.

Examining Tolerance.

If you take a circular disc a metre in diameter (around 40” for the metrically challenged) its circumference will be 3.14 metres. To the nearest centimetre. 3.142 to the nearest tenth of a centimetre. 3/64″ in a length of over ten feet… the decades of diminishment become very small with a rapidity that stops the heart. Or should. With 3.1416 and we’re dealing with amounts that are already invisible. 3.151592 and we have passed the point where even a microscope will help us determine which is longer… add a few more digits and the electron microscope is challenged.

In our daily lives, however, where we walk to the grocers to buy a pound of apples, this ratio is utterly irrelevant. That today we’re dependent on it is only because too many people think in earthly terms of weight, time saved and cost – and not the brilliance of thinking that could bring us so much more.

The Square And The Circle.

Even so, we have the brain-dead calculating ‘π’ to a million decimal places. Yet the number just keeps on going. There is no rhythm to the numbers, they are as random a scattering as any on earth. There is no formula to them, no pattern, no regularity of any kind. Indeed, they are so irregular a sequence, the exact sequence of numbers can be used as a generator of random numbers!

The square we can calculate in an instant: its perimeter – the earthly equivalent of the circumference – is four times the length of one side. The circle, on the other hand, leaves us with empty hands.

The ratio of the diameter to the circumference really does elude us. Or does it defy us?

Putting It All Backwards.

And yes, I know the trick: we start counting in ‘π’. It’s a measurement known as the radian, and is used extensively in engineering. But that only inverts the problem: a single item is now 1/π and what was once a simple unity is now a complex and irrepressible length of numbers that we have no possibility of determining. Isn’t it best to leave ‘π’ to itself? Let it be what it is, and not worry too much over the details? Circles are circles, and the quality of my bicycle wheel is more a matter of a well seated tyre than the process that led to its being made.

Oh, and its being properly inflated, too. Which given that I have a leaky valve; when I leave the gallery and wish to mount my steed the accuracy is no longer what it was. That is to say, I have a flat tyre. Flat tyres aren’t perfectly circular, are they?

There is just so much we need to know. There are people who have a desire to know too much, and in doing so, are oblivious to something that should be obvious to us all: the ratio of the circle’s diameter to its circumference is something we cannot bring down to earth. No force on earth will subdue it, no bomb change its nature. It is something we all have to live with.

Is it any wonder that the circle has always represented the heavens?


Other Posts In This Series:

Part 1: Experiencing Time First Hand.

Part 2: All Hard Drives Look Alike.

Part 3: What Ahriman Wants. (Published privately).

Part 4: Stirring Horn Silica.

Part 5: A Horn Silica Rainbow.

Part 6: Messed Up Beans. (Published privately).

Part 7: Newton’s Rainbow.

Part 8: Untangling The Astral And The Etheric. (Published privately).

Part 9: How To Count Water.

Part 10: Socrates’ Task And Mine. (Published privately).

Part 11: The Square And The Circle. (Published privately).

Part 12: Squaring The Circle.

Please note that privately published posts are available to trusted friends without cost. The content is not intended for the general public and is restricted to those who can demonstrate that they understand the nature – and implications of – Rudolf Steiner’s scientific thinking. That is to say, it is not for the unready.

In certain circumstances, pdfs of these posts are available on request; you may do so by leaving a comment. This will tell me if you can grasp the nature of the post you are enquiring about. The comment itself can be left unmoderated or deleted if requested.


23 thoughts on “Squaring The Circle.

  1. Hey.. I actually managed to follow one of your articles all the way through without getting tied up in too many knots! I didn’t know very much about pi before this, so is informative reading 🙂 This is a great line- “The ratio of the diameter to the circumference really does elude us. Or does it defy us?” That’ll get the old thinking faculties fired up for sure..! Awesome ponderings 🙂


    1. Them knots is there to be tied… and then untied.

      I’m glad it came across as was intended; it wasn’t aimed at the mathematician. After all, they’d only want to know the last three digits in the series …

      Liked by 1 person

      1. Yeah is fascinating.. Pi, and the impossibility of tying it down in a form which make sense to us! Lol I love the idea of the gods just creating it as a way to piss us mere mortals off… I’m thinking that’s a good plot point in a fantasy/sci fi novel!…


      2. I think it’s more they’re trying to tell us something, rather than annoy us.

        As to your Sci-Fi novel, won’t it be more of a case of the truth being close to fiction?


  2. I commissioned my narrowboat on the basis that the skilled boatbuilder Nick Thorpe working in steel with chalk, rulers and by eye worked to a tolerance of just one eigth of an inch over a 70 foot boat.

    At the other end of the scale I understand that the sun is travelling at around 520,000 miles an hour around the centre of our galaxy and will take about 285 million years to complete one revolution.

    In the words of Henry Vaughan:
    “I saw Eternity the other night,
    Like a great ring of pure and endless light,
    All calm, as it was bright;
    And round beneath it, Time in hours, days, years,
    Driv’n by the spheres
    Like a vast shadow mov’d; in which the world
    And all her train were hurl’d. ”

    Perfect…..Vaughan and Newton – Motherhood and apple π?


    1. I would argue at his use of the word ‘hurled’ – after all, speed is not something one notices directly, is it? You have to look at the motorway signs passing to get an indication of how fast you’re going. The motions of the heavens are nothing if not gentle.

      Hence we can determine how fast the sun is going, and for all we know, it might as well be as stationary as our earth feels under our feet 😉 Save that it travels the sky each day.

      We can’t perceive speed directly, any more than we can the passing of time.


  3. G – That’s a bit harsh on Mr V. He was afterall writing in 1650 AD

    Your nudge in the direction of the passing of time led me to this:
    “A study co-written by Dr Andrew Jackson, of the Department of Zoology at TCD, in the Animal Behaviour journal last September shows how different organisms experience time at different rates. Small-bodied animals with fast metabolic rates, such as flies and birds, experience time more slowly than large bodied animals with slow metabolic rates, such as turtles, the study concludes.”
    I think this helps explain that when you are a kid the 6 week school summer holiday seemed to last an etenity but now having a more expansive waist line time seems to fly by more quickly with every year that passes. You might even say its hurtling by………


    1. Well, there were people who understood time, even in Mr Vaughan’s day. Perhaps they’d read his poem and came to the same conclusion as I?

      Kepler knew these things, otherwise he’d not have been able to determine the quality of a planet’s orbit, thus: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit”.

      If you think squaring time is merely to multiply it in the way modern scientists do…

      As to what animals perceive, try this…


  4. My daughter’s cat “Caspurr” (RIP) merely preceived me as the doorman. The more important question is “what was going through the mind of Schrödinger’s cat”?


    1. Having thought about it I can answer at least half my question. In its dead state nothing is going through the mind of Schrdinger’s cat. I could hazard a guess that in its alive state, in a darkened box, the cat is probably wondering if it is dead or alive – much like those on the other side oif the experiment……..;-)


      1. Didn’t Schrödinger invent the idea of a cat to upset the quantum physicists??

        After all, if one solves Schrödinger’s equations, one only arrives at statistical averages. There ain’t anything concrete down there…

        … and Schrödinger’s would be as baffled as any of us.


    1. Any thoughts as to what it was that fascinated you? I do my best to fascinate, my problem is that it doesn’t always work… so it’s always nice to know what did!


  5. You are a polymath, Gemma, and your writings always inform and entertain me. I guess topics, such as this one, always resonate with my old engineer’s brain. So perhaps it is a case of your audience not being so enraptured with all fields?


    1. Thanks for letting me know.

      My hope is that there will be something for everyone, but we are all human, and cannot do everything.

      Much as I might try… 😉


  6. Hi Gemma, just followed this link from a comment you made on a post of mine , as always a joy to digest , my uncle Mr F Allen worked at I.T.W a manufacture of plastic cups and utencils very clever man in the industry travelled all over as an adviser later in life , on plastic manufacturing , well here’s a memory about pi , working at a power generation and cooling engineers as a design draugthsman they always used 3.14 , now coming from an engineering family I was taught about pi very early in life unit circle as well and we used 3.1416. So the company were tolerances set at – .020 to + .050, 3.1416 gives tighter tols, They still insisted on using 3.14 always outside their tolerances , because they used Cad macro’s and not calculators.
    Just a useless memory of pi ?


    1. “because they used Cad macro’s and not calculators” Obviously in the days before CAD could handle more than two decimal places…

      A friend of mine at uni used to do CAD in the old days, and worked in one of the very first paperless offices. The problem was that if he wanted to check a detail on another drawing, it would take 20 minutes for the computer to unravel it and post it on his screen.

      Then he’d have to close it and re-open the drawing he was working on… which would take another 20 minutes.

      It didn’t take long for him to have a pile of A0 drawings to refer to 😉

      The real thrust of this post about pi was that it is irreconcilable. And there’s no way around this: it is a reality that nobody can get around.


      1. It was thinking about PI Aand how it is so RANDOM , that struck a CORD that maid my thoughts go of in a LINE of TANGENT about PI and ROUNDabout memories of DECIMAL POINTS of view .😐


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