Developing an Eye for Form a talk by John Blackwood

Science Conference #3At Stourbridge, UK

9.00am to 11.00 am, Saturday 17th February 2012

…. In the four ….

… kingdoms of ….

… mighty Nature.

Four kingdoms• Our eye knows four kingdoms or realms of

mighty nature• This is a primal distinction we all can make

almost every day of the week – it can be dangerous if we do not!

• To begin with it is the form we see and clearly recognise – no physics, no chemistry, no ….. DNA testing

Mineral, vegetable, animal and human

• Transition forms, usually small• Viruses • Amoeba• Chimps?• ….yet there are …• Four basic stances ……• How is an approach to these four to be made?• Do we need a different paradigm for each.

An issue that the point is primal (reductionism)

• Not until around 460 B.C., did a Greek philosopher, Democritus, develop the idea of atoms. He asked this question: If you break a piece of matter in half, and then break it in half again, how many breaks will you have to make before you can break it no further? Democritus thought that it ended at some point, a smallest possible bit of matter. He called these basic matter particles, atoms.

• Can we ask the other question – of what whole is that part, a part? Koestler’s holons.

But really there are three primal elements in geometry

• Point

• Line

• ….and plane

• Why does this threesome matter rather than the “onesome” of the point and atomicity?

Two mutually definable elements – one free element

An extra degree of freedom

• There is an interdependence of of all three but one takes a special place.

• This is the line, but why so?• Two points always define a line• Two planes always define a line (even if it is

infinitely far away)• Two lines are as free as two teenagers

(imagine themselves to be!)

Hypothesis

• Offer merely a hypothesis• What happens if (similarly, but different, to

Democritus) we ask what happens if we start with the LINE as the basic element

• Does this change anything?• After all the line is more “primal” than the

point – from the geometry point of view – than the point ever can be.

After all geometry is the laws of space

• So whatever lives in space has to obey the laws of space (they would not be laws otherwise!)

• If that is so should we not look at the core, the central, element of space and see what evolves from that?

• Hence if one takes the line-wise as a beginning (rather that the point-wise) could this lead anywhere?

Where to begin – with concept or phenomena?

• These are the polarities we meet every day• We observe and we think• This is directed to the phenomena (factoids)

and to concepts (thinktoids (ouch!) or ideas)• Neither is paramount necessarily – we work

with both

So where to start?

• It does’nt matter, one leads to the other and the other to the one

• It has to be a cycle • A breathing, meeting the inner life and the

outer world, rhythmically.• Anyone for the new Yoga?

From death to life

• We breath in and out, life in death out• What is your carbon footprint, eh?• The death component comes from within –

does it have to be that?• The more we work on this cycle the greater

can be the living component of what we offer to the world – even in our breathing – or so I believe

• But we have to start somewhere in this cycle

Line geometry can be studied in its own right no doubt, later – but we can also ask where do we see the line, the

straight, the linear in nature?

Why not start with trees or even grasses?

• The vertical uprightness of the tree is unequivocal

• The form is built around the vertical, part on earth, part reaching to the sun and air, and part entering into the surrounds

• We have straightness• This is a keynote of the living plant world

One could do beautiful books on this!

• And many have done just that … eg Pakenham

• The magnificence of forests all over the world strikes many

• The dark pine forests, open English beech forests, light, airy Casurina forests…

• Mighty sequoia forests in the US of A and so on – to stand in that silence full of trolls …. awesome!

But what do we know of the line as such – as an ideal element in the

toolbox of geometry?

Here it gets interesting …

• For the line is an unspeakable being, it represents to our minds something virtually unmanageable, the infinitely thin and the perfectly straight and infinitely long

• Yet are there some things to be said of it?• There certainly are ….

For we can ask – what can live in the line, in this infinite straightness?

• The other two elements of course!• These two elements (recall)…are the point and

the plane• How so do they live?• A line can be thought of as an infinitude of points

• But it can also be thought of as an infinitude of

planes• Is there any inherent order or structure in these

points and planes that any line can carry?

Meanwhile back among the trees, the grasses and the bamboo, itself a grass, in fact, among the phenomenological

…

What do we actually see?

• We see a series of nodes• We see these nodes all along the stems• This is where further branchings branch out• The stem gets thinner as it strives to the light

Is there any systematic, or order, to this nodal appearance?

Revisit the line itself

• Are there any behaviors that the points can exhibit on any line?

There most certainly are …

• There are at least three kinds of behaviour, or measures as they can be called

• These three are fundamental to the geometry of any line

Three measures

• These three are known as:

• Circling measure• Step measure• Growth measure

Circling measure

Step measure

Growth measure

Is there any one of these of particular interest to us here?

One of the point measures …• One measure, the growth measure, has what look like two

concentrations on it• These could be described as “end points”

• But between these end points, in the middle, the point spacings change.

• Notice that the spacings on either side get slowly smaller and smaller. Soon they become very small as if to some kind of limit.

• At such limit points we say we have affectively “a point at infinity”

• So this growth measure has two such points at infinity. I call such as these, end points.

So a question arises – do the nodal distances change (and if so how) for

the empirical phenomenal world

Meanwhile, back at the bamboo …

• We notice that there is, about the middle of the stem, is the largest nodal (or point) distance and those close are much the same.

At the ends …• This is closing is noticeable at

the ground but there are not that many visible nodes. Nevertheless they get closer together.

• At the other end, the sun pole or end I think of it as, the node spacings become very small. This particularly evident at the growing tip or point.

A concurrence of idea and phenomena

• This means that as the ends are approached the spacings get smaller and smaller, since the largest is in the middle

• So could it be proposed that the earth pole and the sun pole are equivalent to the infinitudes of the growth measure?

• So we note that the geometry story has a similarity to that which the perceptual artefact has to say

• But does it end there?

No!

We have only just got started!

•Recall that the inhabitants of the line are twofold. There are points, as just discussed, but there are also planes. There is an infinitude of planes in the line just as there are points.

Planar behaviours

• Just as there are three kinds of measure of points in a line there are also three kinds of measure of planes in a line

• These three are equivalents to the three point measures

• What do they look like?

Planar behaviour around a line

• Here we see the line “end on” as it were, as well as suggestions of the planes, end on

One of the planar measures

• If we imagine the growth measure of points is significant – which of the three measures is of importance for the planes?

• It seems that it is the circling measure of planes and not the other two

• This means that the planes spin around the axis like a book thrown open!

Meanwhile again back to the plant …

• Do we see anything representing the planar aspect of the line in the actual phenomenological?

• Is there anything that could be said to define a plane anywhere in the plant form?

• There is and it lies in the fact that branches come from the nodes

• Between any main stem and the branch there is a plane – if we imagine the lines of branch and stem to intersect in as node.

Planar intersection

• At every node we can imagine a plane intersecting the stem and branch – in the middle of the leaf in the sketch –which spins around the stem as the nodes rise up the stem.

What is the “spin”?

• I call the amount the plane rotates about the stem line, the spin (!)

• We need to check if this spin is reflected in the plant world

• This spin depends on the plant species• Many plants seem to have a spin of about 137

degrees (but not all)

Golden Angle Spin

• A number of plants I have looked at, and measured, seem to average out at about this value of 137.5 degrees.

• Those who know their golden sections will know this as the golden angle – which is really only a golden section wrapped around a full 360 degrees

• One such plant is shown in the background

Apparent striving for the golden angle

• This sketch shows how such a plant appears to always be trying to hover about this particular angle. The central plane of each consecutive leaf spinning up the stemstives to make the golden angle the mean or average.

Back to bamboo

• The bamboo spin is not 137 degrees

• The bamboo branches turn close to 180 degrees with astonishing regularity and uniformity with every nodal step up the stem

Ideal forms and specific forms

• Now we have a correspondence of sorts for both the inhabitants of the line with a common artefact out in the garden

• It is not “perfect”, but then no experimental results are [the can’t be, as there will always be errors of measurement]

• Yet the living world of actual species appears to strive towards some ideal form – is this the archetype that it seeks to express?

Archetypal plant?

• My hypothesis is that this may be a beginning to finding something of the nature of the archetypal plant - which Goethe says that he saw and experience in about 1750.

• It will need an awful lot of work with many plants to assess how widespread and universal this notion is

• And it opens up many more questions ………….

A speculation …

Which leads one to ask … ?• Is there a profile towards which the branches are tending

(the canopy) and …• Is the a pattern by which the branches are branching• Is there a relation between these two?• Is the single vertical stem complemented by a horizontal

line at infinity (as the geometry suggests)• Does this mean the plant form (or architecture) extends to

the infinite so that the plant form is actually a structure that embraces the whole of space (now that is holistic!)

• Is this a sort of thinking a macrocosmic counterpart to “string theory” in any sense?

• Does the ….. And so on and so on!

One other thing …

• If this kind of thinking applies to the plant world, it should, in some form, apply to the mineral world

• Does it?• I have not got far but think that the

tetrahedron that works well for the vegetative will simply not do for the mineral

• So what tetrahedral form will?

The big one• Lawrence Edwards did not see in nature the all

real tetrahedron – he once told me• The background shows what is meant here• The thing is full of path curves• But for the mineral world these curves would

have to straighten out, and planes flatten• How could this happen?• What would happen, I thought, if we considered

the infinitely large tetrahedron and the perfectly regular one? So I built a make believe model.

In the back yard

And lo, there emerge Cartesians!

Right in the very middle, three mutually perpendicular axes

And these are the basis for a number of crystal systems, a whole other

story, but the crystal world is the dead mineral world

So I made a model of the big one

• This is impossible of course• But the thing can be approximated at a small

scale by allowing the purely regular to become growth measures

• And yet retain the regularity aspect, a defining and essential feature

This was a start ….

• The mineral world might invoke the largest most regular tetrahedron imaginable

• The plant world demanded a tetrahedron where a little bit of it had become local – just the stem, and even this was only a tiny partial line

• Lawrence Edwards pioneered this work, he gave us the plant tetrahedron

• He worked with George Adams

George Adams and Whicher

Lawrence Edwards worka “spectrum of form”, among many

other things

My first path curve under Edwards instruction in 1976

Varying epsilon, the spiral gradient

Point-wise egg form – David Bowden

Animalic?

• If this story could help grasp the architecture of the mineral and the plant – what about the animal?

• Do we have a tetrahedron for the animal kingdom?

Was there a line of particular significance for the animal?

• Of course there is – the spine!• The animal spine is fundamentally horizontal• Millions of species have an unequivocal

horizontality• The spine also has to main foci, the

reproductive area and neck, throat, sound area

• Through these weaves the segmented spine

Animalic architecture

• But the spine is horizontal (or close to)• It is also 90 degrees away from the verticality

of the plant• 90 degrees is the maximum distance that two

lines can be between each other• This jump seemed to me to show vast

distance between these two kingdoms

Start with the fish …

• The horizontal spine …

………………………………

And then there is the human …

• Another leap of 90 degrees ….• What is the meaning of this?

Forms of multiplication – Cassini