Showing posts with label Geometry. Show all posts
Showing posts with label Geometry. Show all posts

Friday, November 28, 2025

Intimidated by Logic

"No one is intimidated by logic,
except logicians.
...
All our language is composed of brief little dreams; and the wonderful thing is that we sometimes make of them strangely accurate and marvelously reasonable thoughts. What should we be without the help of that which does not exist? Very little. And our unoccupied minds would languish if fables, mistaken notions, abstractions, beliefs, and monsters, hypotheses, and the so-called problems of metaphysics did not people with beings and objectless images our natural depths and darkness. Myths are the souls of our actions and our loves. We cannot act without moving towards a phantom. We can love only what we create.
...
You can never be too subtle,
and you can never be too simple."

Paul Valery (1871 - 1945)

Sunday, November 09, 2025

Electric Universe


"The watcher’s eyes are likely to swivel forward in a sequence of stately turns as the screen’s pixel glows: each quarter-ounce mass of eyeball tugged by six flat muscles, in a glissando slide within the slippery fat lining the orbital cavity. The eye blinks, the widened pupils are in position, and the incoming electromagnetic waves roar in. Ripping through the thin layer of the cornea, they decelerate slightly, with their outermost edges forming a nearly flat plane as they travel inward, carrying the as-yet-undetected signal from the screen deep into the waiting human. The waves continue through the liquid of the aqueous humor and on to the gaping hole of the pupil. The human may have squinted to avoid the glare, but human reflexes work at the rate of slow thousandths of a second and are no match for these racing intruders. The pupil is crossed without obstruction. The stiff lens just below focuses the incoming waves even more, sending them into the inland sea of the jellylike vitreous humor deeper down in the eye. A very few of the incoming electric waves explode against the organic molecules in their way, but most simply whirl through those soft biological barriers and continue straight down, piercing the innermost wrapping of the eyeball, till they reach the end-point of their journey: the fragile, stalklike projection from the living brain known as the retina. And deep inside there, in the dark, barely slowed from their original 670 million mph, the waves splatter into the ancient, moist blood vessels and cell membranes, and something unexpected happens. An electric current switches on."

- David Bodanis, Electric Universe

Friday, November 07, 2025

A Pattern is a Message


"Organism is opposed to chaos, to disintegration, to death, as message is to noise. To describe an organism, we do not try to specify each molecule in it, and catalogue it bit by bit, but rather to answer certain questions about it which reveal its pattern: a pattern which is more significant and less probable as the organism becomes, so to speak, more fully an organism.
...
We are but whirlpools in a river of ever-flowing water.
We are not stuff that abides, but patterns
that perpetuate themselves.
A pattern is a message...
...
...and may be transmitted as a message. How else do we employ our radio than to transmit patterns of sound, and our television set than to transmit patterns of light? It is amusing as well as instructive to consider what would happen if we were to transmit the whole pattern of the human body, of the human brain with its memories and cross connections, so that a hypothetical receiving instrument could re-embody these messages in appropriate matter."

Norbert Wiener (1894 - 1964)
The Human Use of Human Beings: Cybernetics and Society

Thursday, November 06, 2025

Geometry of Music


 "Music is the arithmetic of sounds
as optics is the geometry of light."

- Claude Debussy (1862 - 1918)

Tuesday, November 04, 2025

Cymatic Urphänomen


"She [nature] is the sole artist, creating extreme contrast out of the simplest material, the greatest perfection seemingly without effort, the most definite clarity always veiled with a touch of softness. Each of her works has its own being, each of her phenomena its separate idea, and yet all create a single whole."

Johann Wolfgang von Goethe (1749 - 1832)
 The Metamorphosis of Plants

"The notion of the Urphanomen is an invaluable illustration of the concrete nature of Goethe's way of thinking which dwells in the phenomenon. The primal phenomenon is not to be thought of as a generalization from observations, produced by abstracting from different instances something that is common to them. If this were the case, one would arrive at an abstracted unity with the dead quality of a lowest common denominator. For Goethe, the primal phenomenon was a concrete instance - what he called 'an instance worth a thousand, bearing all within itself.' In a moment of intuitive perception, the universal is seen within the particular, so that the particular instance is seen as a living manifestation of the universal. What is merely particular in one perspective is simultaneously universal in another way of seeing. In other words, the particular becomes symbolic of the universal."

Henri Bortoft (1938 - 2012)

Photographer's note. To help contextualize the relevance of the quotes, I need to point out that the image above contains three (out of a total of about 24) "snapshot" views of a single unfolding cymatic process. Not only does the triptych show only a tiny fraction of what my eye saw through the viewfinder, but the apparent sharpness of the images also belies the frenetic swirling and rhythmic thrashing of the water because my relatively slow 1/40th sec to 1/60th sec exposures smear over finely detailed patterns. But, while we may be unable to "see" the cymatic Urphanomen in its full splendor (and, even if we could, perhaps would understand as little as does Philip K Dick's "electric ant" after it tries seeing all of reality at once), "each of her works has its own being, each of her phenomena its separate idea, and yet all create a single whole." 

Friday, October 31, 2025

Symphonic Geometry


"The human mind is only capable of absorbing a few things at a time. We see what is taking place in front of us in the here and now, and cannot envisage simultaneously a succession of processes, no matter how integrated and complementary. Our faculties of perception are consequently limited even as regards fairly simple phenomena. The fate of a single man can be rich with significance, that of a few hundred less so, but the history of thousands and millions of men does not mean anything at all, in any adequate sense of the word. The symmetriad is a million—a billion, rather—raised to the power of N: it is incomprehensible. We pass through vast halls, each with a capacity of ten Kronecker units, and creep like so many ants clinging to the folds of breathing vaults and craning to watch the flight of soaring girders, opalescent in the glare of searchlights, and elastic domes which criss-cross and balance each other unerringly, the perfection of a moment, since everything here passes and fades. The essence of this architecture is movement synchronized towards a precise objective. We observe a fraction of the process, like hearing the vibration of a single string in an orchestra of supergiants. We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands and millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint. It has been described as a symphony in geometry, but we lack the ears to hear it."

Stanislaw Lem (1921 - 2006)

Thursday, October 30, 2025

Experience, Emptiness, and Luminosity


"One of the points in the traditional Buddhist way of viewing the question of what reality is or what truth is, is that in fact we cannot perceive reality, we cannot perceive truth. This is not to say that there is no reality or truth, but rather that whatever we perceive, if we happen to perceive anything, we see in accordance with some particular language or approach, and we color it with our own styles and ways of looking at things.
...
The nonexistence of ego is not a philosophical matter, but simply a matter of perception. Perception is unable to trace back its existence {to an origin}, so it becomes just sheer energy, without a beginner of the perception and without any substance. It is just simple perception. Perception on that level has three aspects.
...
The first is perception as experience. In this case, experience does not refer to the experience of self-confirmation, but to experience in the sense of things as they are. White is white, black is black, and so forth.
...
Then there is {the second aspect}, the perception of emptiness, which is the absence of things as they are. Things have their room; things always come along with a certain sense of room, of space. Even though they may appear within the complexities of the overcrowdedness of experience, they provide their own space within the overcrowdedness. Actually, overcrowdedness and room are the same thing; overcrowdedness is room in some sense. This is because there is movement involved, because there is dance and play involved. At the same time, there is a shifty and intangible quality, and because of that the whole thing is very lucid.
...
There is experience, then space or emptiness, and then the final aspect, which is called luminosity. Luminosity has nothing to do with bright visual light. It is a sense of sharp boundary and clarity. There is no theoretical or intellectual reference point for this, but in terms of ordinary experience, it is a sense of clarity, a sense of things being seen as they are, unmistakably.
...
So there are these three aspects of perception: the sense of experience, the sense of emptiness, and the sense of luminosity. The point is that with that level of perception {that contains the three aspects}, one is able to see all the patterns of one’s life. Whether the patterns of one’s life are regarded as neurotic or enlightened, one is able to see them all clearly. That seems to be the beginning of some glimpse of the mandala perspective, the beginning of a glimpse of the five buddha energies."

- Chogyam Trungpa (1939 - 1987)
Orderly Chaos: The Mandala Principle

Tuesday, October 28, 2025

Ouroborosian Complexity


"In the end, we self-perceiving, self-inventing, locked-in mirages are little miracles of self-reference. We believe in marbles that disintegrate when we search for them but that are as real as any genuine marble when we’re not looking for them. Our very nature is such as to prevent us from fully understanding its very nature. Poised midway between the unvisualizable cosmic vastness of curved spacetime and the dubious, shadowy flickerings of charged quanta, we human beings, more like rainbows and mirages than like raindrops or boulders, are unpredictable self-writing poems - vague, metaphorical, ambiguous, and sometimes exceedingly beautiful."

 -  Douglas R. Hofstadter (1945 - )
I Am a Strange Loop

Monday, October 27, 2025

Atomic Poetry


 "We must be clear that when it comes to atoms,
language can be used only as in poetry.
The poet, too, is not nearly so concerned with
describing facts as with creating images
and establishing mental connections."

Niels Bohr (1885 - 1962)

Tuesday, October 21, 2025

"To be" is to Inter-Be

"If you look deeply, you will see that there is a cloud floating in this sheet of paper. Without a cloud, there will be no rain; without rain, the trees cannot grow; and without trees, we cannot make paper. The cloud is essential for the paper to exist. If the cloud is not here, the sheet of paper cannot be here either. So we can say that the cloud and the paper inter-are. 'Interbeing' is a word that is not in the dictionary yet, but if we combine the prefix 'inter-' with the verb 'to be,' we have a new verb, inter-be.
...
If we look into this sheet of paper even more deeply, we can see the sunshine in it. If the sunshine is not there, the forest cannot grow. In fact, nothing can grow. Even we cannot grow without sunshine. And so, we know the sunshine is also in this sheet of paper. The paper and the sunshine inter-are. And if we continue to look, we can see the logger who cut the tree and brought it to the mill to be transformed into paper. And we see the wheat. We know the logger cannot exist without his daily bread, and therefore the wheat that became his bread is also in this sheet of paper. And the logger’s father and mother are in it too. When we look in this way, we see that without all these things, this sheet of paper cannot exist.
...
Looking even more deeply, we can see we are in it too. This is not difficult to see, because when you look at a sheet of paper, the sheet of paper is part of our perception. Your mind is in here and mine is also. So we can say that everything is in here with this sheet of paper. You cannot point out one thing that is not here – time, space, the earth, the rain, the minerals in the soil, the sunshine, the cloud, the river, the heat. Everything co-exists with this sheet of paper. That is why I think the word inter-be should be in the dictionary. 'To be' is to inter-be. You cannot just be by yourself alone. You have to inter-be with every other thing. This sheet of paper is, because everything else is."

- Thích Nhất Hạnh (1926 - 2022)
The Heart of Understanding

Postscript. As is likely obvious to even casual visitors to my humble blog during the last week, I am currently immersed in the world of cymatics (although my wife thinks of it as more of an obsession). "Cymatics" refers to the study of sound and vibration; specifically, when the two are combined in a way that creates complex patterns in different media such as sand, water, or - what Swiss physician Hans Jenny (who coined the term) liked to use - corn starch in water. For example, if a metal plate is covered with a thin layer of flour or sand and is made to vibrate at specific frequencies, so-called standing-wave Chladni patterns appear in which the flour or sand collects along "nodes" (i.e., lines or points that undergo minimal vibration), while areas that undergo the greatest motion (i.e., the "anti nodes") are effectively left empty as the vibration pushes the flour or sand away. Chladni patterns are named after physicist Ernst Chladni who performed the first such experiments in the late 1700s.

I thank my left-brain/day-job as a physicist for introducing me to cymatics in the late 1990s when I stumbled across this paper describing what (at the time, were never before seriously studied) complex emergent patterns in vibrating layers of small granular media (e.g., cylinders filled with BBs from a toy shotgun). One line from this paper immediately grabbed my attention when I first read it and that still haunts me (both as physicist and photographer): "These excitations [called "oscillons"] have a propensity to assemble into 'molecular' and 'crystalline' structures." I remember musing, Whoa!, pump energy into an otherwise static structureless pile of 'things' and get self-organized geometric patterns?!? It was during my (absurdly slow, pre-Google days) search for related experiments that I discovered Jenny's work on cymatics, the underlying dynamics of which has a far-from-superficial overlap with the physics of oscillons.

So, having known about cymatics for about 25 years - and having even posted about it briefly in 2006 on this same blog! - why has it taken me so long to photograph it? I have no easy answer to that, just as I cannot explain why I never photographed my dad-working-as-an-art-restorer when he was still alive, which is something that - 23 years after his passing - I now profoundly regret (see Postscript 1 in this post from 2010). The creative process and the muse that guides our path are both mysterious and ultimately unknowable, which is as it should be. So, I'll leave it at that. But, whatever the reason(s) for my flurry of recent purchases of frequency generators, vertical vibration generators, lights, and more plates, goblets and petri dishes than any sane photographer has reason to own (and our kitchen cupboards have room for), I am - at the moment (and for the foreseeable future) - completely and utterly "obsessed" with cymatics. The reason is simple enough to state: cymatics is a quintessentially perfect amalgam of all three of my aesthetic and intellectual passions - physics, photography, and mysticism

The first two separately play obviously critical roles. The physicist-side of my brain is giddy over the vast phase space waiting to be explored: vibration frequency + medium (type + mix type) + vessel (type + diameter + depth) + ... And the photographer side is not too far behind: light (type + source(s) + directionality) + angle-of-view + f-stop + exposure time + ... But it is the idea of "cymatics as creative bridge" between seen and unseen, between energy and pattern, and between physical and spiritual that I resonate most deeply with, and is most ripe with creative possibilities. (For example, it has not escaped my attention that, in a "mystical" sort of way, the energy that the universe ineffably pumps into an otherwise structureless bag full of 'elemental things' gives rise to an emergent multidimensional dynamic cymatic-like sentient geometry called "Andy")

Since I've only started exploring the cymatics-scape universe, I have no idea what patterns await to be discovered and/or how long the search will keep my interest. But, given that I'm still looking for synesth-scapes after being mesmerized by reflective patterns in my mother-in-law's Nambe-like metal salt and pepper shakers in 2009,  cymatics may take a while 😊

For those still reading this, here is a link to a newly revised version of Hans Jenny's opus, Cymatics: A Study of Wave Phenomenon and Vibration. This version includes both volumes of the original work, as well as new chapters that include a biographical sketch of Hans Jenny, a non-technical primer on the physics of cymatics, and commentaries by researchers, sound therapists, designers, and artists. Indeed, I strongly recommend perusing the entire CymaticSource website, since it is a veritable storehouse of additional information, books and videos. (I am not affiliated with this website in any way. But, having recently purchased the aforementioned reprint of Jenny's revised Cymatics volume, I can personally attest to its quality - it is a stunningly beautiful book.)

Monday, October 20, 2025

Multitudinousness Minds


"Mahamati, since the ignorant and simple-minded, not knowing that the world is only something seen of the mind itself, cling to multitudinousness of external objects, cling to the notions of being and non-being, oneness and otherness, bothness and non-bothness, existence and non-existence, eternity and non-eternity, and think that they have a self-nature of their own, all of which rises from the discriminations of the mind and is perpetuated by habit-energy, and from which they are given over to false imagination. It is all like a mirage in which springs of water are seen as if they were real. "

- Lankavatara Sutra (c.350–400 CE)

Sunday, October 19, 2025

Near Symmetry of Nature



"So our problem is to explain where symmetry comes from. Why is nature so nearly symmetrical? No one has any idea why. The only thing we might suggest is something like this: There is a gate in Japan, a gate in Neiko, which is sometimes called by the Japanese the most beautiful gate in all Japan; it was built in a time when there was great influence from Chinese art. This gate is very elaborate, with lots of gables and beautiful carving and lots of columns and dragon heads and princes carved into the pillars, and so on. But when one looks closely he sees that in the elaborate and complex design along one of the pillars, one of the small design elements is carved upside down; otherwise the thing is completely symmetrical. If one asks why this is, the story is that it was carved upside down so that the gods will not be jealous of the perfection of man. So they purposely put an error in there, so that the gods would not be jealous and get angry with human beings. We might like to turn the idea around and think that the true explanation of the near symmetry of nature is this: that God made the laws only nearly symmetrical so that we should not be jealous of His perfection!"

Richard Feynman (1918 - 1988)

Friday, October 17, 2025

Surging and Ebbing


"All mass possesses inertia, all force tends to persist. Owing to this universal property and condition, a body, be it at rest or in motion, tends to remain in the same state, and a force, manifesting itself anywhere and through whatever cause, produces an equivalent opposing force, and as an absolute necessity of this it follows that every movement in nature must be rhythmical. It is borne out in everything we perceive—in the movement of a planet, in the surging and ebbing of the tide, in the reverberations of the air, the swinging of a pendulum, the oscillations of an electric current, and in the infinitely varied phenomena of organic life. Does not the whole of human life attest to it? Birth, growth, old age, and death of an individual, family, race, or nation, what is it all but a rhythm?  All life-manifestation, then, even in its most intricate form, as exemplified in man, however involved and inscrutable, is only a movement, to which the same general laws of movement which govern throughout the physical universe must be applicable."

Nikola Tesla (1856 - 1943)
The Problem of Increasing Human Energy

Thursday, October 16, 2025

The Unimaginable Universe


"On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realized that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a back yard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand...
...
I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet...
...
I saw a Persian astrolabe; ... I saw the rotted dust and bones ... I saw the circulation of my own dark blood; ... I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe."

Jorge Luis Borges (1899 - 1986)
The Aleph

Monday, August 18, 2025

An Unknowable Unfolding


"How did the universe get from matter to mattering?
...
Life comes from life," declared Pasteur.
But, if so, whence life in the first place?
...
The biosphere explodes in diversity,
creating more and more cracks in the
floor of Darwin’s nature until the cracks,
ever expanding, become the
very floor of nature,
and nature herself.
...
The ultimate, an unknowable unfolding, slips
its foundational moorings and floats free.
...
Agency introduces meaning into the world!"

Thursday, August 07, 2025

Language of Mathematics


"Something becomes objective (as opposed to 'subjective') as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics."

- Armand Borel (1923 - 2003)

Friday, July 11, 2025

Living Geometry


"We apprehend Him in the alternate voids and fullness of a cathedral; in the space that separates the salient features of a painting; in the living geometry of a flower, a seashell, an animal; in the pauses and intervals between the notes of music, in their differences of tone and sonority; and finally on the planes of conduct, in love and gentleness, the confidence and humility, which give beauty to the relationships between human beings."

Aldous Huxley (1894 - 1963)

Tuesday, July 08, 2025

Boundless Ambiguity


Paradox Museum, Miami, Florida

"The writing lies before you and always says the same, if you believe in words. But if you believe in things in whose places only words stand, you never come to the end. And yet you must go an endless road, since life flows not only down a finite path but also an infinite one. But the unbounded makes you anxious since the unbounded is fearful and your humanity rebels against it. Consequently you seek limits and restraints so that you do not lose yourself tumbling into infinity Restraint becomes imperative for you. You cry out for the word which has one meaning and no other, so that you escape boundless ambiguity. The word becomes your God, since it protects you from the countless possibilities of interpretation. The word is protective magic against the daimons of the unending, which tear at your soul and want to scatter you to the winds. You are saved if you can say at last: that is that and only that. You speak the magic word, and the limitless is finally banished. Because of that men seek and make words."

C. G. Jung (1875-1961)
The Red Book: A Reader's Edition

Wednesday, June 11, 2025

Creating the World


"We create the world that we perceive, not because there is no reality outside our heads, but because we select and edit the reality we see to conform to our beliefs about what sort of world we live in. The man who believes that the resources of the world are infinite, for example, or that if something is good for you then the more of it the better, will not be able to see his errors, because he will not look for evidence of them. For a man to change the basic beliefs that determine his perception - his epistemological premises - he must first become aware that reality is not necessarily as he believes it to be. Sometimes the dissonance between reality and false beliefs reaches a point when it becomes impossible to avoid the awareness that the world no longer makes sense. Only then is it possible for the mind to consider radically different ideas and perceptions."

Gregory Bateson (1904 - 1980)
 Steps to an Ecology of Mind

Thursday, May 22, 2025

Photographs-Otherwise-Not-Taken, Taken

Inside of Library, University of Otago, Dunedin, New Zealand

"Nows within this now, rather like snapshots in an album. Each Now is separate and a world unto itself, but the richly structured Nows 'know' about one another because they literally contain one another in certain essential respects. As consciousness surveys many things at once in one Now, it is simultaneously present, at least in part, in other Nows. This awareness of many things in one could well exist in a much more pronounced form in other places in Platonia."

- Julian  Barbour (1937 - )
 The End of Time

Note. The admittedly busy title of this blog post obviously begs an explanation. I'll start by saying that it is inspired by a short email exchange I recently had with a photo buddy of mine (the Zen-master, Paul Cotter). In reply to Paul's kind comments about my recent "travelogue images," I countered with the suggestion that my favorite images from the trip are/may-be those I took with my iPhone and not my 21L-sling-bag's-worth of "pro" gear (the details of which hardly matter)! While I am not (entirely) convinced of the veracity of my claim (and others may differ), I have zero doubt that my iPhone gifted me many images that I will cherish in the years to come precisely because these are photographs I would otherwise have not taken! Some examples - click to see full-size:

View of a wall while waiting to be seated at a restaurant

Footprints on a beach in front of another restaurant


A view from inside the Novotel Auckland Airport
while my wife was busy getting us checked in

Frosted window inside restroom at the
Aoraki/Mount Cook National Park Visitor Centre

View inside a restaurant while being led to our table

Upside down view of one of the ceilings at the
Nadi International Airport in Fiji

Another (upside down) view from inside the library at
the 
University of OtagoDunedin, New Zealand
 
A snapshot view of urban geometry while waiting
for my wife to pay the parking meter


A 5 sec exposure of a part of our boat ride to Milford Sound,
stabilized by my iPhone's computational photography algorithms

I have dozens more of these "Photographs-Otherwise-Not-Taken, Taken" images, all of which share this one salient pattern: had I not used my iPhone to capture them (embarrassingly easily by, literally, framing and tapping, and without any of what my wife describes as "glacier-paced compositional machinations"), they would all have been but fleeting moments doomed to be lost in the mists of memory and time.