The Truth of Algebra

This is a piece of fun, playing around with numbers and equations. It need not be taken seriously but has an intriguing result.

We are used to the easy truth:
consider
x + 6 = 3x
6 = 3xx
6 = 2x or 2x = 6
x = 3
simple, but it can miss a hidden mystery .

If we put what we found (x = 3) back into what we started with,
we get
3 + 6 = 3 × 3
which is:
3 + 2 × 3 = 3 × 3      (or 9 = 9)
of course! – This is how our world is,
but something lies beyond this.

When the x enters our world
we can write:
x + 2x = 3x
which gives
3x = 3x
in which x can be any number from minus infinity to plus infinity.

Of course 3 is in x.
x and 3 are now in both our equation and the unknown.
The unknown is infinite and our equation is part of it.

It need not have been three but that seems apt:
an unknown entered into our equation as a number in it, and a piece of fun.

Sue’s Birthday Bunnies

Dedicated to my friends Dick & Sue.

For her birthday little Sue
was given by her Daddy two
little bunnies in a hutch.
She said, ‘I love them! Oh so much!’
She loved them, and they loved Sue,
And they loved each other too.

Bunnies did what bunnies do,
so what a great surprise had Sue
when she peeped inside their door:
her two bunnies now were four.

‘Daddy come! Oh Daddy quick!
(Daddy by the way was Dick)
One and one have just made four!’
Daddy came and Daddy saw
that the present he’d supplied
had gone forth and multiplied.

‘Oh no!’ he said, ‘I gave you two.
Pretty soon we’ll have a zoo!
There first were two, and now two more,
It’s two and two that becomes four.’

Later talking to his neighbour
Dick said, ‘How I had to labour!
‘Sue may be bright and pretty quick,
but no good at arithmetic!’

The neighbour said,
‘Now don’t you fuss,
although I only drive a bus
I study speed and things like that
when in my driver’s seat I’m sat.
A speed of just two miles an hour
If doubled needs a bit more power.
But, and this is hard to scan,
I’ll try and do it if I can,
two miles an hour plus two again,
Is not four m.p.h.’

‘Explain!’

‘The actual sum, as I have found,
Is two miles per hour, plus two miles more
less two divided by the speed of light in miles per hour.
This argument you can’t resist.
I am a Quantum Physicist
not a poet.’

‘I thought you drove a bus!’

Dick’s other neighbour, on his way,
Stopped to pass the time of day.
‘Math and physics show us we
really need philosophy.
One and two and three and four
mean nothing if not joined to more.
They are shorthand, abstract terms,
to count the stars and sticks, and worms.’

‘And bunnies!’ spoke up little Sue.
We really must give her her due.
She knows that numbers are a tool
not bound to any other rule
than Einstein’s relativity
and Heisenberg’s uncertainty.
Schrodinger’s unhappy cat
would surely say, ‘Amen.’ to that.

Hazelnut Forest revisited

In March I wrote a puzzle poem Hazelnut Forest, its title an equation,
λ = 2πħ/p
so here is, as I said would be, the promised explanation.

The forest is the universe when the Word made all things new,
and the Spirit found the first conditions good and proving true.
Its leaves, the smallest particles of which the world is made,
the calling birds swift flying in the dappled light and shade,
are photons that were called to be when light was first displayed.

Its title is a formula, a particle’s waveform,
for everything is energy,
and particles just seem to be
the focus of a mystery,
the fine eye of the storm.

Another poem followed that wondered at the size
of the forest (or the world) as seen by wiser eyes.

How small the forest? We really cannot see.
We cannot give position, speed, time or volition;
to what is all around us, a truly strange admission.
As size get small and smaller, in the atoms heart and less,
in proton, quark or photon, and spacetime’s emptiness,
there is a finite limit bound in uncertainty.
How small is the forest? It’s just too small to see,
for in that finite limit is all infinity.

Men like Werner Heisenberg,
Max Planck and de Broglie,
worked out the math, and many more
have worked at detail and for sure,
where you and I give up and snore,
they plucked cherries from the tree.

A Circle With a Volume, I recall,
the last and strangest poem of them all,
came from Planck’s discovery
that length, like time, just cannot be reduced
infinitesimally small.
No matter what dimensions that we tell,
the smallest there can be, that we call Planck’s Length, L,
gives structure to the rest. There is no spell
that lets us cut fine finer till there’s nothing there at all.

The smallest, fundamental space,
the smallest, fundamental time,
are bound with that uncertainty
that binds the forest leaves.

Centre to edge is less than width,
the wheel’s centre to its rim,
your nose to your ear,
less than the wheel’s width,
less than ear to ear.

But the width of a fundamental
is the smallest distance possible .
Where can its centre be?
How far from its edge?

W. B. Yeats’ troubled poem The Second Coming that I quoted in this poem sums the uncertainty and the resulting fragility astonishingly aptly:

Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,

From the inbuilt uncertainty of this fundamental seed, spacetime burst in an instant, followed by an immense expansion phase.
From fundamental to universal in microseconds.

This volume, this conundrum,
too veiled for us to see,
a mystery its diameter,
its radius an enigma,
the Sybil of Cumae,
time in eternity.

Ah! The Sybil of Cumae! Who was she?
Tis said she asked Apollo, who wanted her to wife,
that she might have, though mortal, as many years of life
as the grains of sand held in her hand.

False promises were made and when her wish was gained
her favours were withdrawn – Apollo raged.
Trapped through the years, her body aged;
kept shrinking in a jar ’til just her voice remained.

And why a hazelnut forest? In the mid-fourteenth century, following a vision, Dame Julian of Norwich compared all creation to a hazelnut held in God’s hand.
Such a tiny thing, encompassing all creation, shown her by God in a series of visions in which she saw the depth and greatness of His love for all mankind.

Hazelnut Forest

This perhaps should come with the warning, ‘Let the reader who can, understand.’
(Why a hazelnut forest? LHC? Who was the Sybil of Cumae?)

Small clues: LHC is the Large Hadron Collider, the largest machine ever built on Earth (27 kilometres across) with which many new particles, some expected, some not, have been discovered. Other clues are hidden in the tags, but not in the correct order (WordPress shifts them into alphabetical order).

Some time in April (I promise!) I shall write an explanation. I also promise (don’t look at my crossed fingers) that the explanation will not itself need explaining.

λ = 2πħ/p

How small is the forest?
How tiny its leaves?
Where the pattern of branches
tosses and weaves;
and the canopy sways,
and the summer winds moan,
until shortening days
say summer is gone.

Halfway in the forest
its deepest heart;
where calling birds fly
to its farthest part,
and the leaves’ rustling sigh
gives place to the sky,
and the height of the trees,
and the birds’ flying free,
and the tiniest leaves,
are the forest to me.

How Small is The Forest?

How far is it from constancy
to Heisenberg’s uncertainty?
A tiny length defines
a volume that we find:
the smallest we can know.
A fundamental distance
that we call Planck’s length (L)
shows there is a thickness
we cannot go below.

A circle has a volume just like a carousel
the volume of a circle is pi times r squared L,
(by this we come to see
there is no ‘true’ 2D
and a circle without volume,
is an anomaly).

A Circle With a Volume

This volume hides a mystery.
How small can it be?
How can it be measured,
a space too small to see
by eye or LHC?

The smallest space we cannot see:
the rings inside, the rings inside,
the rings inside a tree,
is found from four-thirds pi
times its depth to power of three.

This volume, this conundrum,
too veiled for us to see,
a mystery its diameter,
its radius an enigma,
the Sybil of Cumae,
time in eternity.

Radius is half of width,
we know that very well
but half a fundamental
is a word we cannot spell.
Diameter equals radius,
it flickers to and fro,
in the tiniest of instants,
the shortest we can know.

Things fall apart, the centre cannot hold.
Fission and expansion, a cosmos to unfold.

A whole that ever seeks
the shimmer and the chime,
Infinite from finite
in the infinite-finite rhyme:
the Word that ever speaks
at the birth and death of time.

The Language of the Universe?

In the beginning was the Word?

Many say the language of the universe is mathematics, but the language of mathematics is not necessarily numbers. Numbers are a shorthand for words. In quantum mechanics collapsed waveforms is the term for a relationship between particles and their waveforms, in which particles, or indeed any combination of particles (atoms, molecules, chemical and organic compounds, even you) are seen as the focus or point of action of the energy waves involved. In the same way numbers and equations are like the collapsed waveforms of the huge quantity of words that might otherwise be needed to describe them. It is a good analogy. For many mathematical concepts the number of words would be as infinite as the cosmic extent of particle waveforms.

Pi (π), the circumference of a circle divided by its diameter, is one such mathematical concept. Written as a decimal it extends to an infinite number of decimal places, of which the first thirty two are as below:

3.14159265358979323846264338327950…

Should you have any need to remember this, some time ago I came across a mnemonic for the first dozen or so places. I changed and extended it to thirty-two before getting bored. The number of letters in each word is the number at each decimal place.

Now –
I sing a scale excelling,
in mystic voice and magic spelling,
sublimest strains incarnate.

Art in its measures will reveal
an angel’s song for the carousel,
and in eternal harmonies dwell,
O!

Feel free to add more of your own.

The idea of numbers being a shorthand for words is not a difficult concept, after all without words how could we explain what numbers are to children? But there may be more to them than that. Pi is far more than the simple relationship of a circle to its diameter.

Pi

Are numbers and equations
the collapsed waveforms of words?
And is pi’s definition
the circling of the birds
round and wide above the hills?
or the volume of a drop of water from the rills
rolling down to plop into shining highland ghylls?

Then the circle of the sphere
and the rolling of a tear
when a sobbing child cries, ‘Why!’
and the Earth around the Sun in perihelion,
and the wide, wide width of tears is pi.

The quick birds’ wheeling cry,
and the crying tears of pain,
and the earth around the sun,
and the round drops in the rain,
and the signs of endless sky,
the music of the spheres,
and the circle of the years,
tell us why.

Birds circling round their prey
know the distance from their nest,
and swooping down from sky,
sharp claw and shining eye,
returning straight and high,
the circle and the swoop,
the short returning loop,
and the gather of the storm
round the centre still at rest
say more to you and I
than the radius and the circle
that are pi.