The word "pattern" in everyday parlance often means something typical that can be repeated, something characteristically capable of repetition into similar copies (Ulf Grenander, Elements of Pattern Theory, Johns Hopkins University Press, Baltimore, 1996, p. 94). For example, in nature one can observe myriads of patterns in plant leaves or in beehives or in branches of trees or in the honeycomb-like shapes in dragonfly wings. Again, for example, in nature definite patterns can be observed in the daily life of animals: waking at sunrise time, finding water, hunting for food, socializing, and going to sleep at sunset time. In each case, there are observable patterns. So then the question arises as to whether the patterns in human geometric designs simply reflect patterns found in nature. Or whether patterns in human designs are somehow different in noticeable ways.
For an example of a human geometric pattern, consider a pattern in a Penrose tiling. A penrose tiling (inspired by Roger Penrose) is a non-periodic tiling generated by an aperiodic set of prototiles. For a sample Penrose tiling, see the attached image. A Penrose pattern is self-similar, so the same patterns are repeated in larger and larger scales. Such patterns, introduced by the British physicist and cosmologist, are examples of periodic tilings. M.C. Escher, the Dutch artist, is famous for pictures of periodic tilings with shapes that resemble living things.
Humans are part of nature too; in that sense, what we make is also part of nature.
Our technology is more advanced than e.g. the technology of bees, and that allows us to create more exact patterns. The Penrose tiling has less variability in the tiles than the cells of a beehive; as a concept the tiling is even 'exact', but when it is physically made/printed there will also be some variability between the tiles.
Many human designs are inspired by what we see in nature, but some designs come from thought, fantasy and mathematics. On earth, only humans can create a symphony on "Euler's identity".
Based on “Fractal geometry” we can say that patterns in human designs are a mirror of patterns found in nature.
Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Nature is rough, and until very recently this roughness was impossible to measure. The discovery of fractal geometry has made it possible to mathematically explore the kinds of rough irregularities that exist in nature.
Fractal geometry has permeated many area of science, such as astrophysics, biological sciences, and has become one of the most important techniques in computer graphics. For example, fractal patterns have appeared in almost all of the physiological processes within our bodies.
This new way of viewing our surroundings, this new perception of reality, has since led to a number of remarkable discoveries about the worlds of nature and man, and has shown that they are not as disconnected as once thought.
Lambert,
Although, we are part of nature, it is possible for us to introduce patterns that have no counterpart in nature. See, for example, the flying horses pattern in the drawing by M.C. Escher.
The bees could say the same: there is no counterpart of our hives in nature!
This could be said for every phenomenon in nature (e.g. object, event, living being) accepting that every phenomenon is unique in physical expression in at least one scale of analysis or perception
I think that the patterns found in nature are less than the patterns made by humans, since in nature there might be a finite number of patterns, whereas humans can construct for given n patterns a n+1 one!
As for the arguments that we are part of nature, this is true, but it is also true that there are patterns made by humans that cannot be found in nature-humans. For references etc. see http://en.wikipedia.org/wiki/Tessellation
I wonder whether aliens with far superior technology would ask the same question : their creations versus earthly 'nature'; making no distinction between human creativity and the creativity of other earthly animals.
Dear C. Lewis Kausel,
Many thanks for your observations. I am reminded of what you wrote in the attached paper about architecture, especially that the fact that designs of buildings appear to acquire extraordinary meaning and such designs inspire strong attachment in every culture. See the attached pdf file.
The valid points made by Lambert Zijp about beehives aliens can be supplemented with a consideration of design patterns that convey meaning to humans but not necessarily to non-humans. The patterns found in works of art and architectural works are imbued with meaning for human viewers that goes beyond the geometry of such works.
Perhaps the following story will make more clear the point about the meaning of architectural design patterns in the attached article. I remember the first time I projected a movie on a large screen when my dog (a yellow Labrador retriever) was in the room. Some scenes in this movie included a barking dog and my dog walked up to the screen, sniffed the image of the dog and then lost interest in what he was seeing. For my dog, the dog in the movie had no scent and so was not considered real, even though the dog in the movie was running, jumping, and loudly barking. This suggests that humans find meaning in objects that non-humans do not find.
Dear James,
"This suggests that humans find meaning in objects that non-humans do not find".
Yes, but sometimes humans find meanings in objects that other humans do not find.
Moreover, artists may create a piece of art with an intended double meaning, one of which is made obvious for everyone while the other requires additional knowledge.
Finally, the object itself is not always required. Sometimes, humans can find meanings in a simple contour of that object. This is the case, for example, in shadow puppetry.
I think that human mind based on Mathematical arguments can produce more patterns than those permittable for our local universe (our universe realization, a subset of all possible universe possibilities). So, by accepting (me) the above axiom, I can accept the fact that we can 'smell' more patterns than those that can be found in our universe.
Yes Marcel, but we can build the concept of a linear space, which dogs cannot do.
Probably the animals can 'feel' more patterns, but this does not answer the initial question: Although we can recognize a subset of all patterns of our local uiverse we can imagine mathematically how all the patterns of every possible universe could be.
Quote Demetris : "we can imagine mathematically how all the patterns of every possible universe could be."
I think you are grossly overestimating human imagination. Also, I don't see what mathematics has to do with for example the horse tiling of Esher (picture uploaded by James Peters).
Even IF every pattern can be described mathematically, you assume that current state of the art mathematics is sufficient; that there will be no further developments in mathematics, giving rise to new patterns.
*No Lambert, I mean as Mathematics evolve we will be able to mathematise more and more patterns, thus converging to a better level and so on... see it as a dynamic evolution process, not a static one... I don't know details for the current situation, other people are more experts in pattern theories than me.
*Marcel the term ALL has to be treated as a limit after a great elapsed time from now...
I think we are a long way from having found all patterns in the chaos around us.
A vortex street:
@Lambert: You said "Also, I don't see what mathematics has to do with for example the horse tiling of Esher (picture uploaded by James Peters)." I think You should learn that tessellations are just group theory! See fro example:
Jose Maria Montesinos Classical Tesselations and Three Manifolds
Jinny Beyer Designing tessellations. The secrets of interlocking patterns
Wilhelm Magnus] Noneuclidean Tesselations and Their Groups.
@Costas : Yes, I am aware of tessellation being part of mathematics. But describing the appearance of a horse is not easily done with mathematics.
@Lambert: I do not know which "appearance of a horse" you mean. In Escher's drawing it is just the basic tile!
@Costas: I meant Pegasus instead of a horse. It has manes, feathers, eyes...
Even when just regarded as a basic tile (without further details), the Pegasus outline is not easily captured in a mathematical formula. If you think otherwise, I look forward to seeing your mathematical Esher formula soon in this thread ;-) !
The mathematical formula of Pegasus will turn out to be a beast too :-).
@Demetris Christopoulos: ...we can imagine mathematically how all the patterns of every possible universe could be.
There is a way to do what Demetris has suggested, starting the Escher tiling that Lambert has kindly posted again (from my previous post). To do this, consider 2D set patterns. A 2D set pattern is a collection of sets of points in the Euclidean plane such that the members of the collection have common properties. For example, let A be the set of points along the edges of a single Pegasus horse (from Lambert's post). Let X be a set points in an Escher tiling containing a repetition of a Pegasus. Then the collection
\[
P(A) = \{B \subset X: B resembles A\}.
\]
In other words, a single Pegasus generates a set pattern. Since the number of Pegasus is unbounded in the Euclidean plane, the Pegasus set pattern contains an unbounded number of copies of the Pegasus represented by the set A. So it is possible to talk about all set patterns in the Euclidean plane (and there are lots of those). Another example of a 2D set pattern is shown in the attached image showing the edges of a dragonfly wing. In this case, the set M is the pattern generator. Then this edge set pattern contains all sets of points such that each set of points in the pattern contains at least one point with the same gradient orientation as the points in the set M.
@Lambert: You have a distorted image of what mathematics is. Surely does not only deal with formulas of some real or imaginary entity. Furthermore, algebraic geometry which deals with solutions of polynomial equations, we are not sure what is that we are talking about! What represent a variety? May be be the flame of Heraclitus! Any way, if I cannot provide a formula for your pegasus, that does not mean that the problem is with mathematics. I would rather say that that problem might not be well defined. Using triangle approximations I can approximate your image as close as I want.
Quote Costas: "You have a distorted image of what mathematics is".
Why do you slap me in the face? You might even be right, but YOU lack evidence to base that qualification on. Or is it just because I said that the shape of a horse/Pegasus is not EASILY captured in a mathematical formula?
You write "I can approximate your image as close as I want". OK, do it, professor!
And I want it pretty close and 'elegant'. You can use any polyhedron you like, but I bet a mathematical description of the outline of Pegasus will turn out to be a beast too...
So, no more chit-chat, now it is hands-on. I'm not a mathematician, but I will take on this challenge. That is, if YOU put the effort, I will too. May the most elegant solution win...
P.S. Upvotes of your insult are not mine, nor will any downvotes be, because I don't use the downvote feature.
Dear Lambert. First of all I did not have the intention to "slap you on the face"! Many people may have distorted image of what is mathematics. See the thread on "is mathematics invented or discovered. Now let me focus on "the slap on the face". Mathematics, in my opinion, deals essentially with "mathematical concepts". If some concepts happens to be expressed by a formula, it is OK with me and with all. But not all concepts can be expressed as a formula. I may have "the concept of your horse" without any formula about it (This is your distortion, you believe that in mathematics every concept is expressible by a formula). Any way if my opinion offended you, I apologise, I did not have this intension!
As for the triangulation of "your horse" this is a problem of computational algebraic geometry, and they are many interesting algorithms for that. Personally I do not have such experience, so "Professor cannot solve your problem!)
Dear professor Costas, thank you for your warm reply.
I will respond to some issues raised, tomorrow, but for now (since we are living in the same time zone): night night!
Lambert
Dear Costas,
So the 'horse' is a concept, and thereby part of mathematics. True, I fail to see that; my distortion as you call it.
Mathematics is not the first discipline that comes to my mind when I think of a horse; I would think of biology or art.
The 'concept' of the cup of coffee in front of me also has little to do with mathematics; more with cultural anthropology, physics or even culinary art.
Yes, the shape can be tessellated. One could also do frequency analysis of the image, regard the image as a matrix, compute principle components. In fact, since the image was jpeg compressed, some mathematics has already been applied: discrete cosine transform.
But just regarding the horse as a 'concept', or formalizing it somewhat like what James did (subset A belongs to the horse and subset B does not), is not really helpful. It does not give any clue about what it looks like (shape / color) nor about its meaning (what is a horse / why does it have wings).
Surface distortion by a floating paperclip:
Dear Lambert, I understand what you mean. I would like to generalise a little of bit in order to all satisfied. I would like to consider what is an object in category theory. In category theory an object (conceptual or real) is determined by using the Yoneda Lemma: If we know all the ingoing or outgoing arrows to an object then we know the object! For example, suppose you would like to define the life object "Drossos". If you try to analyse me then what you might achieve is a non-lived form of "drossos". If we are working in Biology this is not enough! If we apply Yoneda's Lemma and know e.g all ingoing arrows to "Drossos" , e.g. what effect the environment, the family, the educational system, etc. (all these are regarded as ingoing arrows) had on "Drossos", then we can know the object "Drossos". If we take you "horse" then every arrow
f:T--->H, is variable element of your horse, e.g. a curve. If we consider all such curves then the "picture of your horse" will be revealt! as you can see I did not use any "formula". Similarly you can do triangulations. Since I do not know such algorithms, i will attached the triangulation of Beethoven instead":
Hi Costas,
but this is just the visible surface. What if all internal biology and its dynamics is added to mathematically describe the 'living' Beethoven...
By the way, the picture of Beethoven is not of a surface triangulation, but of a tessellation with quadrilaterals.
Sorry to have high jacked your thread James; I go back to topic now.
Costas remarked that the number of patterns found in nature is less the number of patterns made by humans, because the universe is finite, and humans can construct from pattern N, a new pattern N+1.
I instinctively agreed, but now that I think about it...
Lets assume a pattern does not have to be physically realized in order to exist. It is enough when a person thinks of a pattern, because it exists in her/his head, maybe only for a fraction of a second.
But does a pattern that CAN be thought also exist? No, it first has to be thought.
How many patterns can the human population think of per second. I don't know, but I'm pretty sure there are more patterns in finite nature.
First Lambert: Beethoven is not tesselations! It is rather "triangulations" even if you use some quadruples. For the tessellation you have to pinpoint the basic tile!
About patterns: Humans can combine patterns to make a new one. Nature cannot do this!
For Marcel: This picture is just the form of the surface of Beethoven. If you want a living
Beethoven. then you have to find all the incoming arrows or the outgoing or properties. For example he was deaf, etc. These attributes describe the living Beethoven! That is why Category Theory should be important for Biology and living entities. I devote a song to Lambert:
@Costas : Thank you for correcting me regarding the use of the word 'tessellation'.
But that song...?
"Falling in errors is what's wrong with me... you must find me severe punishment..." etc.
Why don't you just say it in plain English instead of hiding behind Greek lyrics?
You say: "Humans can combine patterns to make a new one. Nature cannot do this!".
Throw a pebble in a pond and watch the pattern. Now throw TWO stones: you get an interference pattern. A combination of patterns resulting in a new one.
But the crux is in the word CAN. Humans CAN create an infinite number of patterns, but humans do not do so. In fact, humans can NOT create an infinite number of patterns, because their lives are not infinite, and creating a pattern takes time.
@C. Lewis Kausel : In terms of variety, I think you are right.
@Lambert: OK, you have a talent to distort everything! I send you the song as a joyful way to summarise the story here in RG. I am not hiding anything by sending the song!
Even that is for you something bud! But let me remind what happened:
You started by saying that Escher drawing does not have any relation to mathematics! The I mention that it has, and it is just a tessellation. You answer, that you new about tessellations. Then you describe Beethoven picture as tessellation,
There is also your view that mathematics consist of entities that can be expressed by "formulas", without even say what kind of formulas you are talking about! A formula of a formal language? Just an ordinary mathematical formula?. The I send to you that this is, in my opinion, a distorted image of mathematics. You answer to me, that I "slap you on the face"!. I do not want to go on with these thinks. It seems to me that we do not have a common ground to achieve a human communication. So please let stop here. further discussion does not add something positive to the RG discussion.
Dear Lambert, what do you drink and you do not share it with us? Please share your favorite 'finding' at least with the descendants of Dionysus... We shall certainly understand you!
@Demetris : Well I will not drink Ouzo for a while :-). But I think it better to rest the case and go back to topic. Probably the irritations would not have occurred when Costas and I could see each others face and hear each others voice. I have seen it before, e-mailing (forum postings) can escalate because humor can be misunderstood and words are weighed on a gold-scale. Only sticks and stones can hurt me, not a 'concept'.
Can someone come up with examples where it is not easy to determine whether the pattern was created by humans?
I hope I don't step on yet another person's toes by posting 'Lavender Mist' by Jack the Dripper :
Hello friends, in Malaysia and Indonesia, we like designs on fabric, and we call them Batik. Some of the patterns are obviously from nature: flowers, leaves but with some artistic parts added-on. Some are free flowing arrays of color combinations that resemble our dramatic sunsets. Batik is characteristic of the beauty found in the countries where it is produced.
Sent from mobile.
Lambert,
Here is another lavender mist image that combines nature and art.
@James : Actually, your lavender is a subset of mine; I found it! (see below). If I would run image registration software that I have at my disposal, it would immediately get stuck in a local minimum, yet my eyes found it within a minute!
Can you actually see that it is made by a human, or might it as well be a piece of marble/granite or a microscopic crystallographic image?
@C. Lewis Kausel : In another of James' threads, I wrote that (to me) a pattern is anything but randomness. In other words, can some signal be found amongst arbitrary values. In that sense, almost everything you see and hear are patterns. I think James is restricting patterns to signals that have some kind of repetitiveness in them, but let's ask him:
James, is 'Lavender Mist' (no repetition) a pattern?
@Lambert and C. Lewis Kausel,
You have raised a crucial issue: What is a pattern? There is more than one answer, if we take into account pattern formation in nature. Here are a few approaches to answering the question:
> Artist pattern: repeated decorative design. Here are a few examples:
--places of worship such as Notre Dame or mosques.
--Lavender mist (seeming disorder with underlying order provided by overlapping edges and merging colours).
--paintings with repeated edges.
> Natural patterns: any recognizable order. Here are examples:
--ocean waves rolling up a shoreline.
--refraction of white light by water molecules on a rainy or misty afternoon.
--plant leaves.
> Mathematical patterns: any set of numbers or objects or things that related to each other by some rule or rules. Here are some examples:
--Penrose pattern Rule: repeated prototiles.
--Fibonacci sequence rule: starting with 1, 1, each new member of the sequence is the sum of the preceding to two numbers to obtain 1, 1, 2, 3, 5, 8, ...
--Shape pattern rule: repetition of shape shadows and shapes (see, for example, the attached goblet shape pattern).
@Lambert and C. Lewis Kausel,
The goblet pattern in the previous post also conforms to the idea of an artistic pattern with the repetition of a decorative design (every goblet holds liquid).
And there is quire a bit more that can be said about natural patterns. Here is a quote from R.P. Feynman that gets at the heart of natural patterns:
Nature uses only the longest threads to weave her patterns, so each small piece of the fabric reveals the organization of the whole tapestry.
For a deeper view of natural patterns provided by Feynman, see the attached pdf file.
@James
That pdf does not seem to be written by Richard Feynman, a pity...
How do you know that the goblets hold liquid? It seems all Ouzo has evaporated.
Lambert,
Many thanks for pointing out the need (my omission!) to mention the author of the paper that quotes Feynmen, namely,
J.A. Scott Kelso, Dynamic Patterns. The Self Organization of Brain and Behaviour, MIT Press, 1997.
Lambert,
Good question. And the answer leads to a "goblet pattern". One can recognize a goblet pattern by its repeated design that includes an ornamental, elongated cup-shaped container that holds some liquid. At this point, some chilled wine wine would welcome. For example of a goblet pattern, see the attached image.
In answer to your question, if a goblet belongs to a so-called goblet pattern, then that goblet holds some liquid. If you agree that the goblets ln the previous post belong to a goblet pattern, then those goblets hold lilquid. Cheers!
@C. Lewis Kausel,
Mosaics in ancient shrines exhibit beautiful tiling patterns. Examples artistic tiling patterns can be found in the mosaics in the holy shrine in Mashad, Iran (see attached image).
Cheers to you James!
I fail to see liquid in the golden goblets too... You post very low-resolution pictures; maybe that's the problem. Or do I not grasp some metaphor?
Lambert,
Given a goblet pattern, each goblet in such a pattern holds liquid. But that does not mean that a goblet in a such a pattern has liquid in it. A goblet that does not hold liquid cannot belong to a goblet pattern (by definition).
We can also imagine a leaky goblet pattern: the repeated design in a leaky goblet pattern consist of (fancy :-)) but leaky goblets that do not hold liquid (for long).
Cheers.
@C. Lewis Kausel and Cj Nev,
I am guessing that the issue about human nature being separated from nature grew out of the discussion about some patterns designed by humans which have no counterparts in nature. But I do think anyone has contended that human nature is separated from nature.
The room allocation scheme in a Hilbert hotel is an example. A whimsical view of a Hilbert hotel is shown in the attached image.
Dear C. Lewis Kausel,
Please give an example (or examples) of industrially-made forms.
@C. Lewis Kausel,
A good example of a form (metalworks made) is the base of the tower shown in the attached image. The base has splendid symmetry, sloping edges, and an upward spiral (sort of a metal tree).
@ James:
You write:
- One can recognize a goblet pattern by its repeated design that includes an ornamental, elongated cup-shaped container that holds some liquid.
- ...if a goblet belongs to a so-called goblet pattern, then that goblet holds some liquid. If you agree that the goblets in the previous post belong to a goblet pattern, then those goblets hold liquid.
- Given a goblet pattern, each goblet in such a pattern holds liquid. But that does not mean that a goblet in a such a pattern has liquid in it. A goblet that does not hold liquid cannot belong to a goblet pattern (by definition).
Where is the liquid in the two goblet arrangements?
Are you joking? Is this religious talk? A free-mason secret thingy maybe?
'Google images' was not helpful : hardly any goblet arrangements with liquid in them.
Master James, please enlighten this grasshopper.
@C. Lewis Kausel
Yes, also industrially made objects fall down and disintegrate eventually, i.e. they find their equilibrium and become chaos.
Sad, isn't it. I'm afraid art becomes trash too if not 'repaired' once in a while.
As to cell-phones, I don't have one and try to stay away from them. They intrude my freedom.
Ergonomic? Seems like it when I see people punch elaborate SMSs with just their thumb. But for me, punching three times to get the letter I want is frustrating. And a small screen. And another GUI when you switch trademark... yuck!
This is an example of batik for ladies. Generally, Malaysians are quite creative and artistic.
Miranda,
Would you say that batik design patterns mimic patterns in nature?
Yes James, it's a bit abstract. But basically it's based on flowers, leaves, sunsets, water falls, water 'flows', streams.....
@Cj, thanks for supporting our clothing industry :)
Miranda,
Here is another example of a batik design pattern that imitates nature to some extent in terms of horses, seemingly galloping in the wind.
Miranda,
Perhaps you will be interested in the attached study of "mbatik" (motifs in batik design patterns). Apparently, there is some justification in viewing batik motif generation in the context of fractals.
Dear friends, It's Beautiful. Even though, I'm not myself and dazed by so much work today, I can still see the beauty. This is the Year of the Horse, according to the lunar calendar.
Our previous Prime Minister, Dr Mahathir, loves horses. Because his name 'Ma' means horse :)
Miranda,
Many thanks for your observations. It is very nice that the horses motif for the batik design pattern in my previous post dovetails with the lunar calendar.
Of course, m-batik design patterns belong to a class of patterns introduced by Prof. Ulf Grenander in the 1990s. The basic component of Grenander patterns is a pattern generator that is a set that is near every other set in the pattern. In the horses batik design pattern, a horse with a flowing main is the motif. Knowing that, one can imagine with the extension of horses batik pattern would look like.
Another example of a motif batik pattern is attached to this message. In this case, the butterfly is the motif that generates the batik pattern.
Dear Miranda, Lambert and C. Lewis Kausel,
Another example of a motif set pattern where the generator is the image of a horse (with a rider) comes from Escher. See the attached image. Perhaps it would be possible to use Escher's pattern as a model for a motif batik pattern. An attractive feature of such patterns is that they represent imaginative extensions of familiar things in nature.
Dear C. Lewis Kausel,
Yes, the illustration from your book is suggestive of a motif generated pattern, where the upper edges (profile) of cardinal's hat can be viewed as a generator of the archway edges in the vault above the cardinal's hat. The set of points along the edges of each archway have the same gradient orientation as the outer edges of the cardinal's hat.
Does this interpretation come close to your original idea in depicting environmental harmony?
Dear C. Lewis Kausel,
The design vocabulary is wonderful and quite appropriate.
@C. Lewis Kausel:...is this a pattern of nature?
In terms of the pattern that combines the shape of a Bishop's miter and structure of a Gothic vault in the attached image, it is possible to view the archways and the tall headress worn by the Bishop in three different ways:
1. art pattern: repetition of a design (in this case, curvature of mitre and archways).
2. nature pattern: recognizable order in the relation between mitre and archways.
3. both.
Lambert,
Returning to your earlier post showing a gold paperclip making waves (ripples) in a liquid. Please tell me more about this image: was the paperclip dropped in the liquid?
And what sort of camera was used?
In terms of our earlier discussion about patterns, it looks like the combination of the paperclip and the waves it causes, is a hybrid pattern:
1. natural pattern: recognisable order of the waves emanating out from the paperclip and the paperclip itself has a recognisable order.
2. mathematical pattern: repetition of the basic parts. If we consider the innermost wave as a set of points, then the other waves resemble the innermost wave. In that sense, the outer waves repeat the shape of the innermost wave.
Do you agree?
C. Lewis Kausel,
Your suggested view of historic structures as patterns of ingenuity is very interesting. You may find the following doctoral thesis interesting:
http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=16&ved=0CEAQFjAFOAo&url=http%3A%2F%2Fclok.uclan.ac.uk%2F8589%2F1%2FMark%2520Alan%2520Haydock%2520July09%2520the%2520influence%2520of%2520railways%2520on%2520the%2520devolpment%2520of%2520accrington%2520and%2520district%2C%25201848-1914%2520Degree%2520of%2520Doctor%2520of%2520Philosophy%2520unpublished%2520July09%2520University%2520of%2520Central%2520Lancashire%2520unknown274.pdf&ei=ZNkXU8DKFdGVrgHphICQAg&usg=AFQjCNHsT4ZqnWRFuBIwPk1iz-4AmUAcrQ&bvm=bv.62577051,d.aWM
Followers of this thread have expressed interest in natural patterns. So perhaps what E.H. Cohen and Q. Zaidi have to say about the salience of mirror symmetry in natural patterns will be of interest. See the attached paper.
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