I agree with Marcel that nature does not produce symmetry following 'mathematical rules of theoretical perfection'.

Not because there are imperfections in the symmetry (I consider those imperfections to be execution errors), but because there is no STRIVE to perfection in nature. It is a passive process: the less successful variations are weeded out. Of course that weeding out leads (after many generations) to well optimized survival machines, but I reject the notion of active strive towards perfection.

So why the (bilateral) symmetry? As a computer-programmer, I would say: re-use of identical code (in this case DNA), i.e. efficiency of coding. Why not tri-lateral or some other higher order of symmetry? No idea!

I'm sure Marcel (Centre d'Ecologie Fonctionnelle et Evolutive) can come up with some explanation ;-).

Dear James,

I don't think nature produced symmetry following 'mathematical rules of theoretical perfection'. For instance, the two sites of a human face are not perfectly symmetric to allow f. i. 3 D vision. What is your opinion about the patterns found in the eyes of insects or patterns found on the surface of butterfly wings? I think the two sites of an animal will always differ in detail, no perfect symmetry because of constraints in biological expression.

Very interesting. I'm no expert and I can not tell whether the natural pattern on giraffes for example refers to the focal point at infinity. But I believe to be contributing to remember that the artist Escher explored many of these patterns in their work.

The link below gives more details on the work of Escher

http://www.mcescher.com/

Dear Alexandre,

I would have started the Escher collection with a more colourful image.

Cheers

Dear Marcel,

You have raised a number of interesting questions (and issues).

For a Platonist such as Kurt Godel, one might view nature itself as a geometer, merrily painting its patterns on countless surfaces such as the body of a giraffe. The natural tessellation on the surface of a giraffe does appear to stretch toward infinitely small patches along the edges on, for example, the neck of a giraffe. The issue of "stretching to infinity" is more pronounced in the markings on the body of a zebra. In general, nature plays tricks with space-filling shapes that do not form a tessellation (spatial covering without overlapping shapes) but do fill space in an exotic way. For example, consider how the leaves of a tree fill space.

Your question about the patterns found in the eye of an insect or on a butterfly wing is excellent. Insect eyes are made up of 100s of tiny optical sensors and form a tessellation in 3D space. I am amazed at the structure of insect eyes, how they work. I would say that the markings on a butterfly wing fill a 3D space with similar adjacent shapes. Perhaps we can view what nature does as a form of 3D tessellation. Do you agree?

Dear Alexandre,

Yes, Escher was fascinated by covering the the plane with shapes that get smaller and smaller (toward the edges of a region) and tend toward infinity. The attached image shows a tiling (tending toward infinity at the edges) with a fish shape. For more examples, see

http://www.google.ca/search?q=images+infinity+%22M.C.+Escher%22&lr=&as_qdr=all&tbm=isch&imgil=W9hDDRVppCOonM%253A%253Bhttp%253A%252F%252Ft3.gstatic.com%252Fimages%253Fq%253Dtbn%253AANd9GcQdrgfALcp5sN39UpGGTRvy-e10IY7dnMGmjscT9b7Tju0KehoK%253B459%253B464%253BgOwwP_lMI4FrHM%253Bhttp%25253A%25252F%25252Fpirate.shu.edu%25252F~wachsmut%25252FWorkshops%25252FEscher%25252Finfinity%25252Finfinity.html&source=iu&usg=__eQN2doCqZAnHPEQ9RcJOHMeU9Is%3D&sa=X&ei=thKbU5GxEoOcyASOoYGIAg&ved=0CCYQ9QEwBA&biw=1210&bih=773#facrc=_&imgdii=_&imgrc=W9hDDRVppCOonM%253A%3BgOwwP_lMI4FrHM%3Bhttp%253A%252F%252Fpirate.shu.edu%252F~wachsmut%252FWorkshops%252FEscher%252Finfinity%252Fsquare-limit.jpg%3Bhttp%253A%252F%252Fpirate.shu.edu%252F~wachsmut%252FWorkshops%252FEscher%252Finfinity%252Finfinity.html%3B459%3B464

Deart James, I wonder if this self-similar pattern is inside the proximity theory or if it is closer to fractal theories, like Mandelbrot's, since when we have this property, then arise questions about fractal dimensions for example.

Dear James,

Good question!

It may not be the case that all symmetric or tiling works of humans do have natural counter parts at the moment. Most are made or imitated from looking at nature as Escher said : “I spent a large part of my time puzzling with animal shapes”, meaning his works probably were inspired by shapes and structures he saw from animals and others are made purely from imagination of a structure or using different transformations of 2-D and 3-D spaces on known geometric objects to create a new structure. It will then be the job of later societies to figure out what natural object that structure may represent or even to give a mathematical meaning to the work.

Creating such designs purely from imagination is like developing a mathematical theory of its own system of axioms and rules of deductions for consistency but not all such consistent structures of mathematics may indeed have real world model at the moment.

I agree with Marcel that nature does not produce symmetry following 'mathematical rules of theoretical perfection'.

Not because there are imperfections in the symmetry (I consider those imperfections to be execution errors), but because there is no STRIVE to perfection in nature. It is a passive process: the less successful variations are weeded out. Of course that weeding out leads (after many generations) to well optimized survival machines, but I reject the notion of active strive towards perfection.

So why the (bilateral) symmetry? As a computer-programmer, I would say: re-use of identical code (in this case DNA), i.e. efficiency of coding. Why not tri-lateral or some other higher order of symmetry? No idea!

I'm sure Marcel (Centre d'Ecologie Fonctionnelle et Evolutive) can come up with some explanation ;-).

Dear Demetris,

Self-similar patterns fit comfortably inside both proximity space theory as well as in fractal theories. From a proximity perspective, self-similar shapes are descriptively near near other. That is, if we include shape descriptors among the features of points in, for example, a tessellation such as Escher's square limit tiling, observe that there are edges with matching gradient orientations. Such edges are descriptively near each other. Again, for example, if we consider colours among the features of the points in the shapes in a tiling, then one looks for shapes containing points with matching colours.

Many of the coloured triangles in Escher's square limit are descriptively near each other.

In fact, a space containing points in a fractal pattern can be endowed with a metric, a pair of closure operators, a Efremovich proximity relation and a descriptive proximity relation. Then self-similar shapes in a fractal pattern constitute collections (families) of near sets.

@Dejenie A. Lakew: ...not all such consistent structures of mathematics may indeed have real world model at the moment.

Agreed! There are mamy examples of structures of such structures (e.g., lines in the plane vs. line segments in one of Escher's tilings) have no real world counterpart. Can you suggest some more examples of mathematical structures that have no real world counterpart?

Quote Marcel : "In nature, what is the function of 3D tessellation when only the surface of the three-dimensional shape is visible?"

You are right, it is a 2D tessellation of a 2D surface, embedded in 3D space.

Dear James,

Why creating aesthetic shapes in three dimensions (e.g. 3D tessellation as James mentioned) in nature when a living being only can scan or observe the surface of the 3D shapes?

Dear Lambert,

Perhaps the theoretical shapes not observed in nature on Earth exist on another planet (ca. 200 billion stars in our galaxy and 40 billion stars with a solar system)? Constraints on chemical design definitely does not allow any physical structure more or less expressed in the individual mind, either as an idea or a dream...

Lambert and Marcel,

There is a more general view of tessellation. That is, one can consider a tessellation of an n-dimensional space by a countable family of tiles. The tiles have non-intersecting interiors. Notice that the markings on a giraffe provide an example of a tessellation of R^3 (3D space). This is the case with the space-filling markings on a giraffe, even though we only can scan the surface of a 3D shape.

The more general view of tessellation in an n-dimensional space provides a source of models of physical structures such as grain structures in steel. For more about this, see

I.S. Praha, P.P. Zlin, Bernoulli cluster field: Vernonoi teessellations, App. of Math, 47 (2), 2002, 157-167:

http://dml.cz/bitstream/handle/10338.dmlcz/134492/AplMat_47-2002-2_7.pdf

I think you are mistaken,James. The markings on a giraffe provide an example of a tessellation of a curved surface (2D), embedded in 3D. Unless you want to take into consideration the length of the giraffe's hairs. Tessellation of 3D can only be done using 3D 'atoms', e.g. cubes and tetrahedrons; it cannot be done with 2D atoms like squares and triangles.

Quote Marcel: "Perhaps the theoretical shapes not observed in nature on Earth exist on another planet".

I cannot deny that. Even on earth not every being is bilaterally symmetrical: worms, plants, sea-stars. However, bilateralism is very common and I wonder why...

(I certainly wouldn't mind an extra hand!)

Lambert,

Agreed! The markings on a giraffe represent--stretching a point--an approximate 3D tessellation. I was looking at the photo, not taking into account the bumpy surface induced by the giraffe hairs. On the other hand, again stretching a point, a tile can be any thickness. Then, if the 3D tiles cover the surface without overlap, then we can talk about a tessellation. Do you agree?

Sorry James, I need to say NO again. The markings on a giraffe constitute a 2D tessellation. If one allows for hair-length, it would be a curved surface with a 'hair-length' thickness, and in that sense, a 3D tessellation. However, I very much doubt your 3D depth data (did you clip 2mm of their hair, in order to see what the pattern is like?).

So, for now the pattern data are 2D, unless you volunteer to shave the giraffes.

Lambert,

I agree that the markings covering parts of a giraffe do not constitute a 3D tessellation. If we what we see in the digital image, namely a plane surface partly covered by the markings, then it seems the markings do not constitute a 2D tessellation. I say this because the plane surface shown in the picture is not completely covered with the markings. If we consider on the region on one side of a giraffe (excluding the crown of the neck and the head) shown as a plane surface in the picture, then, Yes, we have a natural 2D tessellation of the region.

Marcel and Lambert,

Perhaps you will be interested in how actual 3D tessellations are put together. Here is am animation that shows the construction of a cubic tessellation:

http://www.tessellations.org/animation-foxy.shtml

Here is an aquarium fish tessellation:

http://www.tessellations.org/animation-aquarium.shtml

The problem with the fish tessellation is that the fish are flat.

Marcel and Lambert,

Here is an animated Penrose tiling that I just found. It offers examples of 3D constructions that are closer to the discussion that we have been having about giraffes:

http://www.youtube.com/watch?v=2ZVKY2BOU0k

Do you know of any other examples of 3D tessellations?

Take a look at the clip entitled

What things really exist?

This is an interview with Roger Penrose. He is arguing that there is a mathematical existence in addition physical existence and mental existence. A small part of the mathematical world that encompasses the physical world. This discussion is very close what Marcel mentioned about other planets.

Also, take a look at some of the animated constructions of 3D tessellations.

Dear James,

How does Penrose tiling work for four dimensions?

Aside from the visual geometric nature, is there any counterpart for the Penrose Tiling in terms of a computational approach? For example, given a structure is there any equation that will give me an affirmative answer: yes or no, if a structure is tillable? If I had to guess, there is a similarity with Difference Equation theory, because it is both iterative and computation.

Dear James,

If Optimal Space is for design is the purpose for repeated patterns, then what is the purpose of variations in pattern designs? I am curious, why not have square patterns on a round shaped object, is it because it's not optimal use of space?

@Tri Atran : How does Penrose tiling work for four dimensions?

Great question! I am guessing that a Penrose pattern can be viewed in the context of a finite-dimensional Hilbert space. Then the construction of a 4D Penrose pattern would be carried out (but not visualised) by definiting a homeomorphic mapping from a set of vectors in 3D Penrose pattern to a set of vectors in a 4D Hilbert space. I mention Hilbert space because it has some nice properties that make it amenable to what you want, namely, complete inner product space.

I just now found a video showing the construction of a hyperbolic tessellation:

http://www.youtube.com/watch?v=xJYY8v2pNw8

The example construction is given in Euclidean 3-space.

Dear James,

Many of higher dimensions of mathematical theories that we love to work on them do not have real world counterparts at the moment. One of the sensations of mathematical theories that attracts many theoretical physicists - string theory of physics for instance is a logically consistent, valid from its foundations and in its arguments tries to explain, describe and validate the quantum universe and make compatible with the macro universe by introducing magnifications of dimensions, is one such example. The theory is developed purely from mathematical perspective, navigating the free universe of imagination using the vehicle of mathematics, but yet to prove or disprove indeed it describes reality. But again, when I say " .... there are no real world models at the moment ... ", it does not mean it will never have one. Lobachevsky said : " No part of mathematics however abstract it may be that will not be applied to real world ".

http://www.superstringtheory.com/index.html

Dear Dejenie,

It is remarkable just how much of higher dimensional mathematics does not appear to have a counterpart in the real world. Yet, keying on your quote from Lobachevsky, we have the tantalising prospect of seeing how abstract mathematics can be applied in the real world.

Here is an article you may find interesting:

J. Milnor, Hyperbolic geometry: The first 150 years, Bulletin of the Amer. Math. Soc. 6 (1), 1982, 9-24:

http://www.math.ecnu.edu.cn/~lfzhou/others/hyperbolic.pdf

You find it interesting to find a way to tessellate a region of the plane with the fugure eight knot shown on page 14.

On page 1, it is mentioned that in 1830 was testing what he called his "imaginary geometry" as a model for the real world.

Dear James,

Thank you for the interesting expository article of John Milnor on Hyperbolic Geometry and the π function, citing the pioneering works of Lobachevsky on the field. As you pointed out from his statements about " ...imaginary geometry ...", he argued that " if the universe is non-Euclidean, then from his unit of distance calculations, he arrived at a conclusion that our solar system is minimal in expanse " - very interesting observations!

Dear James, I learned proximity theory from your posts here in RG and I thank you. But, concerning the original post, I think that the view of dynamical systems and chaos can give us many results, numerical bounds like:

http://en.wikipedia.org/wiki/Feigenbaum_constants

and also the Lorenz attractors:

http://en.wikipedia.org/wiki/Lorenz_system

that can lead our analysis a step forward every time we face self similar patterns.

Quote Kamal: "What about our brain? the two halves are not symmetrical!".

True enough; the intestinal tract with its organs is neither.

Deear Kamal and Lambert,

The issue of the brain is worthwhile considering. We can view the brain in many different ways. For example,

brain as integrator: whatever the eyes see is integrated into a composite picture. This capability we have to assemble a picture from what the individual eyes see is amazing!

Returning to the issue of the nearness of symmetric structures tending toward infinity, it does appear that pictures constriucted from optical signals provide a picture with focal points, fading toward the edges, giving the impression of tending toward very low light intensities along the edges of each image.

brain as harmoniser: whatever we deduce from sensory signals (visual, tactile, sound, taste, smell) is harmonised to create a mental picture of the world around us. As a harmoniser, we view the brain as a vehicle for constructing abstract tilings built from received sensory signals.

brain as pattern discoverer: mental pictures constructed by the brain fit into a relator space. A relator is a set of relations such as an ordinary garden variety proximity relation. The brain recognises proximities between mental pictures and discovers mental picture patterns.

Perhaps you have examples of the products of the brain that can be viewed as abstract tilings. Over to you, Lambert and Kamel and Cecelia and Professor Drossos and others.

The products of the brain are nerve activity and hormones. I don't view those as (abstract) tilings.

Here is a 3D brain tiling:

Lambert,

Just consider how many aspects of this topic (products of the brain) there are to consider: lots. To see this, consider

'

L.C. Sam, Public Images of Mathematics, Ph.D. thesis, University of Exeter, 1999:

http://people.exeter.ac.uk/PErnest/pome15/lim_chap_sam.pdf

Sam writes:

More often, most artists and some psychologists (e.g.; Horowitz, 1983; LeDoux, 1998) tend to relate image to 'mental imagery', and choose to refer it to "mental contents that have a visual sensory quality" (Horowitz, 1983, p.3) or some kind of mental picture.

However, Thompson (1996a) argues that an 'image' is more than a mental picture.

Instead, he characterised his meaning of 'image' broadly to include "experiential

fragments from kinesthesis, proprioception, smell, touch, taste, vision, or hearing"

(p.267) and also "fragments of past affective experiences, such as fearing, enjoying, or puzzling, and fragments of past cognitive experience such as judging, deciding, inferring, or imagining" (p.268). Thus Thompson's definition includes affective as well as sensory elements (starting on page 70).

Kinesthesis, proprioception, smell, touch, taste, vision or hearing; they are all inputs to the brain, not outputs. As for fearing, enjoying, puzzling, judging, deciding, inferring or imagining; I would call them internal processing of the brain, again not outputs.

The output of the brain is motor-neuron activity and release of hormones. Of course that output could result in a drawing of a tiling.

I like this pavement:

Lambert,

Great post! The pavement example is a good example of a partial tessellation. It is partial because whole tiles are not repeated along the edges. But one can easily imagine the extension of the pavement image so the edges are complete tiles.

Incidentally, you can see from Escher's Square Limit tiling, a tiling tends to work if the tiles start full size in the middle of a 2D region and get smaller in reaching out to the edges. So edge tiles tend toward infinity.

Do you have access to any other examples of tilings?

Some hyperbolic tilings:

http://www.josleys.com/show_gallery.php?galid=262

I have observed similar behaviors in recurrent double (or n-dimensional) sequences over finite alphabets, notably if the alphabet is some finite p-group and the recurrence is a homomorphsm of p-groups. See for example:

https://www.researchgate.net/publication/257048181_FpFp-affine_recurrent_nn-dimensional_sequences_over_FqFq_are_pp-automatic

https://www.researchgate.net/publication/257387282_The_Thue-Morse-Pascal_double_sequence_and_similar_structures?ev=prf_pub

https://www.researchgate.net/publication/228929838_Linear_recurrent_double_sequences_with_constant_border_in_M2_%28F2%29_are_classified_according_to_their_geometric_content?ev=prf_pub

and other papers cited by them. Strange enough, those structures have a priori nothing with tilings or with perspective at infinity - all these behaviors are implicit there, and wait to be visualised by construction!

Article FpFp-affine recurrent nn-dimensional sequences over FqFq are...

Data The Thue-Morse-Pascal double sequence and similar structures

Article Linear Recurrent Double Sequences with Constant Border in M2...

I forgot that there are also some Images online, see here:

http://tilings.math.uni-bielefeld.de/people/m_prunescu

Mihai,

Many thanks for the links to papers on p-groups. Also, the tilings on your web page are great. The attached impressive tiling comes from your web page.

That picture reminds me of the 1D cellular automaton features, exhibited by Conus Textile.

Dear Lambert and James,

Lambert's comment touches a good point in the discussion. The recurrent double sequences in question are indeed (also) automated sequences, as shown in the cited articles. For the notion of automatic sequence, see the monograph

Automatic Sequences: Theory, Applications, Generalizations

Jean-Paul Allouche, Jeffrey Shallit

Cambridge University Press, 21 iul. 2003 - 571 pages

Mihai,

You may find the cellular automaton carpets CDF from Mathematica interesting:

http://demonstrations.wolfram.com/CellularAutomatonCarpets/

The sample cellular automaton carpet was produced using this CDF. Have you used a CDF? Do you have access to Mathematica?

James, I used only own programs. I work with Microsoft's Visual Studio Express, which is a free-ware. I don't have access to Mathematica. However, I do not doubt that once one has a formula / algorithm, it is funny to implement it in various software frames. What I doubt, is if Mathematica has the possibility to make general recurrences, where the cell to color depends of (say) five neighbors: three on the same line, one on the top, and the last left from it, and so on... Hand-made programming has the advantage to let you free to do whatever you want...

After working some years with such sequences I found Wolfram's book "A new kind of science". It is a nice book, but of course it puts more questions as it answers. This is the state of art... Maybe he shouldn't call it "new kind of science" because it is just science. If we do not know enough in some direction, this does not make this direction less scientific.

Now some words about the objects: as far as I know, it is open which recurrent double sequences are automatic. Recurrent double sequences are Turing complete, and most of them are not automatic. It is known that those which are given by affine rules over finite dimensional vector space for some finite field are automatic. But it is open if all group homomorphisms applied to a p-group, (where p is a prime) produce automatic n-dimensional sequences. I conjecture this, but te problem is so far open. I didn't find any counterexample and I use now this Occasion to recall this open problem.

Mihal,

Good post!

In my previous message I gave a link to a Computable Document Format (CDF) file that can be used to construct cellular automata as well as tilings in the plane. You can download a copy of the CDF player to your PC. The Player is available at

http://demonstrations.wolfram.com/download-cdf-player.html

There is no charge for the Player. After you have installed the CDF Player on your PC, then use the Player to open a copy of the CDF from my previous message. Then you can start constructing some interesting tilings and being assessing the potential of Mathematica. There are 100s of CDF files available on the web. Many of the CDFs offer excellent implementations of some very interesting mathematics.

Thank you James! I didn't work Friday, but it did today. It's installed now. However, Now I Need just some time for training.

Dear Mihai,

I hope your use of the CDF player goes smoothly for you. Most of the the CDFs that can be downloaded from Wolfram are very interesting.

Sure there are such structures in nature! for example, some cactae have arrangements of their thorns in a loxodromic way. See this picture of mine.

Dear Arturo,

The pattern (arrangement of the thorns in a loxodromic way) in the cactus image is astonishing.

An arrangement of the parts of something in a loxodromic way results in what is known as a groove pattern.   See, for example, the groove pattern in the double-stranded DNA on page  29 of

P. Bucek, Ph.D. thesis:

http://is.muni.cz/th/106452/prif_d/Bucek_Pavel_Errata_Final.pdf

Honeycombs in nature have generated honeycombs in geometry and hence tessellation. I suspect that this answer has already been given because it seems childishly obvious to me.  If that is the case, I apologize in advance.

@Nelson Orringer: Honeycombs in nature have generated honeycombs in geometry and hence tessellation.

As far as I know, the image of the honeycomb in your post is a new one and brings to mind some very interesting pattern-making activities of bees.     Apparently, bees are known as cell-excavators and they distribute themselves as densely as possible.    One of the interesting things about honey comb cells is that the apex of each cell is one of the trihedral vertices of a dodecahedron.    And the cells themselves are elongated rhombic semi-dodecahedra.   For more about this, see the the chapter in the book referenced here:

http://books.google.ca/books?id=9OoKAAAAYAAJ&pg=PA315&lpg=PA315&dq=math+%22honey+comb+cell%22&source=bl&ots=QFXonLjOv&sig=YHzH844x6EEZtNjNz_HTfzy5DhY&hl=en&sa=X&ei=opTFU92NI9WgyAStwYHYDg&ved=0CDoQ6AEwAw#v=onepage&q=math%20%22honey%20comb%20cell%22&f=false 

A shorter link to the same page is

http://www.mocavo.com/Proceedings-of-the-London-Mathematical-Society-Volume-16/221230/324

All right, Jim, if honeycombs have not received mention before in this thread, I will "loop the loop" by returning to your original question and mentioning the Alhambra, where honeycomb architecture abounds. Here is a famous example in the honeycomb vaulting in the Hall of the Abencerrajes. 

@Nelson, what does 'ison  does not pre' mean?

Merely a typographical error, Lambert. Sorry. I meant to say that the vaulting of this hall has also been compared to cave stalactites, but that this "comparison does not pre"suppose symmetry, as Jim´s question requires. However, my computer mysteriously erased all but fourteen letters.

An anthill consists of pieces of leaves, needles of a conifer, branches, clods (a complicated system of ins and outs and a fragile construction). There are few homes of ants of different species, whose families live in few anthills, connected by the tracks. 

The cake "Anthill" is harmonious and tasty.

(Yes, Computer works in mysterious ways)

That anthill cake reminds me of the Tower of Babel: not exactly a symbol of harmony...

In Brazil one summer, I saw a strange, tawny, two-storey high structure by the side of the road.  My hosts informed me that it was a termite hill. I prefer Irina´s anthill cake.

@Nelson, @Irina and @Lambert,

Bringing honeycombs, ant hills, termite hills and towers of babel structures into the conversation does lead us in some interesting directions.    In each case, there are remarkable symmetries (all natural) and a repetition of certain structures that yield an overall perception of patterns.   A pattern, after all, is some structure with lots of symmetries (companionable collections of objects that are alongside each other) that we  can describe.    Something we can hang our hat on.

With Nelson's ant hill cake, for example.  I see lots of almost perfect convex sets.

http://www.math.udel.edu/~angell/ch1.pdf

In Nelson's ant hill cake, the quadrilateral in the front, lower edge is an example of a convex set.

You could also consider the presence of the Fibonacci spiral in nature, eg. sunflower seeds, plants, cells. There is a tessellated effect here.

Dear James, I photographed this absolutely symmetrical shell.So wonderful, I think,it's like a butterfly.

And this is one.

Dear Marcel,

Your post about asymmetries in natural patterns such butterfly wings or the two sides of an animal is excellent.   Zebra striping on the right and left sides of a zerbra is another example of asymmetry.   And I agree with you that the right and left sides of a human face  are usually asymmetric.  

The image of the butterfly in @Irina Pechonkina's latest post is another example of natural patterns that display asymmetries.   For example, the left and right wings of the butterfly are slightly different.    This lack of uniformity makes natural patterns interesting.   It seems that when nature repeats something in a pattern, it repeats the natural pattern in different ways.

There is no doubt about your observations.   And I agree that there is a lack of uniformity in natural patterns and that nature tends not to conform to the axioms and rules of mathematics.

Do you have another examples of asymmetric patterns in nature?

Changing (curved) space surrounding formations I believe accounts for differences or asymmetries in otherwise symmetrical (mathematically-conforming) patterns throughout all of nature.

@Cj Nev: Changing (curved) space surrounding formations I believe accounts for differences or asymmetries in otherwise symmetrical (mathematically-conforming) patterns throughout all of nature.

Your point is well-taken, since the slight distortions (stretching and shrinking actions) appear to be induced in natural patterns on a large scale by the curvature of space.

A very interesting question!

I would use fractal or recursion to characterize the tiling patterns. In this regard, ht-index may help quantify or measure the complexity of the fractal or recursive patterns; see the paper on ht-index

https://www.researchgate.net/publication/271831032_Ht-index_for_quantifying_the_fractal_or_scaling_structure_of_geographic_features

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.

Article Ht-Index for Quantifying the Fractal or Scaling Structure of...

Dear Bin, Congratulations on a much needed work, insightfully raising fractals clearly and simply to their next quantifiable level (or scale, both horizontally and vertically, etc., out of the straightjacket of power laws) and characterizing Mandelbrot's fractal as "the ideal" in the same brilliance Mandelbrot relegated spheres, cones, circles, smoothness, and linearity to more accurately a roughness pattern.  Your work should prove, if not already, seminal in advancing this science.  Thank you.

Dear Cj New,

Thanks for your nice comments!

Note that I have given a relaxed definition of fractals: a design or structure is fractal if and only if the scaling pattern of far more small things than large ones occurs multiple times (Jiang and Yin 2014, Jiang 2015). The reader might ask why I redefined fractal. This is to follow Benoit Mandelbrot who changed the definition of fractal from strict fractal like Koch curves to statistical fractal like coastlines. The definition by Mandelbrot is still too strict (because of power laws, excluding other heavy tailed distributions), and I therefore further changed it (essentially based on fractal dimension) to the new definition based on ht-index (Jiang and Yin 2014). Under the new definition, all previous fractals such as Koch curves and coastlines remain fractal. This new definition capture fairly well human intuitions on fractals; see an example below on the Mandelbrot set:

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.

Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77.

https://www.researchgate.net/publication/270634544_Headtail_Breaks_for_Visualization_of_City_Structure_and_Dynamics?ev=prf_pub

Also I re-interpret the notion of self-similarity as being the head/tail division recursively occurs in the nested rank-size plots (see an example below), or the small head and long tail recurs multiple times. In other words, the head is self-similar to a whole, or sub-whole, again and again in a recursive manner.

The nice wording "the straitjacket of power laws", yes indeed for the definition by Mandelbrot. I have moved away from it, and used more loosely defined heavy tailed distributions including power law, lognormal, and even exponential, as long as ht-index > 3.

Article Head/tail Breaks for Visualization of City Structure and Dynamics

Dear James Peters,

I have a question related to the following publication, which year it was firstly published?

B. Grunbaum, G.C. Shephard, Tilings and Patterns, W.H. Freeman and Co., N.Y., 1957

I have seen 1987, 1990, via Google and Amazon. It matters for me, because I want to know whether it is before or after Benoit Mandelbrot's classic The Fractal Geometry of Nature (1983). Thanks. By the way, is it possible to notify me as soon as you have added a response to my question?

Dear Bin Jiang,

Many thanks for your interesting question.

The first edition of 

Branko Gr\"{u}nbaum an G.D. Shephard, Tilings and Patterns, W.H. Freeman and Co., N.Y., 1987

was published 4 years after Mandelbroit's The Fractal Geometry of Nature, 1983.   A review of the first edition appears in

http://www.ams.org/journals/bull/1987-17-02/S0273-0979-1987-15600-X/S0273-0979-1987-15600-X.pdf

In terms of tilings, you may find the floor tiling in the Salon de Carlos V in Real Alcazar, Seville (see attached image) that appears in

http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/New%20Problems/paintings.pdf

Wow, very interesting tilings.

Given a tiling or pattern, simple or complex, can we identify its units likely in a recursive manner? These units are mainly according to human perception, or following Gestalt principles. I suspect that software that is able to create the sort of tings should be able to identify the units or pieces. Anyone can suggest the kind of software? Thanks.

Hi, I believe intuitively or naively that there are far more small pieces than large ones for a whole giraffe pattern. I wonder if any researcher had measured the pieces for a giraffe. The sizes would enable us to further examine the ht-index to better understand the complexity of the giraffe pattern. Any suggestions? thanks.

In the previous paper that was based on three European countries data (France, Germany and UK), we found that there are far more small street blocks than large ones in any country: the small ones constitutes cities or natural cities, while the big ones being the countryside.

Jiang B. and Liu X. (2012), Scaling of geographic space from the perspective of city and field blocks and using volunteered geographic information, International Journal of Geographical Information Science, 26(2), 215-229. Reprinted in Akerkar R. (2013, editor), Big Data Computing, Taylor & Francis: London.

https://www.researchgate.net/publication/46585944_Scaling_of_Geographic_Space_from_the_Perspective_of_City_and_Field_Blocks_and_Using_Volunteered_Geographic_Information?ev=prf_pub

This is what I have just found using keyword "giraffe skin" in Google images:

Article Scaling of Geographic Space from the Perspective of City and...

I prefer the original fauna. Go back and have a look at the amazing presentation of living giraffes.

Dear James Peters,

The notion of the proximity (nearness) should go beyond images, in which nearby pixels are merged to generate meaningful patches. Given point, line, and polygon objects, we should be able to apply the proximity operation as well. For example, natural streets are generated from nearby segments with least deflection angles (following Gestalt psychology); see the illustration below:

Jiang B., Zhao S., and Yin J. (2008), Self-organized natural roads for predicting traffic flow: a sensitivity study, Journal of Statistical Mechanics: Theory and Experiment, July, P07008.

https://www.researchgate.net/publication/1918773_Self-organized_Natural_Roads_for_Predicting_Traffic_Flow_A_Sensitivity_Study?ev=prf_pub

Also, natural cities are generated from nearby edges (of a TIN - triangulated irregular network) whose lengths are shorter than a threshold defined by head/tail breaks:

Jiang B. and Miao Y. (2015), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, 67(2), 295 - 306.

https://www.researchgate.net/publication/259914895_The_Evolution_of_Natural_Cities_from_the_Perspective_of_Location-Based_Social_Media?ev=prf_pub

Article Self-organized Natural Roads for Predicting Traffic Flow: A ...

Article The Evolution of Natural Cities from the Perspective of Loca...

@Hanno Krieger

I prefer the original fauna. Go back and have a look at the amazing presentation of living giraffes.

I understand your preference, but this "shoe" image captures fairly well (in fact in a better sense) the scaling property of giraffe skin, in which there are far more small pieces than large ones.

@Bin Jiang: Hi, I believe intuitively or naively that there are far more small pieces than large ones for a whole giraffe pattern. I wonder if any researcher had measured the pieces for a giraffe.

Excellent post!

Pattern formation and natural patterns (particularly giraffe patterns) is the focus of

M. Walter, Integration of Complex Shapes and Natural Patterns, Ph.D. thesis, University of British Columbia, 1998:

file:///Users/jfpeters/Downloads/ubc_1999-389979.pdf

See Section 2.5, starting on page 36,  where an estimate of the plausible date for the start of pattern formation for giraffes is given.    Importantly, it is mentioned that it is safe to assume a linear growth in the giraffe pattern formation process (p. 36).

Giraffe patterns are considered in detail in Section 3.4.1, starting on page 58.   Voronoi measures for giraffe and leopard patterns in section 3.5, starting on page 63.   See, especially, Fig. 3.14, page 66.   Spot areas are summarized in Table 3.2, page 67.

Of particular interest vis-a-vis the question about whether small spots or big spots are more natural, is the study of pattern generation without growth (Section 6.7, starting on page 121; see, especially, Fig. 6.5, page 122) and pattern generation with growth in Section 6.8, starting on page 124.    It appears that smaller spots are found in smaller surfaces of the giraffe, namely, legs, neck and head (see Fig. 5.3, page 95, Adult giraffe).

Marcelo Walter is on RG:

https://www.researchgate.net/profile/Marcelo_Walter

Thanks James F Peters for the excellent sources!

Small spots are with small surfaces, and big spots with big spots. This is the same for street and field blocks: small blocks in cities, and big blocks in countryside. This is a speculation I made in the following paper; Section Discussions on the study (the third paragraph):

Jiang B. and Liu X. (2012), Scaling of geographic space from the perspective of city and field blocks and using volunteered geographic information, International Journal of Geographical Information Science, 26(2), 215-229. Reprinted in Akerkar R. (2013, editor), Big Data Computing, Taylor & Francis: London.

The scaling pattern seems universal in geography and biology.

Dear James

thank you for inviting me for this discussion which is indeed interesting. In my PhD I explored the synthesis of natural visual patterns and used the giraffe as a case study.

The biological hypothesis for the different sizes of spots is related to the timing when the pattern formation process starts. Pigment cells (melanocytes) migrate from the neural crest during embryonic development to a specific location of the embryo. Therefore, the closer the body part is of the neural crest, the faster the cells reach their final destination and therefore more time to develop pattern and consequently larger spots. In the lower body parts and legs, melanocytes take longer to reach and only start the patterning process later, with smaller spots.

This migration process also explains the common white belly seen in many mammals such as cows, dogs, and horses (the pigment forming cells stop before reaching the belly)

Dear Marcelo Walter, thanks for your insightful comments?

With respect to the following comment:

"Therefore, the closer the body part is of the neural crest, the faster the cells reach their final destination and therefore more time to develop pattern and consequently larger spots."

This process you have given seems opposite to street and field blocks. In other words, fast developed parts (which are cities) tend to form small street blocks, while slow developed parts (which are countryside) tend to form large field blocks. Interesting, isn't it?

@Marcelo Walter: … the closer the body part is of the neural crest, the faster the cells reach their final destination and therefore more time to develop pattern and consequently larger spots.

Welcome to the discussion  and many thanks for joining this thread.

From  what you have written, the  key elements are time and proximity (closeness) of a body part to the neural crest.

The length of time and closeness to a starting position come into play in the study of the crystal growth process.    It has been observed that when crystals start growing from several sites in space, crystals start growing at the same rate in all directions and stop growing when they come into contact.   A crystal emerging from each site in the growth process is in the region of space closer to a particular site than to all other sites.   In effect, the growth of crystals is strongly influenced by the length of the time in the growth process and by how close each site is to neighbouring sites.  

Taking this a step further, those sites that are closer together yield small crystals, since the growth process is shorter in duration because of the blocking action by neighbouring crystals filling the space around a site.    Crystals that start in sites that are further apart have longer growing cycles and get bigger than those crystals originating from sites that are closer together.  

These observations appear to be in line with the observations about the growth of giraffe spots.   Spots that have a longer time to develop over a large region of space such as the giraffe back get bigger than those spots in small regions of the space with a shorter growing time.    Do you agree?

Research on crystal growth and the growth of polygonal regions in Voronoi tessellations is part of the study of space covering meshes in

T.M. Liebling, L. Pournin, Voronoi diagrams and Delaunay triangulations: Ubiquitous Siamese twins, Documenta Math., 2012, 419-431:

http://www.math.uiuc.edu/documenta/vol-ismp/60_liebling-thomas.pdf

See Section 3, starting on page 422, for more about Voronoi diagram patterns and the study of crystal growth.

Indeed, it seems that crystal growth process and giraffe spots have the same basic underlying mechanism with respect to distance and space. The main difference, it seems, is in the place where the process starts. Whereas for crystal growth the process begins at the same time everywhere on a given domain, for giraffe spots (and any other mammal with patterns) the patterning cells all start at the neural crest and have to travel over the body until they reach their final destination. Only when they reach this destination, they start working.

If we take the rosettes of a Jaguar, for instance (http://www.ipsnews.net/2012/11/brazil-embarks-on-cloning-of-wild-animals/) the individual variation over the body is a result of this travelling of melanocytes. Closer to the neural crest the rosettes had time to go through a full cycle of patterning and return to black; in the middle part of the body we see the full rosette with small black spots inside; at the lower parts (legs and belly) we see the black spots which would develop still as full rosettes. To me, it is amazing :-)

In perceiving this kind of patterns in which there are far more small pieces than large ones, human minds tend to concentrate on small pieces, or highly density parts. That is why cities rather than the countryside received high attention. In the same vein, eyes, nose and mouth received far more attention than elsewhere in the human face (see the following figure). It is the scaling pattern of far more small things than large than that make cities imageable; the major reason that the image of the city, or mental map is created in human minds. This is the major point I argued in this paper:

Jiang B. (2013), The image of the city out of the underlying scaling of city artifacts or locations, Annals of the Association of American Geographers, 103(6), 1552-1566.

https://www.researchgate.net/publication/230802698_The_Image_of_the_City_Out_of_the_Underlying_Scaling_of_City_Artifacts_or_Locations

A short or concise version:

https://www.researchgate.net/publication/233922898_Why_Can_the_Image_of_the_City_be_Formed

Article The Image of the City Out of the Underlying Scaling of City ...

Article Why Can the Image of the City be Formed?

@Bin Jiang: ... [we] tend to concentrate on small pieces or highly density parts.

Perhaps you will agree that our tendency to concentrate on small pieces of a scene stems from our habit of looking toward distance (seemingly, infinitely far) parts of a a scene.    This habit of ours (letting our visual perception travel to infinity) provides a basis for perspective in works of art such as the attached sketch of a stage scene by Serlio in 1584.

James, your observation reminds me of the world view from the New Yorker living in the 9th Avenue:

May I show you examples for vanishing points in nature (last autumn in northern Germany and in Island 2013), which guide the eyes to infernity.

Does this count?  The purple lines are phi spirals.  There are multiple arms and they're tiled, i.e., they pivot about a single pole.  Other samples only cover.

Promoting convergence: the phi spiral in abduction of mouse corneal behaviors.

"We have, therefore, reached a point at which it is possible, and even probable, to conclude, not merely that the Phi spiral (a new mathematical conception) is the best formula for the hypothesis of Perfect Growth, and a better instrument than has yet been published for kindred forms of scientific research; but also that it suggest an underlying reason for artistic proportions, and provides an exquisitely delicate standard by which to appreciate divergences and variations of different kinds."

~ Cook, The Curves of Life

@Jerry Rhee: spirals, by themselves, are not tillings or tessellations , but they do emerge from tilling structures. Phi spirals indeed arise from sunflowers, or cactae, where the tillings are the subunits of the whole structure, namely the seeds of the sunflower, and the distribution of thorns in a cactus.

Thanks Arturo...but the basal cell arrangements, itself subject to systemic growth processes, are what constitute the spiral and they are tiled.  Aren't the skin patterns a result of cell arrangements, too?...and the different patterns on cactuses?

Where do you see phi spirals on sunflowers and cactuses? 

@Jerry Rhee and @Arturo Ortiz Tapia, you have both made some very pertinent observations.   Apart from the issue of tessellations or tilings, nature does present enormously interesting geometric structures. 

Continuing what Aruro observed, you may find the 14 polymorph tessellations (from Wolfram) interesting.   These are orderly compositions of 3 regular and 8 semiregular tessellations.   For more about this, see

http://mathworld.wolfram.com/Tessellation.html

The question for followers of this thread is the following: Are there examples of tessellations in nature?

@Jerry Rhee and @James F Peters. I do not have for the moment any picture of a sunflower of my authorship. However, I would like to share a photograph showing what I previously stated.

Here is the complication with discussing how signs are like natural things.  Our perspectives also matter.

The following are side by side comparison of an ostrich corneal endothelium using SEM imaging and the color is my own of mouse corneal endothelium labeled for the nuclei (in blue) and Cx43, a membrane-bound channel protein in red.  The corneal endothelium is a MONOLAYER.  Yet, the red label illustrates the 3D arrangement.  Still, with a proper perspective, it can be seen as a 2D image.  That is, things are "roughly" planar and "roughly" hexagonal.

The beautiful ostrich cornea SEM is from:

Scanning electron microscopy of the corneal endothelium of ostrich, João Antonio Tadeu PigattoI Angela Aguiar FranzenII Fabiana Quartiero PereiraI, Ana Carolina da Veiga Rodarte de AlmeidaI José Luis LausII Jaime Maia dos SantosII, Pedro Mancini GuedesIII Paulo Sérgio de Moraes BarrosIII

Arturo,

the phyllotactic image, although very beautiful, is a result of the phi ratio (the golden angle) in combination with a growth algorithm (which dictates production of periodic nodes in the sequence of a generative spiral).  The parastichies (nearest neighbor perceptual judgments) that are apparent as shapes of spirals are not phi spirals.  Simply layer a phi spiral (logarithmic spiral that expands by a factor of phi every quarter turn) onto the image and it will be apparent immediately.

@Jerry Rhee and @Arturo Ortiz Tapia: wonderful examples.   The ostrich corneal endothelium geometry is astonishing!   Equally astonishing is the pattern of seeds in a sunflower.

Perhaps you will both agree that these are natural approximations of similar polygonal structures in geometry.

James,

What's even more astonishing is why there should be a phi spiral displayed at a mesoscale at all...that is, whether there are tesselations at a smaller, uninvestigated scale that is responsible for production of such a perfect mathematical form.

Clearly, this violates Marcel and Lambert's comments above:

"I agree with Marcel that nature does not produce symmetry following 'mathematical rules of theoretical perfection'."

"Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a fore-ordained goal, is like the operation of destiny...The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real."

~Peirce

   Just an offhand idea at how to make an infinite plane tiling. Draw a rectangle, fill it with any design whatever (preferably polygonal), and make sure that any boundary lines in it arrive perpendicularly at the sides of the rectangle; then, reflect the drawing on all four sides of the rectangle. I haven't done it, but I believe it Works.    

 @Raul Simon: infinite plane tiling.

Your idea is excellent!   Have you considered starting with a set of generating points (chosen either randomly or deterministically) to construct a Voronoi mesh to provide a plane tiling?

Dear James: 

   I'm afraid I don't know what a Voronoi mesh is...

   Thank you for upvoting my answer.

Dear Raul,

please have a look at this URLs.

http://en.wikipedia.org/wiki/Voronoi_diagram

http://www.google.de/imgres?imgurl=http://astro.cornell.edu/~dmunoz/cfd_images/cylinder_mesh.png&imgrefurl=http://astro.cornell.edu/~dmunoz/cfd.html&h=303

Thank you, Hanno.