Patterns
&
Sequences

Clearly

human awareness of groups and shapes

and movement and repetition

they discover in the patterns and rhythms of the world

begin with their formation in the womb,

pulsing with two pre-birth heartbeats,

and experiencing the bio-chemical rhythms

that accompany a mother's sleep cycles.

Then, post-birth,

a whole new world is revealed for them to make sense of.

Suddenly there are some seriously interesting rhythms

between things that seem to be opposites,

like the blinking in and out of waking and sleeping,

like day and night,

like hot and cold,

like hunger and satisfaction.

There are regular patterns of change

coming in through all the senses,

motions like the rocking of a cradle or a pram.

And then there are lullabies,

sung with a calming heartbeat,

using a sequence of sounds

of melodic frequency relationships

to soothe and relax inner turmoil.

It takes a while before they learn

to record what's going on using symbols and signs.

And at first they have no names for geometrical shapes

when they first encounter circles,

squares and triangles,

and playing with a ball teaches much about the nature of a sphere

and the dimensions in which it moves

without ever questioning the number of its sides.

It is mathematics that is the language

we use to describe these things,

and write them down on chalkboards and paper,

and of course, once maths had ways in which

to express and exchange mathematical ideas,

that opened up yet another intellectual, ideational world

that could encompass infinities.

Numbers may have first been used for practical problem solving,

for counting, dividing and sharing,

and for trading and telling time,

but by playing with the number signs

and then adding to them,

all manner of patterns could be seen in the numbers themselves.

And from the simplest rhythms of multiplication,

maths developed in complexity and subtlety,

as did its manifestation in the worlds of sight and sound.

The things humans have built show this clearly

in their geometry and decoration,

and there's even a whole mathematical field

devoted to how tiles fit together,

while all the music of the world through history

comes from variations of pattern in rhythm and frequency.

But humans are also happy to enjoy their patterns

in less precise ways in the world around them.

Birdsong doesn't need to be constrained by precision,

and the sound of waves breaking on a beach

doesn't need to have rhythmical precision to be relaxing.

Now rhythms and patterns are comparatively easy to visualise,

but the same can't be said about sequences.

The repetition of rhythms is a short closed loop

compared to the more open ended nature of many sequences,

some of which can have some odd and unexpected characteristics.

Primes seem to be strangely difficult to predict,

yet have a habit of appearing in pairs,

and seem to display clear patterns in ulam and sacks spirals,

though not clear enough to make them predictable.

The fibonacci sequence doesn't look too interesting in a straight line,

but is readily visible all around in the shaping and formation of the natural world.

Number sequences can be seen as displaying

the extraordinary alignment of simplicity and complexity

to be found in creative growth,

a combination that can be seen in the development of fractals,

and perhaps their best known example,

the well known beauty of the Mandelbrot set.

But these sequences are not just beautiful,

they solve problems,

such as the best way to consider the measurement of a coastline.

Fractals have developed into chaos theory

as a way to examine complex interfaces.

Benoit Mandelbrot said

Nature exhibits not simply a higher degree, but an altogether different level of complexity

and the immense complexity of calculation in these areas

has really only been possible

due to the development of computers,

allowing the machine to deal with the number interactions

while the humans approach things through

imagination and visualisation tools.

Equations can be turned into shapes,

the visualisation of which can in turn

show us the inner workings of natural processes,

or become transformed

into engineered objects,

and there are many problems that would have been impossible to solve

without the use of imagery.

Whether it be in cosmology,

topology or knots,

our mathematics will always reach

further than our ability to see it

or integrate it into the world around us,

a gentle reminder to remain ever humble,

and aware of the vastness of what we do not know.