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Geometry in Nature – Dornach Oct 2013

________ ____ ©John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 [email protected], Morphology.org

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Lecture two

Thur 10-10-2013, 9.00am to 10.30am

Geometry in Nature – Tetrahedrons Three talks to Mathematics conference at the Goetheanum

by John Blackwood

The first presentation considered the ideas and natural expression of line, foci on the line,

rhythms in the line and ended with the question that asked “in what context was this all this

happening?”.

It may have been noticed that there was also the attempt at something of a cognitive process

or methodology pursued. This process had to do with the breathing between the conceptual

and the perceptual – in an attempt at a reconciliation between the two. The world is not

divided, except for our consciousness as we currently experience it. This can change.

A summary of the second presentation follows:

Planar path curves

• This talk attempts to cover:

• A very brief introduction to some planar path curves.

• The different tetrahedrons that appear to be at the background the forms of the four

kingdoms – or some of the kingdoms at least.

• I speculated that there could be four such configurations, all very different, but

nevertheless transformations of each other (however radical) .

• If it was correct for the plants – as Lawrence Edwards contended – perhaps such a

basic structure would work for the other kingdoms too ….

•

• Why tetrahedrons?

• But why tetrahedrons at all?

• Because, just as I believe the line element is the core element in form structures,

these lines are not in isolation.

• If the basic form in space is the tetrahedron then this line element (in all the

kingdoms) must be part of some such larger form context. It can‟t be on its own. So

for the plant stem (a line) there must be five other lines – to even begin to get near a

tetrahedron form! Where on Earth (or Heaven) were they? And where were the four

points. And the four planes.

• So – as I saw it – the visible „spines‟ in all the kingdoms had to be part of a greater,

yet imaginable and thinkable, context.

• Edwards explored this extensively and fruitfully for the plant world …

•

• Plant tetrahedron



2

• Edwards work had demonstrated the plant application to my satisfaction (more

later), that is that there was a tetrahedral structure that could encompass the plant

kingdoms basic morphology.

• The tetrahedron he worked with had six lines, four points and four planes as basic.

• Here is shown just the two real lines, one immediate and local, the other at the

infinite periphery, and both mutually perpendicular.

•

• … and with the four points added in

• Two real lines (black), static

• Two real points (black) , static

• … and two imaginary points (red), in

motion.

• Find the four planes (two real and two

imaginary) ... ?

•

• A mineral tetrahedron

• So I said to myself …

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

apply to the mineral world too. Does it?

• I have not got far but think that the particular tetrahedron that works well for the

vegetative will simply not work at all for the mineral – but a transform of it might. It

had to!

• So what tetrahedral form will manage the

architectures of this static, hard, dead and solid

kingdom? This I wondered about…

•

• Path curves

• Before attempting to look at that, I thought to

review a basic path curve construction. These are

curves which inevitably arise when such as a plane

transforms into itself. What does this mean?



3

• A graphic sequence may help to grasp what is going on in this process. This is

written up in Ch 10 of my book (Geometry in Nature).

• It is outlined here for just one case – and only for a particular case in the plane.

•

• Alternate translation and rotation

• This sequence of constructions attempts to show how a simple movement in an

invariant triangle creates an entire path curve field covering the entire plane with

forms.

• We see alternate rotations (of lines)

and translations (of points) … but

ordered by the field in which they

subsist.

•

• Triangle in a plane

• Select any random triangle in a flat

plane.

•

• A path curves emerges …

• A partial path curve begins

to emerge from point

infinitude A and line

infinitude c, sweeping

towards point infinitude C

and line infinitude a ….

(source and sink come to

mind).

• From triangle to path curve field

• This sequence of constructions attempted to show how a simple movement in an

invariant triangle creates an entire path curve field covering the entire plane – in two

dimensions.

• (In parentheses - there can also be a similar system in a point. In other words there

are two kinds of path system in two dimensions!)

• But what of three dimensions?

• I was interested to see how these path curves worked out in tetrahedrons (in space),

not just in triangles (in planes).

•

• Triples

• All these lines, foci, rhythms had to be part of a larger context.

• Planar path curves depended on the movements of point/line pairs.

• In space the curves would have to depend on the ordered movements of all three

elements together. I called these point/line/plane triples, or just triples for short. This

was mentioned at the end of the last talk.

• Throw one of these into tetrahedral space and how is it obliged to behave?

• One small picture may help ...

•



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• Pooh sticks

• We could think of the flow of waters in a river

or stream or creek or tsunami.

• A stick thown into the stream is obliged to go

with the current.

• This is Pooh Sticks bridge. Pooh and his friends

played Pooh sticks. Sticks were dropped in the

water on one side of the bridge and they all

rushed to see whose came out first!

• From the many swirls and eddies it can never

have been a forgone conclusion – yet each stick

would have its own definite path (it might even

be a path curve?)

•

•

• Basic tetrahedron

• This is perhaps the form in which we most readily

imagine the general tetrahedron.

• I thought to start with the triples behaviours within

any all real tetrahedron … here all the 4 + 6 + 4 =

14 elements are totally fixed.

•

•

All real tetrahedron

• The “real” one

• Lawrence Edwards once remarked too me that he did not see the all real

tetrahedron anywhere in nature.

• The background of this slide shows what is meant – for the all real form will have

four real points, four real planes and six real lines, all fixed and static – going

nowhere.

• This form will still have the possibility of growth

measures or rhythms of points and planes in all six

of the lines and can be saturated with path curves.

•

• Movement in the real one

• Further the thing is full of and surrounded by, an

infinitude of path curves, which will be determined

by how much point/line/plane triples move within

the space and connected intimately with the



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rhythms, of points and planes in the encompassing lines. And all in 3D!

• But we still do not see these curved forms in nature … at least I havn‟t yet.

•

• The really, really BIG real one!

• But for the mineral world these curves would have to be

able to straighten out – and become the linear, precise

configurations typical of the beautiful world of crystalline.

• How could this happen morphologically/geometrically?

• What would happen, I thought, if we considered an

infinitely large tetrahedron which was, at the same, a

perfectly regular one, an equilateral tetrahedron?

• Such a form is of course barely imaginable – in fact it is

quite cosmically universal … so I resorted to a subterfuge

to help with this imagining ….

•

• A model of sorts

• I built a shrunk model of this form – or attempted to.

• But an infinitely large model is – obviously – a total, even absurd, impossibility – of

course!!

• So I imagined the thing as if the four points and planes were radically contracted but

that the central forms remained relatively unchanged, or yet only slightly changed.

• This was possible if there was imagined to be a growth measure between each of the

two points rather than measures of equal distances.

•

• From curve to straight …

• If the points of the tetrahedron are infinitely far away then one can hardly put them

on the paper – but if differentially contracted then they can appear on the page along

with points mid way between them.

• In one of the lines I imagined four such consecutive points (between the initial two)

that were almost the same distance from each other.

• Then I calculated the cross ratio for these four consecutive points.

• But what is this “Cross Ratio”?

•

• Cross Ratio

• The Cross Ratio is a ratio of ratios. Whoever

discovered it must have been quite a person as it

is fundamental to all linear calculations in a line.

• Cross Ratio = (AC/CD)

• (AB/BD)



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• Six distinct cross ratios – Milne

• A brief extract from a 1911 geometry book by Rev. John J. Milne (not the Winnie

the Pooh writer), may be of interest, p5 of Cross Ratio Geometry gives:

• … six different values for the CR, given x is one of them.

• Six different numbers

•

• Cross Ratio = 4

• Let the distances between them be a, b and c.

• If a = b = c then the Cross Ratio (CR) will be equal to 4. Test this. Let a = b = c =

100 mm.

• If CR = ((a + b)/c) / (a /(b + c)), in one case

• Therefore: CR = (100 + 100) / 100) / (100/(100 +100)

• Hence: CR = (200/100) / (100/200)

• CR = 2 / 0.5). = 4 exactly

•

• Model – slight variations

• But if the two infinite points decide to become a mere 1000 mm apart then this

changes the cross ratio – but by surprisingly little. That is if one assumes that a = b

= 100 mm. Then c (or z say) becomes a little less. This is what I worked with.

• I found that c would need to be about 92.3 mm rather than 100.

• This led to a CR a little greater than 4.

•

• 4% error for the first step

• If again CR = ((a + b)/c) / (a/(b + c))

• Therefore: CR = (100 + 100)/92.3) / (100/(100 + 92.3)

• Hence: CR = (200/92.3) / (100/192.3)

• CR = 2.166847…/ 0.520020…).

• CR = 4.166846

• So although not quite representing equality of

measure, I considered it satisfactory to work with

(about 4% error for the first step).

•

• Model beginnings

• Where then to begin with the modeling? •

• Special skew lines

• I built a miniature model of this tetrahedron form –



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or attempted to.

• This commenced with two skew lines of wooden dowlling, 1000 mm long, mutually

perpendicular, and positioned 707 mm apart (as calculated for a regular tetrahedron

of 1 meter side lengths – it only needs Pythagoras theorem twice over)

• These were spaced apart with lengths of threaded steel rod. This made adjustment

reasonably precise.

• Then three sets of these were made and were brought together so that the end points

coincided in the four points … next image …

•

• Miniature model •

• Three pairs of skewed line pairs

• The full model was based on the four real

points all being 1000 mm apart (I meter).

This also seemed a usable scale.

• Each 1000 mm line segment was divided in

two and the joins of the of the skew lines

linked at a center point.

• Here I “cheated” because I did not want the

integrity of the metal rods broken and thus I

let the central lines not quite coincide. The

three lines (or rods) went through drilled

holes a small block of wood, a little

displaced from true center (by about 3 mm – see inset at right).

•

• Marking the growth measure • Next came the marking of the growth measures in the lines. Initially

I assumed all six to be identical. I already had one point of the

measure - the central point in each line. Two further points were

then 100 mm on either side of this central point.

•

• Lines across the space

• I then asked myself how might the joins (lines that is)

of point pairs of the skew line pairs give planes at the

very center of this pseudo cosmic form.

• Working wth only one pair of skew lines, emphasised

in blue

• The saddle

surface become

very clear.



8

• Next image shows all these three surfaces. I now used coloured cord.

•

•

•

•

• In the back yard

• Line complexes

• These line complexes (this might even be

the correct technical term [Nick?]) swept

through the entire space of this tetrahedron

– and beyond.

• Or did they? Recall this figure is

pretending to be infinitely large – so we

can hardly go beyond that. (How can there

be anything beyond the infinite – discuss)

• There was no beyond!

• These curves, or surfaces, were contained within this vast space. But what did they

really look like?

• I tried to get nearer to this to see what these path surfaces looked like – by the

device of filling the surfaces in with strips of coloured paper. This brought them to

light quite clearly I thought. See next, and next animation

•

• What do we notice?

• A Cartesian set of axes right in the

middle of the thing. Is Descartes a

mere subset of this global

tetrahedron framework? I only ask!

• At the very center three mutually

orthogonal planes and lines –

which we could call x, y and z

planes, perpendicular to the x, y,

and z axes.

• Three sets of identical surfaces

(red, green and blue) which – on

closer inspection – all turn out to

be saddle surfaces.

•

• A Cartesian set of axes right in the middle

•

• Saddle surfaces

• What was going on here? What were the surfaces doing? How did they relate to

each other?



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• I found the answer intriguing to say the least. A single surface (say blue) met two

pairs of the tetrahedrons lines in a saddle form.

• But there were two other saddle surfaces.

• This meant that each surface connected with the other two in the tetrahedrons outer

lines. While in the center these three surfaces were mutually perpendicular, at the

periphery they were (pairwise) at 180 degrees to each other. What a twist! See next

slide.

•

• A blue cross … • The blue plane through the center is

actually a doubly twisted surface

meeting four of the lines of the

tetrahedron at the periphery. The

dotted curves show the saddle.

• Abstracted here is a twisted blue

cross form which meets 2/3 of the

lines of the tetrahedron.

•

• Integrated planes

• What further intrigued me was the

way in which each central plane

(Cartesian to all appearances)

connected with the periphery.

• An attempt was made to graphically

display this.

• What emerged was a threefold cross

…

•

• Three “crosses”

• The red and green “crosses” can be portrayed similarly – with mutual

perpendicularity at the center (when all three are superimposed).

•

• Three fold mineral cross?

• And when put together within the tetrahedral form … • Note that each coloured bar of the cross twists 90 degrees from one side of the

tetrahedron to the other. Perpendicularity rules again!

• Another rendering is offered next …

•

• Did this architecture mesh with the mineral world?

Minerals •

• Cartesians as a subset!

• It was this mighty architecture

which I thought might apply to the

mineral crystal structures on earth.



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• For lo, there emerge the Cartesian axes!

• And, yeah, right in the very middle forsooth, three mutually perpendicular axes

stand forth unbidden!

• And these three axes are (perhaps) the veritable basis for a number of the seven (or

is that six?) crystal systems.

•

•

• Mutually perpendicular

• Single flat blue plane through the middle, for an

infinitely large tetrahedron – while the other two

(red and green would be mutually perpendicular

to this one – and also flat.

•

• Seven crystal systems (or six?)

• There are apparently seven basic crystal systems

– according to (some) crystallographers.

• These are called, in the literature I have seen:

• ISOMETRIC

• TETRAGONAL

• ORTHORHOMBIC

• HEXAGONAL

• TRIGONAL

• MONOCLINIC

• TRICLINIC

•

• Account for seven

systems?

• For the infinite and

regular tetrahedron to be

the basis for the solid

mineral form it would

have to be able to account for all of the architectures of these seven systems.

• Could it do so? Could it even begin to do so?

•

• Three systems

• Three of the systems were based on the Cartesian axes.

• To accommodate these three all that was needed was

for there to be different rhythms or measures in the

tetrahedrons line pairs – or so I thought (and still think).

• ISOMETRIC: The rhythms in all pairs of skew lines

would all be the same.

• TETRAGONAL: Two rhythms would be the same and

one rhythm for one skew pair would be different.

• ORTHORHOMBIC: All different rhythms, that is all

three pairs would have different rhythms.

•

• Isometric, tetragonal, orthorhombic.

• For the infinite and regular tetrahedron:



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• This form covered the ISOMETRIC system where, if a side length is a, then all side

lengths are the same: that is a = a = a, and angles are all mutually perpendicular,

also the case for the next two systems. • The TETRAGONAL demanded: a = a ≠ c units.

• The ORTHORHOMBIC required: a ≠ b ≠ c units.

• How might this work? Consider only the tetragonal system …. next graphic.

•

• Tetragonal forms

• Two sides are equal, the third greater or smaller.

•

• Tetragonal, a = a ≠ c

• Taking only the tetragonal form – how is this to be

understood within the tetrahedral context?

• If my supposition that the side lengths are related to

rhythms in the tetrahedrons six lines is correct – it would

mean that (for the tetragonal) that there would be one pair would have two rhythms

the same and the other two pairs would have different (to the first pair) rhythms

within them.

• This is crudely illustrated next.

•

• Tetragonal system and “cosmic” rhythms in the lines at

infinity

•

• Local view of a tetragonal

unit cell

• And when one looks at what

might be going on locally

(that is, in the EARTH

sphere) – this is what one

should see – a typical TETRAGONAL form,

structured by the vast (dare one say COSMIC?)

regular equilateral tetrahedron.

•

• Other four systems – a work in progress?

• In a similar way the other two, ISOMETRIC and ORTHORHOMBIC, might be

accounted for by the infinite regular equilateral tetrahedron.

• But what of the four remaining crystal systems?

• Were there other infinite all real tetrahedrons with some measure of regularity? The

“scalene” tetrahedron seemed to me to be a bit too flexible – but who knows!

• Part of the appeal of this idea was that there was an innate regularity – at least of

some degree. A couple of other forms are suggested next. (This is not in my book

but has been hovering on the edges of my mind for a while.)

•

• An isosceles tetrahedron

• One such partially regular tetrahedron would be when two opposite sides are equal

but the other four are equal to each other but not to the first two.

• Could this be called an “isosceles tetrahedron”



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• What happens to the angularity? Are the skew lines able to remain mutually

perpendicular to each other – for it is not just linear measures we are concerned

with, but also angularity between the axes.

•

• A scalene tetrahedron?

• A third form could be when all three pairs of skew lines are not equal to each other.

• At this stage this is part of “work in progress”, or research if one will. Nowhere near

any conclusions as yet … just exploring ...

•

• And the plant kingdom?

• If the forgoing special tetrahedron was anywhere near successful for the Mineral

world – what about the plant world.

• Here I mention the form, fields, parameters and some examples of the next special

tetrahedron.

• Edwards called this the semi-imaginary tetrahedron…..

•

Semi imaginary tetrahedron •

• Infinite and local

• What do we have to do to convert the all real tetrahedron so that it becomes the

semi-imaginary semi-real tetrahedron?

• One of the line pairs remains real, but one of the lines remains at infinity and the

other becomes local, giving a polarity. Could this be seen as a combination of the

cosmic and the earthly?

• The other two line pairs become imaginary or complex and hence can be thought of

as in continuous motion. They constantly bridge the infinite and the local. One line

pair rotates in one direction and the other line pair rotates in the other direction

(clockwise and anti clockwise – or respectively, deasil and widdershins in old

money).

•

• Barbara and Lawrence

• In 1976 Lawrence and Barbara Edwards spent

about six months in Australia.

• Erwin Berney and myself were extremely

fortunate to spend one evening a week

working with him on plant geometry, for the

entire six

months he

was in Oz.

This was education. I was 36. Who says you

ever stop learning?!

•

•

• My first path curve under Edwards

instruction in 1976

• In these days it was all done by drawing – unless



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one could afford $40,000 for an HP computer – which did little more than wire

frame layouts anyway. •

• George Adams and Olive Whicher

• An elevation view of this form had already appeared in

Adam‟s and Whicher‟s Plant betweeen sun and earth.

• Adams saw this field structure as that which surrounded the

growing germ part of the plant.

•

• This was a start ….

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

• The plant world demanded a tetrahedron where a little bit of it had become local –

just the stem, and even this was only a partial line, delimited by end points.

• Lawrence Edwards pioneered this work, with the buds forms of plants, though

working also with the form of the heart and embyology.

• He worked with George Adams.

•

• Texts

• Edwards core text on this was The Field of Form (1982), republished in 1993 as The

Vortex of Life.

• The form fields he outlines can be described using a very small range of parameters

– yet these are capable of covering a huge range of forms.

• These parameters are:

• Lambda, this gives the profile outline form (λ)

• Epsilon, this give the slope of the path curves on the surface of the form (ε)

• Rhythm or stepping, this gives the spacing of the

nodes on the curves on the surface (a?)

•

• Lawrence Edwards work • What the path curves, in space, give us is the

profiles of forms as they appear to us. Lambda

ranges from λ = 0 through to λ = 1 and through to λ

= ∞.

• This is only part of the picture however….

•

• Full range of form factor, λ

• λ varies from 0 to infinity through positive and negative values.

• Through positive there are the convex egg forms and through negative are the

concave vortex forms.

• This can be seen as a form spectrum, no lesss a spectrum (in my view) than the well

known colour sprectrums.

• The extremes are when λ = 0 (cone form) and when to λ = ∞, i.e.(inverted cone

form).

• Notice that half the picture is missing in each case – for above and below the end

points shown are the parts of the field which traverse the infinite!!

•

• More of the field of form – “egg and vortex structure”?

•



14

• This shows more of the field

• – but is somewhat distorted

•

• Varying lambda

• How we might see some of these forms

in the physically visible world

•

• Varying epsilon, the spiral gradient

• Not only can the profile be given a form

factor but the slope of the curves can be

characterised numerically by the

parameter

Edwards called,

epsilon or ε.

• How we might

see some of

these forms in

the physically

visible world

when epsilon

varies (keeping

lambda the same)

• Varying rhythms or node spacing

• There is still another variation possible and this has to do with the node spacing

along the curves themselves.

• This can be characterised as a rhythm … •

• Scale – almost anything! • Add in a simple scale factor to form the shape definitively in concrete physical

space and the range of forms that can be accommodated is vast.

• They can range from tiny gyrogonites (less than a millimeter in height) to the huge

bunya pines of Australia. •

•

•

•

•

•

•



15

• Point-wise egg form – model by David Bowden

• Typical…

•



16

Plants

•

• Vegetative expressions

• This beautiful little shape that the geometry

inevitably gives for this local/infinite tetrahedron

can be compared to a great number of natural

plant structures …

• Some very few examples are offered – but there

could be – and there are – thousands.

• A few worked examples are shown below.

•

• Isopogon flower head

• Does the semi imaginary tetrahedron apply?

•

• Isopogon flower head – curve fitting (1988)

• Went for a walk in the New South Wales bush sometime in late 1980‟s.

• Photographed this beautiful little flower head.

• Did a curve fitting exercise.

• Observe the connection of idea with observation – next slide

•

•

•

•

•



17

• Concept and percept

•

• Protea bud?

• Another wonderful example came

my way when a high school

student friend, Jenny Ellis,

brought in what we thought was a

protea bud – she knew my interest

in such things.

• With this large bud (the small square is

1 cm by 1 cm) I tried to take the

analysis a bit further.

•

• An analysis?

• I thought that if this form was an

expression of a path curve system then

it would have to meet a number of

criteria.

• If the path curve or S curve was true

then to meet the plant tetrahedron it

would have to approach the infinitudes

(or end points) and go through at least

two points on the forms surface.

• The practice was to first estimate the

position of the central vertical line.

•

•

• … continued

• Then to estimate the two end points (top end point was usually quite clear but the

bottom was a guestimate – but one could get quite good at this).

• Then to pick two points on a clockwise curve as far apart as reasonably possible.

• Then to pick two points on a anti-clockwise curve, again as far apart as reasonably

possible.

• Then a program was written which – if these were S curves or were path curves –

would put a curve through the two chosen points and the end points and give the

profile outline of the bud, all in one go.

• I had my doubts that this would work at all well ….

•

• Fibonacci numbers 5 and 8

• But I was astonished how close the two profiles were to each other.



18

• The program plotted the other

curves (in both directions). This

was possible if the number of

curves was found to be a pair of

consecutive Fibonacci numbers.

And they were – in this case the

number of “spirals” in a clockwise

direction (green) was 8 and the

number in an anti-clockwise

direction (red) was 5.

• 5 and 8 are consecutive Fibonacci

numbers (in the series 1, 1, 2, 3, 5,

8, 13, 21, 34, etc, etc).

• An interesting question which I

have not taken the time to explore

would be if the clockwise number

was always the higher Fibonacci

number – and what that might

imply.

•

• Profile closeness

• Ideally the outer profiles should

overlap exactly. They did not

(quite) but were, to my mind,

convincingly close.

• It had been found by

experience that near the

end points , since they

were effectively

infinitely

unapproachable, the

profile would not fit so

well.

•

• Vindication! • All the curves in both

directions were not a

bad match to the petal

tips.

• This was “cooking with

gas”!

• Did it work more

generally?

• Was this a universal

archetype for plant

form?

•



19

• Cycad cone

• Another example was the cone of a cycad found in a

garden in a local suburb – this cone would be about

40/50 cm high.

• The following images show the process I adopted –

without comment – but similar to the protea.

•

• Cycad analysis methodology

•

•

•

•

•

•

• Just to confirm radial symmetry – the cycad view from

above

•

• Pineapple

• Sometimes it is self evident which

node points to choose to track a

curve.

• The nodes can appear to follow more

than one curve in the same direction.

• I personally have followed the

practice of the “smallest

quadrilateral”, with the “closest

distances”.

• What is meant by this? The

pineapples at right will demonstrate.

• I chose to use the lower

interpretation – which gave

consecutive Fibonacci numbers of 8

and 13.

• What were the top numbers? You

will have to find a pineapple and

check!

• How well did this fruit conform?

•

•



20

• Pineapple

analysis (2008)

• 13 Clockwise

curves (green):

• Epsilon = 1.42

• Lambda = 1.19

• 8 Anticlockwise

curves (red):

• Epsilon = 0.42

• Lambda = 1.20

• Notice that the

curves depart

from the

pineapple as the

end points are

approached.

•

• Mammilaria cactus

• This little cactus was also a

good candidate – or so it

seemed.

• Top views are shown with

the nodes joined by freehand

curves.

• There were 13 anti-clockwise

and 21 clockwise.

• The plants “Fibonacci”

response was often the first

thing to check out.

• Analysis follows …

•

•

• S curves and profiles

•

•

•

•

•

•

•

•

•

•



21

•

•

•

•

• Mammilaria

analysis (2008)

• This looked

reasonably good for

some of the curves

and nodes.

• But the end points

have been avoided!

• Only part of the

profiles can be

claimed. This is no

more or less than

usual however.

•

•

• Charophyte gyrogonites

• It was due to a mention by Peter Glasby about the work of Adriana Garcia at

Wollongong University on the charophyte gyrogonites that led me to look at the tiny

heads of these water plants, a form of algae.

• These little plants have become important through their fossils and the evolutionary

interpretations these can lead to. Whole conferences are held on these small plants

and their remains!

Animals •

• Animalic -tetrahedron of the third kind?

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

then about the animal?

• Do we have a tetrahedron for the animal kingdom?

•

• Egg profiles

• Many an animal starts out from an egg form.



22

• The duck’s egg

• This is good to start with as it was one

of the examples Edwards wrote about

in his seminal works.

• He reports on two ducks eggs. With

rather unusual markings.

• On a farm in New South Wales I

traded a mechanical toy for a ducks

egg. We both thought we had got a

good deal.

•

• Egg of duck …

• Although the markings do not usually

show up on ducks eggs this sample

showed them quite clearly, clearly

enough to analyse. I have wondered if

such visible markings are even a sign

that the duck is having some problems,

as they are not frequent

apparently!

•

• Duck egg profile and

spiral analysis

• By selecting just two

points on the spiral like

curves we can calculate

that:

• Lambda = 1.25

• Steepness, epsilon = 1.018

•

• Birds Eggs

• Ostrich, Seagull and Chicken



23

• Ostrich egg

• This image and profile (in red) is due to work of

Nick Thomas.

• As it is close to an ellipse (not far from a circle)

it will have a lambda approximately equal to 1.

• Note also the hint of a spiral line (Marked with

two crosses (in red). I did not get time to analyse

this but would say epsilon would be about 1 as

well (angle not too far from 45 degrees).

•

• One final bird egg … from Oz

• Can you estimate lambda, epsilon and rhythm?

• (show sample).

• Lambda can be estimated, but since there is no

evidence (in the shell itself) of spirals, no

estimate of either epsilon or

rhythm is possible.

•

• Emu data, and calcs

• Note that one does not usually

attempt to plot the endpoints, in

this context they are infinitudes–

hence inaccesible anyway.

•

• Mammals

• Most mammals are placental but

a few do lay eggs.

• In Australia there are two in

particular: the echidna and the

platypus.

• Although it was a bit of an

adventure even to find pictures

of eggs of these creatures I did

manage to find profiles good

enough to analyse for their

outlines.



24

•

• Echidna

• This was more interesting (in a way).

• I could find no images initially. Then I

came in contact with Mervyn Griffiths

who lived in Canberra. He, as a researcher,

had written whole texts on the

monotremes. He gave me a photo of half

an echidna egg!

• Imagine my surprise when, in

conversation with him, and I showed him

my own and Edwards work he says, “that

looks like the work of the second

hierarchy”!

• At that stage I had no idea of his interest in

Steiner‟s work. He was a deep student but worked very much on his own. •

• Echidna egg - details

• Calculations.

• Plotted profile.

• Calculated and empirical – synchronicity.

•

• Platypus

egg

• The egg of

the

Platypus, I

found in

an

Australian

Geographi

c

Magazine,

in a great

article by

Jack

Green. • Lambda

was nearly

1.



25

•

• Sea urchins

• Here was an

animal which

still

mimmiked the

plants

orientation –

yet also

moved.

• This example,

a. pallidus had

a weighted

mean lambda

of 1.226 and

an epsilon of

infinity

(vertical).

•

• Salamacis

Sphaeroides

• Salamacis

data/calc

•

• Salamacis –

predicted

field curves

• For a given a

fivefold

symmetry, a height and maximum

width, and a lambda of 1.486 and a

definite rhythm, this could then be

an approximation to the field

behind the Salamacis form.

•



26

• Spine points on the test profile

• Inset: note the white dots of the actual test are very close to the blue dots as

calculated (yellow are merely intermediary)

•

•

• Software model for salmacis

sphaeroides

•



27

• Then there is another egg case …At left beautiful

example of a double spiral in a clockwise direction.

Whose egg is this? At right is a section of the form – both

ends are quite different in their morphology.

• Reptiles

• Salty at the Taronga Zoo Monday before last.

• Crocodile eggs

•

• A near ellipse

• For the top egg in the previous slide a “standard” ellipse from the Word software

gave a reasonably good fit simply by eye.

•

• Eggs as animal precursors

• To me there was little doubt that the young animal world (or some of it) managed

some very good geometry – without the help of any computers or calculation! The



28

formative impulses (whatever they are) in these creatures were very capable

geometricians .

• This was in the egg forms. I began to think of the egg as a kind of repetition of the

plant stage that the animalic can sometimes (that is often) go through.

• Consider the egg form as a profile. It is the very same as the bud and tree form

profiles. It has the same radial symmetry viewed end on (this is the same radial

symmetry of the plant).

• So the initial skin of many a juvenile animal has the plant signature.

•

• Linear in the animal

• But where does „the egg‟ go from here? What further forms do the animals take?

• Are they still egg like? Yes and no methinks. Something else is working into the

subsequent forms.

• So where to start this next investigation? If the line is still to be regarded as

important as a key element in the animal world – where was it?

•

• Was there a line of particular significance for the animal?

• Of course there was – the spine! Recall first talk.

• The animal spine is fundamentally horizontal

• Millions of species (literally and numerically) have an unequivocal horizontality.

• The spine also has two main foci, the reproductive area and neck, throat, sound area.

• Through these weaves the segmented and nodal 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

distances – if one can use a spatial

metaphor here – between these two

kingdoms, just as there is vast distance

between the very natures of plant and

animal.

•

• Around the line of the spine

• What is it that surrounds the spine of the fish.

How can the spine – as it is inside the body of

the fish – be made visible outside (so to

speak).

• I became aware of something known as the

lateral line ….

•

• Start with the fish …

• And its horizontal spine …

• Partly because the fish could look vaguely like

a re-oriented pineapple!

•



29

• Re-orientation of the pineapple?

•

• Can the Brazilian pineapple form become

transformed to the Japanese pineapple fish

form?

• Or does that go way too far?

•

• The fish and the line

• I noticed, looking through Rudie Kuiter‟s

work that a number of fish almost

represented little more (seemingly) than a

horizontal floating line segment.

• Sometimes they were uncannily straight!

• More straight line

intervals …

•

• Around this line …

• Was there a basic shape

that could be put around

the line (interval) which captured

morphologically the profile of “the fish”? An

ellipse form was tried out ….

•

• Fork tailed large eyed bream …

• This (ellipse) even looked somewhat plausible

around the bream…

•

• Sleek unicorn fish

• But there was no way this unicorn fish

could be claimed as an ellipse.

• The plant had already included the bud

form so why not the fish.

• This was explored a little …

•

• Compare with the bud/egg form

• A series of egg like forms were plotted and

the unicorn compared



30

• Note the nearest comparison above

•

• Best so far …

• This was not too bad a comparison but

was far from satisfactory – and it still

assumed a bilateral symmetry

•

• Lateral lines

• By this time I was noticing the

pervasive lateral line …

• …. usually curved but not

always. • For my scheming mind I

needed for this line to be

relatively straight.

•

• Eastern nannygai

• This little fish had a very

nearly straight line lateral line.

• Not only that but it was clearly

marked with a series of light

coloured „dots‟!

• I assumed this reflected

externally , the position of the

spine.



31

•

• Salmon cutlets

• But was there a way of checking this

latter assertion? • Back to the fish markets …

• And salmon cutlets!

• Here were sections of fish which

might reveal how the internal spine

lay with respect to the external

lateral line.

• How so?

•

• Lateral line on a salmon

cutlet

• Along a series of scales the

lateral line was clearly visible

•

• Salmon section • Note the vertical bilaterl

symmetry.

• Note the horizontal asymmetry.

• Note the division of the flesh above

and below the horizontal line throught

the spine.

• There seemed to be some coincidence

with the lateral line, the plane cutting

through the spine and the spine itself.

• I felt justified in making the working

assumption that this coincidence was

real.

•

• Other cutlets

confirmed this!

• Mulloway and

trevally

• Back to the fish

spine

• There is an amazing website of

a company offering fish skeletons.

• I include a very few samples of those they had

on offer.

• Note that there is a moderate straightness in

most of their fishes spines shown here.

•



32

• Golden scales

• The sheer beauty of the carps scales gave pause! • But there was more … the carp‟s lateral line

was as clearly marked as day itself!

• Some points are identified …..

•

• A tetrahedron of the third kind?

• A number of factors could now come together.

• If the line of the spine could turn through 90

degrees from plant to animal then so too would,

presumably, the tetrahedron that carried these.

• What would this architecture appear like?

•

• Speculation

• A first sketch …. July 2009

• The Human? •

• The Phantom?

• Was there a tetrahedron here too

– for the human? There had to be

… surely

• In some sense the human form

itself had to be the ultimate

tetrahedron, the one from which

all the other kingdoms stem,

through various layers of radical

metamorphosis.

• To look at the human form was to

look at an amazing tetrahedron –

but what on this earth did it look

like overall? With the physically

visible did one seek the invisible – as so often was necessary with all the other

forms.

• Was it all there before our very eyes – the ultimate revealed secret? Is this the form

of the Phantom – after the Fall and and prior to the Resurrection? (Because we have

not resurrected it even if Christ had achieved this at the Mystery of Golgotha).

• These were huge mysteries …

• As you can see I have not got very far with our tetrahedrons!

•

• A fish architecture

• My work on the fish architecture will be explored a little tomorrow.

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