Geometry in Nature Fishy stuff

Geometry in Nature Dornach Oct 2013 Lecture three FRI 11-10-2013, 9.00am to 10.30am Geometry in Nature Fishy stuff 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?. The second presentation hypothesised that all of the kingdoms were able to be seen as having an underlying architecture and that this was an expression of a distinct tetrahedral structure, that is a distinct transform of the tetrahedron and that was unique to the kingdom considered. The third presentation is to do only with one form experiment that I tried, related to aspects of the fish form. A summary of the third presentation follows: This talk attempts to cover: The notion of a leap for the forms allowing the merely living (plants) to active mobile soul engagement (animalic). That from plant to animal there is again a 90 degree leap, from the vertical to basically horizontal. That there is an evolutionary memory (as it were) of the plant in the spiralations of the scales of the fish. Always a new start all kingdoms go through revised repetitions of their earlier incorporations. Animal form? I started this little exploration with the assumption that there was indeed a special tetrahedron, of some sort for the generally animalic, the sentient, the conscious however dim or alert. Further that it had to be a transformation of the tetrahedron for the plant world (or was that back to front?). It seemed to be that there could be something took over the architecture and took the plant spine and made it become horizontal. 90 degrees again! In terms of physically visible evolution we can imagine a huge leap from plant form (mainly vertical) to animailc gesture (basically horizontal). Even the patterns living in the living plant are revisited to an extent in the skin patterns the scales of the fish. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 1

But can we find a geometry that indicates a continuity between the two? And will it lead from radial symmetry to bilateral symmetry? Radial symmetry. Another significant change between these two kingdoms is the change in the basic symmetry. The plant kingdom has a basically radial symmetry. Viewed from above along the spine, the circle, or rotation, presents itself as the plants raw symmetry. Observe tree forms from a helicopter! The background to this slide is a Google Earth image of the Botanical Gardens in Sydney, Australia. Bottom left is the Sydney Conservatorium of Music. Bilateral symmetry The symmetry becomes bilateral in the animal kingdom for, when viewed from the front, there is usually a definite reflection symmetry manifest. Ask this fish. All bilateral To get some of these images I had to resort to a trick. Not being able to capture stills, short video clips were taken. Then it was relatively easy to frame the direct front (or back) views of the fish as they turned. Getting a front view Gotcha! Fish forms? If the scalar pattern of the fish was to be what I thought it could be then there was surely the necessity for the world of the path curves to be structuring the forms. If Edwards is right it is there in the plant world without a doubt. What about in the the fish world though do these curves also reach into the animalic world? Are they enough? John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 2

Could the basic fish form body be described by any path curve profile? To begin with I simply looked at the profile as seen from the side of the animal. Choosing a fish This exploration took me to the Fish Markets of Sydney quite an institution that I normally did not have cause to visit. I had no real idea where to start but I soon noticed that many fish only had small scales. Also that the lateral line, a feature I quickly became aware of, was not very straight in most cases. So I found myself looking for fishes with a more or less straight lateral line and large(ish) scales. Scourge of the Parramatta River No one seemed to mind me busily taking photographs of their displays and stalls either! To cut a long story short, I found an ideal feral fish, the dreaded carp. It was, nevertheless, a beautiful animal even in death. A golden body covered in the most amazing scales with very clearly defined markings along the lateral line points which could hardly be straighter. Starting points I had now to build a picture of all the criteria the fish form would need to fulfill and developed a whole string of assumptions that I needed to allow myself. I did not know if any of these assumptions was true but one has to start somewhere so here goes. Some assumptions I assumed that the fish lay horizontally (most of the time although the pipe fish made me wonder!!) in its environment. Here was the linear, the line like. These particular pipe fish were like gymnasts! I assumed that the spine was reflected in the lateral line of the fish a feature I knew nothing of before this study. I assumed that the fish form would have two end points and that these were on the spinal line one near the head and the other near the tail (but before it began to flair out). What was the fishy tetrahedron? That the tetrahedron for this complex form would have the spine as one of the lines and that its skew mate would be orthogonal to this spine and also that it would be local (not infinitely far away as with the plant). One of my first sketches is in the background. this crude drawing was done about mid 2009. A tetrahedron of the third kind? At first I assumed that the fishy/animalic John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 3

tetrahedron lost the circling measure in the top line, as I presumed that the two points would now coincide (big mistake). This would give the special case of step measure, in the top, line while retaining a growth measure in the lower line (of the spine). I worked with this basic tetrahedron - initially. Could it be like this?. and with associated fields in the front and back planes The path curves turn out to be bilateral and as if they might describe a fish front profile. Or so I thought. But, try as I might, I could not make them work. These curves (in the plane) are given by Edwards in his Projective Geometry, 1985, p218,fig 154. Another model (3 rd kind?) So I thought to make a model (again!). Doubts By this time I had my doubts as to whether this architecture could work at all. So I had another closer look at the fish (never ignore the phenomena!). And lo! Koi (carp) The curves cut right across dorsal and ventral lines and did not try to avoid top and bottom edges. So they must be some kind of continuous spiral. Even if the fins intruded through this skin. Abandoned model Hence this form of the model was abandoned, but I did now have a conceptual model to try to work with. (The model ended up deteriorating in the garden ) For the measure or rhythm in the top line had to be a circling measure, rather than a step measure. I mention all this as it is important to see that it is not good to presume that what one initially thinks will fit the case will necessarily actually fit! Then I assumed that: That the fish profile, from the side, would match up with a single planar path curve and that curve was able to be given by the invariant triangle. That the spinal nodes were an approximation (in the middle) to a growth measure. That the front (and hence rear) views of the fish were of a bilateral symmetry. That this front profile was of the form of an ellipse (or close for some fish were obviously egg shaped). That the spiraling curves on the fish body formed by the nodes of the scale positions are three dimensional path curves. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 4

That these spirals (both clockwise and anti clockwise) would not be regular logarithmic spirals but a spiroid form of some sort (that is an asymmetric spiral). Would this architecture work for the carp? Testing, testing Every one of the assumptions made had gradually to be tested at least to a certain extent and for this one sample of the carp. For if it didn t sort of work here on just one beautiful fish, then there was little point in pursuing the hypothesis. It only had to work once to enable a start to be made on further work on the theory. Thawing out somewhat. The first thing was to take orthographic photographs of the beast thus to establish some concrete data. This involved a lot of fiddling and setting up I was not in the habit of photographing dead fish. And I somehow did all this without my wife knowing at all. I was a bit nervous how she would respond to this large creature in the freezer! In the freezer the body had bent somewhat and I had to allow it to unfreeze sufficiently to be able to straighten it out. And after messing around taking pictures in the hot sun in the backyard we could hardly eat it... Orthographic views early attempt These were my starting data as an engineering draftsman Does the invariant triangle fit? Now I had to check the side profile to see if it could even remotely respond to a path curve analysis. This took quite a few trials. A number of the assumptions came under scrutiny here. (see background). John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 5

Body envelope within an invariant triangle I had to assume the position of the spine was aligned with the lateral line. I had to assume the positions of the end points or foci (near head and near tail). I had to assume a position for the orthogonal top line (and was initially unsure whether it should even be above or below the creature) and marked as point P 3 /P 4 on the previous sketch. idea of the growth measure (also assumed) along the spine. I eventually found an envelope in which I believed the fish body would be contained (except head and tail of course). Some kind of invariant triangle What this was attempting to say was that the skin of the fish body profile was a tiny part of an entire path field spanning all space. And that it was the same kind of field as the invariant triangle in the plane as shown in the second talk but assuming (!), in this one case, an isosceles triangle. So many assumptions.! Body cross section The next step was to see what the front profile could be approximated to. Ellipse section front view This front view resolved itself into an approximate ellipse with a major axis of 76 mm and a minor axis of 48 mm. This would have to be the maximum cross sectional area. Evolutionary transforms? As an aside, I wondered whether the animals cross section would give a clue as to its chronological incorporation into our physical world. Fish sections vary a lot. But there may be a morphological sequence here suggesting a precise chronology (that is, non fortuitous). Circle, ellipse, egg and Early fish had, I understand, a rounder simple cross section even circular. This carp appeared nearly elliptical. A further fish I studied appeared as an egg form profile and a good one too. Then there were other fish with re-entrant aspects, the cow fish. Beyond Projective? John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 6

Was this last a geometry that belonged to something further than projective geometry as such and if so what was it? This was a big question for me as I could not see the morphology of the sentient creatures being limited to the projective, wonderful though this was. Here I hit a conceptual brick wall I knew I did not know enough math! Reverse engineer? Having now found two profiles to work with I thought to attempt a reverse engineering exercise to see if I could find a field of form (Edward s term) that married with the fish itself. My first rough layout sketch is shown here A more detailed improvement What kind of surface spirals? The next step was to see if I could find spiral forms in the planes through head and tail areas, and the top line which might generate the spiral curves on the skin surface. It was obvious that these forms could not be as straight forward as the plant forms regular logarithmic spirals. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 7

But what were they? Special spirals spiroids What was the spiral shape I was looking for? I thought it might be what I came to call a spiroid, that is, something like a spiral but not an equiangular spiral as in plant form fields. Now to test it. Projecting scale points The scale points were projected on to the front plane that slanted down from the upper line through the front of the fish (red dots highlighted with the purple triangles). Now the question was could my spiroid form concept match this empirical curve? Failure again! Try as I might I could not get a reasonable match. None of the spiroids (red circles) in the envelope of spiroids would pass through my red data points (red dots). Multiple trials (2009) I must have tried a dozen times.. Another idea Nothing worked even half reasonably. What was I missing? Dumbo. Then it occurred to me that the circling measure in the top line did not have to be based on rays from the point at equal angles. I had in fact been employing a very special case, and assuming it had to fit. But it would not. Spiroid field (red curves) The angles about the point did not have to be equal. And this was a case that Edwards had shown me years and years before and which had given a quite pretty picture. At that time I had no idea that this beautiful picture might actually be complicit in some natural architecture. Original spiroid drawing sometime last millenia John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 8

This was a case of a circling measure of points (in a line) and a circling measure of lines (in a point) but where the point was not related specially to the center of the circling measure of points (if that makes sense). 1/11/2009 So I tried this kind of spiroid except that now centers of the circling measures were limited to being perpendicular to each other. Still this was something of a special case too but allowing for a reflection symmetry. Symmetrical spiroid Now I took only a few data points at significant places at top, bottom and on (approximately) either side. My new spiroid would have to fit at least these few if the hypothesis was to be vaguely valid! The drawings were getting larger all the time! (Take complete original to Dornach?). If I remember rightly this worked first time! Back plane too? If this worked then so should the projections on to the back plane. Did they? If they did then it could mean that this funny peculiar but much more general spiroid may find a place in front and back planes and so help with understanding the skin surface of a ubiquitous natural form the fish. Those years ago I would never have even dreamt it could actually apply to the real world! Subsets of subsets This structure has a similarity to the plants basic field structure but is far more general encompassing a myriad more form fields due to the flexibility the fields and curves now have. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 9

This would mean that the set of forms that was applicable to the fish forms contained, as a subset, the forms of the plant world, just as the mineral forms seemed to be a subset of the plant forms. Intersecting spiroid cones 1/11/2009 The skin surface as the resultant of two intersecting spiroid cones. Forms through tangents and tangents are lines. Our thesis is still within the initial brief which asserted the primacy of the line. Form through tangent points Needless to say such a curve can be plotted via its lines and points and planes. This drawing (background) shows the pointwise construction. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 10

I did not get to the plane-wise which would be difficult anyway. A cognitive picture I was now closer to an expression of or so I thought the cognitive picture as it is expressed in Steiner s early philosophical/ spiritual work, i.e. his Philosophy of Freedom. For this says, to my understanding, that reality meets us when we truly unite concept and percept. But this only happens through a constant weaving between the two worlds which are initially separated for our current consciousness. This practice has to be the new Yoga, for it is a breathing between two kinds of Maya leading to an eventual resolution of the two initial illusory experiences. Is this the transformed Magi and Shepherd paths? Concept + Percept = Reality! Are we getting there even just a little way towards it? 3/11/2009 state of the art! Is this then the fishy tetrahedron? The curve shown John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 11

would be just one of thousands that sculpt the skin surface of the fish. These curves would also be part of the infinite field which surrounds the fish and in which it is embedded (so to say) and dare one suggest sustained and created in the first place. The fishes formative field would not be derivative but the fish body itself would be and embedded in its field. Fishy field of form!? Tetrahedron of the third kind? Was this then the tetrahedron of the third kind? Or at least was it at an early stage in its evolution? A single sample! But this was only one curve on one surface of one fish of one species. So many more questions arose How true to the whole form was this skin field how did it fade into head and limb structures, abrubtly or seemlessly or what? What of other species of fish? How to include the evident curvature in many if not most of the lateral lines in the fish population of the world? This strongly suggested there was a further and deeper step yet to be taken. However I thought to try other fish species. Mullet So it was back to the fish markets! Again I sought a fish with an inherent straightness and reasonably sized scales. I found the sleek Sea Mullet! John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 12

No lateral! I did the photography that would give me the orthographic views of this animal. It quickly became clear that this creature did not even have a lateral line! I cooked it, partially skinned it no sign of a lateral line. It still had a spine of course. Then I looked in the books. And I find stated that here was a species with no lateral line. I still thought an analysis was worth pursuing. So I attempted to get the orthographic views I needed Mullet orthographic views This was as far as I had got to. Trout Then came an offer from Simon Charter in England, as I had mentioned the trout as a possible candidate. He sent me some exquisite trout photos. These I attempted to analyse A thing of beauty Lateral line John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 13

An analysis This is where we had got to about this time last year. But more data was needed at least the trout had respectable and identifiable lateral line! Mystery of sentient form This seemed to be only a tiny step into the outer morphology of the sentient creature. John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 14

Any full animalic tetrahedron would have to include the curvatures in the spine this was clear from many a fish species, even if the carp (and triut) lateral line was largely straight. And what would bring about the head, rhythmic and tail form geometries which had well and truly begun to intervene with the very earliest of fish forms? For me this was a big research question how does soul intervene morphologically? Line or curve? If spines could curve was the line, the straight line, the fundament I thought it was? Was the curve the thing? Strings anyone?! Were spines really macro strings? What raw structure would give the basis for bird, mammal, reptile among many others? And then there is the human! And another leap of 90 degrees. What is the meaning of this insistent and repeated orthogonality? What other major step allows the human spirit entry into an earthly form? John Blackwood, 93 Warrane Road, Willoughby, Sydney, Australia. 02 9417 6046 jblckwd@ozemail.com.au, Morphology.org 15