@James, thanks for nice thread. You have mentioned the Golden ratios! Nice remembering for myself. The first time I have met the term Golden Section was when I was studying Fibonacci and Golden Section Search./book of David G. Luenberger-Introduction to Linear and Nonlinear Programming/.Fibonacci search method has a certain degree of theoretical elegance itself.

Regarding the golden ratios in the painting of Mona Lisa , YES, that first attract me! Definitely YES!

The beauty or structure is my answer,this is the answer of an engineer, not a mathematician!

Structures of course!

Yes endosed

The question is related to the research of Eric Kandel (Ann Neurol. 2012 Nov;72(5):A7-8. doi: 10.1002/ana.23785. The biological mind and art. http://www.nytimes.com/2013/04/14/opinion/sunday/what-the-brain-can-tell-us-about-art.html?pagewanted=all).

More specifically, the question is related to the architecture of the visual system and the “beauty of geometry”. Depending on your philosophical position about the origin of mathematics (brain invention, external world or a discovery) you can think that the structure of the visual system and the responses of neurons are result of:

(i) Natural selection on brain structures,

(ii) An external optical order or

(iii) A more fundamental structure of the space-time, which has affected both the visual cortex AND the external reality.

Maybe mathematical beauty is just the emotional response to a “click-burst” of parietal, temporal and occipital pattern-recognition networks (smile)!

@Juan, or you might be more realistic and not go for just one "extreme" philosophical position, but instead suspect that every one of those three explanations each plays a role.

In my opiniom, the structures are the most important.

Juan,

from the article you cited, it is apparent that there are many sources of inspiration for mathematics vis-a-vis the aesthetic character of things impacting on our senses. From the same article, it seems that the brain is a creativity machine.

Then would you agree that from the observations made by Eric Kandel, the beauty of sensed structures catches our attention and inspires mathematics? In other words,

can one then conclude that at least some of the time, a source of mathematics stems from observing beauty in natural structures?

The structure may be very complex in nature but always it contains several correlated and non-correlated beauties(Mathematical Models) inside this. As we approach for segmentation and transformations just for making the structure fit for us for processing.

I think the only way to solve any problem is to understand it as set of possible solutions. The patterns produced by nature or designed by us are always a set of beauties(Because we understand structures as mathematical beauties.)

G.H.Hardy was a pure mathematician.Basically working on number theory and the theory of summability. His opinion is perhaps from the angle from which he was looking at mathematics.The beauty of the results in number theory and summability are more attractive. But that does not dilute their structural importance, especially in number theory. He is very much correct in his explanation of beauty as pure mathematics can be wothfully compared with poetry from the philosophical point of view.

There will be no beauty if there was no structure!

It is always beautiful when experimental data/precise measurements proove mathematical predictions...

Dear James,

Your question is very interesting. There is a lot of poetry in math, mainly, in pure math. Nevertheless, usefulness of math is out of any doubt. I think that in this formula

cos(x) + i sin(x) = e^(ix)

there are both poetry and usefulness. Even a lot of magic. Consider that non-visible electromagnetic waves had not been discovered ignoring the formula above. Without knowing electromagnetic wave existence, nor wireless, nor TV, nor computer nor quantum mechanics had been discovered or invented.

Please use the attachment feature and post pictures when possible. See example below for attaching picture, web link & YouTube video. Thanks!https://www.google.com/search?q=two+pillars+in+front+of+Solomon's+temple&tbm=isch&tbo=u&source=univ&sa=X&ei=Ni7uUrX1HcjKsQSE6YFo&ved=0CC0QsAQ&biw=1506&bih=673

Samson & Delilah:http://www.youtube.com/watch?v=hmITjMy_fTY "If I had had my way, I'd tear this old building down", Bob Weir of GD. In my opinion that was poetry for "thinking-out-of-the-box" or being unconventional.

Zaal Kikvidze · 40.41 · 211.08 · Ilia State University

"It is always beautiful when experimental data/precise measurements prove mathematical predictions"

I know exactly what you mean! Check this match between the theoretical and experimental XRD rocking curve for a GaAs mono-crystal: http://www.flickr.com/photos/85210325@N04/9430820747/in/set-72157635172219571

Here is my sense of "art", "beauty", "pattern" in science and experimentation, the natural crystallographic strain field within a ZnSe mono-crystal. X-ray Diffraction (XRD) is the science of deciphering patterns in Fourier space. The resolution achievable in reciprocal space is phenomenal. The "as-solidified" Nano structure of ZnSe viewing the (224) plane! EPD (Etch Pit Density) pseudo-color maps are fascinating as well.

based on Plato idea, the beauty could have source in simplicity, in this sense beautiful patterns could comes first in doing mathematics,

>....based on Plato idea....

The language of mathematics, including its semantics, evolves over time. One source of mathematical beauty for me is the distillation of vague experiential concepts into language as clear as humanly possible. One example is the development of computation theory in the 1930s. Several models emerged, and it was mathematics that assured us that they all captured the same thing.

Hi James,

Bilateral symmetry is going back close to a billion years. See attached.

Our brains are wired to "detect" things that look symmetric, as, this points to a "potentially healthy" organism. Same applies to certain ratios, such as your PHI (golden) ratio. I am sure that, you can come up with many other ratios, that are detected as "beautiful," since this information passes along genetically as a hard-coded object recognition database.

According to the attached extensive theory by Oxford, bilateral symmetry develops based on certain cell division, and cell signaling methods. So, it is natural to expect that, when our kidney, brain, face, hands, etc. are developing, they take certain shapes. We, humans, are wired to detect these natural patterns and classify as "beautiful." In the end, yes, you are right, it is all mathematics :)

The development our body from an embryo can be defined as a set of such complicated mathematical transformations that, it is a mathematical and natural beauty. To make it even more beautiful, we can detect these natural patterns in our visual cortex within milliseconds.

Dear Professor Jim Peters,

In 1985, Serge Lang in his book, The Beauty of Doing Mathematics, wrote:

“Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?"”

In my opinion both the beauty of patterns and the structures in patterns have played important roles in progressing mathematics, from the beginning till now. They are mixed like milk and sugar. In other words, none of them is separated from the other. Sometimes mathematical facts are derived from the structure and later on the beauty concealed in the structure is felt, and vice versa. The starting points are the outcomes of different points of view.

I think the answer to this question depends on context. Of course, the recognition of patterns leads to general assertions about whatever is the object of our attention, in pure mathematics, or in mathematical physics. Some time ago I studied DNA sequences for HIV-1 AND HIV-2. It is hard to recognize patterns in those sequences, however, after an analysis of their complexity it can be inferred that they have a structure, which then can be described using the methods of automata theory. Of course those sequences have a structure that gives rise to all that beauty we find in life all over the planet. Another example: try to program a Turing machine to perform Eratosthenes sieve as in this reference http://www.cs.cornell.edu/Courses/cs4820/2013sp/Handouts/481TM.pdf I did not find any beauty here, in the code, but in the result.

Our brains are attracted by those perceptions that show a sort of order or symmetry, which can be described in mathematical terms and then generalized. However that order or symmetry does not necessarily appear in the fundamental structures or processes that produce that symmetry.

In times when Art was considered (just as Science) instrumental to the knowledge of the world and/or to educate people it was not a problem to define what Beauty was, simply a beautiful piece of art was a piece able to convey the latent message in order to be correctly intended at different levels of perception and personal culture.

The magnificent 'Discorso sulle Immagini sacre e profane' wrote by Cardinal Gabriele Paleotti for setting the role of Art in the perspective of the Trento concilium (end of XVI century) clarified very well this point 'a painting must be intended by people who cannot write and read as well as by super-experts in art and intellectuals'.

Caravaggio was the most fit interpret of Paleotti manifesto and his paintings (endowed with an incredible complexity of theological meanings set on different layers of explanation) are as well incredbly vivid representations anyone can immediately perceive as moving.

With the death of any recognized didactic role of Art and the onset of the horrible dictum of 'Art pour l'Art' simply we opened the doors to ugliness and stated Art was basically useless. In Italy we have the very vivid 'experimental proof' of this process: while any house built by a very ignorant peasant til two centuries ago made the landscape more beautiful, the large part of more recent buildings destroyed the harmony of the environment and produced anxiety and ugliness.

In science , especially in mathematics, our 'audience' is much more limited, only a very tiny part of the population can appreciate if a theorem is 'beautiful' or not and when we ask for 'simplicity' (the same concept set forth by Paleotti) we refer to a vanishingly small self-referential community. Nevertheless, in my opinion, we can make an effort and try to recover the 'communication wideness' to differently educated people to judge about the Beauty (that for me as Roman Catholic simply corresponds to the amount of Revelation of hidden Truth) of a piece of science : a piece of science is beautiful if it allows to be 'caught' , at different levels buth with the same basic meaning, by a wider audience, attached is a work of mine about this theme claiming for the beauty of meta-analysis in clinical studies connecting this kind of beauty to theater that is the form of art most easily linked to science.

I think, there is no objective answer to this question. For each of us there are other priorities. However, above individual opinions are very interesting. Thanks so this discussion, we can better understand the nature of the problem.

As a "source" of mathematics, beauty has priority or more import than structure. Structure without beauty has no import; whereas beauty without structure has every import.

@James, thanks for nice thread. You have mentioned the Golden ratios! Nice remembering for myself. The first time I have met the term Golden Section was when I was studying Fibonacci and Golden Section Search./book of David G. Luenberger-Introduction to Linear and Nonlinear Programming/.Fibonacci search method has a certain degree of theoretical elegance itself.

Regarding the golden ratios in the painting of Mona Lisa , YES, that first attract me! Definitely YES!

The beauty or structure is my answer,this is the answer of an engineer, not a mathematician!

James,

Mathematics is used to reflect our thoughts and define them unambiguously to others. We use mathematics to represent the patterns we 'see' in nature e.g. The Golden Section. Some see beauty in the elegance and simplicity of Newton's laws or Maxwell's and Einstein's equations. However, nature is not turning out to be as simple as people first imagined and the mathematics needed to encompass this complexity is becoming less elegant. There is now a risk that mathematical elegance is a requirement for any acceptable future theories without any other justification than 'beauty.' Scientists may be driven by the mathematics rather than their own insight on reality.

Dear James,

…well, the sense of beauty in math might have a mathematical neural explanation more abstract than the explanation the sense of beauty in art has. Whereas we can perhaps explain the beauty of visual structures with ordinary geometry in primary visual cortex, we need a more abstract thinking to explain why we feel mathematics is also an art.

The sense of beauty in math comes from a sense of symmetry and unity that make us see a mathematical expression as something universal and simple. The source of joy should be described in terms of neural firing, coincident behavior of spiking patterns or something like it. This gives us the feeling that we get something fundamental not only about the external reality but also about the inner mechanisms of brain and mind that handle space (parietal lobe). The highly efficient economy of energy in this information processing or encoding is perhaps the source of the feeling of joy when we understand a formula or mathematical structure.

I think it is the combination of the two. The beauty and the structure are so important to form a complete image.

@Fernando, it is a good paper! J¨urgen Schmidhuber did a nice job!

@James, I think that Mathematics as theory of numbers or logic is mainly built on logical statements which could be pictured following a cognition path. Since our brains need shapes and forms as material reference, mathematical thinking needs organizational sets. The mathematical combination or assemblage is looked more beautiful if it follows a well harmonized pattern or set up. Indeed computer is the valuable tool which let us visualize the harmony of many mathematics, not enabled a century ago!!

I agree with Ljubomir : "The beauty of structure is the answer,this is the answer of an engineer, not a mathematician!"

Fred, Juan, Fernando and Ljubomir,

One of the unusual and interesting ideas is the notion of interestingness as the first derivative of subjective beauty in

J. Schmidhuber, Simple algorithmic principles of discovery, subjective beauty, selective attention, curiosity & creativity, Lecture Notes in Artificial Intelligence 4755, Springer, Berlin, 2007, 26-38, with a preliminary version downloadable from

http://arxiv.org/pdf/0709.0674v1.pdf (cited by Fernando).

Hello James, Thanks for sharing this nice thread. Each has his special view, It's the beauty of the well structure...Sure we feel the beauty of patterns, all of non specialized should feel it. But as a specialists, we try to find the great structured mathematical model that created this amazing image.

Well, Hardy was an idealist. Structure is Bourbaki ideal influenced by structuralist philosophy. Also very idealist. But a lot of modern mathematics is neither structured nor beautiful. Think of a long proof in graph theory for instance....or a complex diagram in category theory. Not to mention the classification of simple groups theorem....

Fernando,

Here are some patterns I noticed in your ticket from the bus:

top line:

LIN: 542: 5+4+2 = 11 *

INT.: 0046 = 4+6 = 10

000677: 6+7+7 = 20

Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,...

From the digits: 1, 1, 2

From the digits (before summing): 2, 5

Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, ...

From summng the digits of LIN: 11

From the digits (before summing): 2, 4, 7, 11

line 3:

N#00023391: 2+3+3+9+1 = 18 **

12:46 1+2 = 3 *

4+6 = 20

1+2+4+6 = 13 *

03/02/2014: 3+2+2+1+4 = 12 *

Fibonacci numbers from summing the digits: 3, 13

From the digits (before summing): 1, 2, 233

Lucas numbers from summing the digits: 18

Lucas numbers (before summing): 2, 1, 3, 4,

line 4:

VALOR: $ 3.97: 3+9+7 = 19

Lucas numbers (before summing): 3, 7

Fibonacci numbers have a closed form solution defined in terms of the golden ratio.

See

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

I think that the beauty 'in poetry, image, mathematics, human allure, animals, nature..." is perceived through the harmony of the pattern(s), their concordance, synchronicity, intelligence and/or organization. Even in very complicated theorems, there is the beauty of the reasoning harmony and logic which describe them

There is beauty in seeming "randomness" as well besides "structure". Depends on the perceiver, doesn't it? Case in point, the bus ticket!

Bringing attention to Carlos' statement "However that order or symmetry does not necessarily appear in the fundamental structures or processes that produce that symmetry", untill we perceive it so?

In many cases, we first discover structures in an observed object such as the shapes in the moving edges of an ocean wave or in the tiles derived from the Fibonacci sequence on a piece of graph paper. Only after structures have been identified, then it is possible to find something pleasing in the perceived structures. In fact, it often case that one must do some work, to find beauty in perceived structures. This is the case in the attached image, where we first find structures in the shrinking tiles derived from the Fibonacci numbers. This Fibonacci tiling example comes from

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

Once we have identified interesting structures that peak our interest and stimulate our curiousity about where the perceived structures lead, then we may discover beauty in the evolving structures. For example, from a tiling derived from the Fibonacci numbers shown in the previous thread, one can starting filling in the spaces between the corners of the tiles with curves and derive spirals and arrive at interesting shapes. In other words, by working with the perceived structures, it is possible to discover something pleasing. In short, by working with perceived structures, we might find beauty in the evolving structures. See the attached image that appears in

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

The is a bit like starting with Fernando's bus ticket, which has many structures (numbers, digits of the numbers, words, sequences of the digits, shapes, spacing and sizes of the letters in the words, and so on). From our initial perception structures in the parts of the bus ticket, we can begin separating and extending the structures to discover something interesting. In other words, even a bus ticket has structures with inherent beauty. And the structures in the bus ticket, it is possible to discover some interesting patterns that lead to some interesting mathematics.

Excellent thought matter James! This should be the same principle applicable in any data display or web page design as well. The concepts of aesthetics should be approximately similar, yes? Many of these patterns appear in crystallographic (stereographic) projections i.e., the Fourier transform of the real space in the crystals.

Ravi,

Yes, the principles of aesthetics are approximately the same. Do you have an example of patterns that appear in crystallographic projections?

Here is an example of the use of the "Wulf net" & "steriograms" to determine the symmetry elements in a crystal through X-ray diffraction and crystallography: http://www.doitpoms.ac.uk/tlplib/stereographic/wulff_construct.php

Please consult the following reference for more about the fundamentals of XRD.

http://www.flickr.com/photos/85210325@N04/10515372183/

>....The question here is not whether mathematics is beautiful but rather whether beautiful patterns are the starting point of mathematics.....

There is such similar pattern in the shape of the universe and galaxies and has been the subject of study of many a mathematician, physicists and others for ages now, hasn't it? Patterns are the easiest concept for the human brain to visually comprehend, in my opinion. Therefore we would natually tend to it. It would be interesting to see this concept from those that do not have the advantage of the visual sense when creating art.

Ravi,

Interesting post! Please give one or more references concerning the patterns in the shape of the universe and galaxies.

James! Just my visual comprehensions from galactic images on the NASA web site and other Carl Segan programs. I'm not an expert in that field, I confess.

Ravi,

Further to what you wrote about patterns in galactic shapes, take a look at the attached image of the spiral galaxy M81. Apart from the problem of explaining the spiral arm structure of spiral galaxies, some interesting geometry can be found the concentric ellipsoidal shapes, starting with the center of a spiraling galaxy and reaching out to the visible perimeter.

Now take a look at the concentric ellipsoidal shapes in the spiraling of the galaxy in the previous post. For more about this, see

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

Cool diversions! Love the potential scope of this line of inquiry.

Ravi,

Again, you have hit proverbial nail on the head with a reference to microscopy. For most of us, microscope images are more accessible and just as interesting as snapshots of galaxies. In terms of looking for patterns in microscope images, I am working on a form of micropalaeontology, inspecting, comparing and looking for patterns in microfossil images, where the average diameter for a complete, very interesting "animal" is 4 to 8 mm.

I agree with Hemanta, that is beauty of structures. Taj Mahal is not known for its exotic carving etc. It is simplicity of structure that strikes most from close quarters and from a distance.

James! We have a real time X-ray microscopic imaging tool that can image at about 10-40um per pixel. If you have a good sample with structure in its subsruface that you need to inspect tomographically, we may be able to help. Send me the sample and I'll place it into the beam for you. However, you must have facilities closer to you with the capability.

Firoz! The fantastic work of art with simple patterns & symmetries (predominantly the "four-fold") by the forgoten artisans of the Taj Mahal! "Ustad Ahmad Lahauri is generally considered to be the principal designer". Instead we attribute such creations to dead and deposed (by flesh & blood) tyrants as their ultimate expression of affection for a dead one in their harem of many.http://en.wikipedia.org/wiki/Taj_Mahal

http://en.wikipedia.org/wiki/File:Taj_Mahal_2012.jpg

Dear Professor Mohammad Firoz Khan,

The Taj Mahal is a stunning source of beauty of structures and directly relates to the question for this thread. Consider what Aristotle wrote: The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.

In the attached paper on symmetry in 3D geometry by N.J. Mitra, M. Pauly, M. Wand and D. Ceylon, it is observed that symmetry preserves a certain property such as geometry symmetry in an object such as the Taj Mahal under some operation applied to the object. Reflections, translations, rotations either singularly or in combination are examples of symmetry operations. A geometric object is symmetric with respect to transformation, provided the object is invariant under the transformation action. By studying symmetric objects in the tiling patterns in places such as the Taj Mahal or the sacred Imam Reza shrine in Mashhad, Iran or the Alhambra in Granada, Spain, it is possible to discover a wealth of beautiful tiling patterns. For more about this, see the attached pdf file.

Ravi,

X-ray tomography has lots of promise, if one is looking for structures and patterns in small objects such as microfossils. Usually, there is software that makes it possible to specify thickess of an x-ray slice and the number of slices. The end result is a multilayer view of an object that can used to construct a 3D view of the object. Is this the case with your tomography setup? Do you have a 3D view of an object that you have obtained with your device? what sorts of objects are x-raying?

We have not used our device yet for tomographic (radiographic) applications. Many others have used such device exclusively for tomographic imaging. Most of our focus has been for topotomographic observations using the XRD rocking curve method from sample surfaces upto depths of 10-20um. Gyorgy Banhegyi may know better. See the following LI discussion for details:

http://www.linkedin.com/groupItem?view=&gid=2683600&type=member&item=270002962&qid=fe9f9e8b-2b9c-4a7f-96d9-914380b238c9&goback=%2Egde_2683600_member_5837865190187610112%2Egmr_2683600

Ravi,

Many thanks for the link. X-ray tomography offers something important that is not obtained using reflected light microscopy, namely, surface depth and structure.

James! I agree. There are many concealed wonderful patterns below the surface in depth that have not been fully explored yet. Exciting!

This is an example (attached) of the micro lattice strain fields in the top 1.1um depth of "as-solidified" Nano structure of a ZnSe single crystal wafer looking down the (224) crystallographic plane. The pseudo color makes it look prettier to the eye by displaying the 3rd dimension, amplitude, in the 2D map. The XRD images were scaled to match the actual sample dimensions. We've used some interesting variations in pseudo color scales to create better visual contrast for viewing.

Just in case you're interested, here's the original data of ZnSe (along with many more in the play list) in a YouTube video "A Scroll Through Reciprocal Space". A record of probably each vibration that advancing solidus surface experienced at the time of solidification in the ZnSe crystal & music for your enjoyment: http://www.youtube.com/watch?v=dFCQS8oUyT0&list=PL7032E2DAF1F3941F

P.S. The small topograph NE of the (224) reflection is the view down a different (hkl). Not indexed yet by the data originator. Feed-back appreciated much always!

Ravi,

Many thanks for sharing the link to the data. I am very interested in this from a pattern recognition point of view. I know that it is possible to view a digital image as a metric descriptive proximity space. Points in an image can be described with feature vectors and sets of points that can be compared with a similarity distance measure. I do this in my new book on a Topology of Digital Images (becoming available from Springer this month).

James! "Pattern recognition" is my hot button as well. At the risk of being repetitious, here's one of my attempts at displaying using "art" the potential of "pattern recognition" in XRD Microscopy and materials Nano structural ID. These "as-solidified" and "as-received" Nano structures are as unique to each crystal as is our individual DNA. "Pattern recognition" in our image processing & analyses is quintessential! Our interest is in designing software to recognize such patterns and identify materials in situ using a NDE tool like Bragg XRD Microscopy.

BTW I could certainly use some critique of our new logo design using "patterns" in XRD. What is your preference, this way or rotated 180 degrees? Should the "red cross-hair" come off?

Dear Jam,

All that you write about beauty of mathematics, it is very clear to me. My father Vladimir Yurkin was the artist (the member of the Moscow union of artists) and basic provisions about which you write, I heard long ago from him. It is interesting that he like mathematics as a hobby. He drew mathematicians at mathematical faculty of the Moscow State University. He helped other artists correctly calculated the position of objects and figures from the point of view of a golden ratio on a canvas etc.

Yours,

Alexander

I think 'beauty' refers to familarity with life. The greatest familiarity of life is its' "of-itselfness". We are constantly trying to capture the of-itselfness of nature and to reproduce it. Mathematics in seeking to capture natural symmetry employs recursiveness that nature seems intuitively to possess alike a hologram that contains all aspects in every corner. I think it is a shape in motion that produces an eccentricity that is difficult to visualize or define in its native form. In this respect we seek what is an identical beauty to both the whole and the conditions that define structure mathematically. In high school I used to create doodles over and over again of faces within eyes within faces ad infinitum. I now believe, rather than as physicists believe that the universe is expanding, that the it actually grows smaller over vast ages, adhereing only to ratios that characterize a shape, a volume possessing prison that elaborates volume and existences. Perhaps in the study of pattern recognition the mind has zoomed right by what it seeks without stopping to reflect on what it has accomplished, meaning and innateness within expressions. There is possibly a danger in this to lose sight of an actual forest, a relativeness, variability, that permits rather than governs...there is more than an of itselfness, universe, to the world, but an exact history and path of emergence that is not constructed of stationery trees. It is within the facsimile of 'universe' wherein beauty is perceived that we should accomplish a slower pace and better knowledge.

to add: If interested the link to recent publication ("Determining the Determined State: A Sizing of Size From Aside/the Amassing of mass by a Mass")is attached.

Article Determining the Determined State : A Sizing of Size from Asi...

Dear Alexander,

Great post! Many thanks for sharing with us the combination of art and mathematics in your father's life. The perception of the orientation, shape and distribution of the space-filling curvature of object surfaces is greatly aided by a knowledge of geometry. The story of your father's lifelong interest reminds me of the close ties between Euclid's work (especially his postulates about lines), Riemann's introduction of manifolds and more recent work on differential geometry.

@Marvin Kirsh: I now believe, rather than as physicists believe that the universe is expanding, that it actually grows smaller over vast ages, adhereing only to ratios that characterize a shape…

This very astute observation of yours reminds me of your paper about the shape of an egg, namely, how an egg shape can be extracted from a x-section of a hyperbolic cone. This is one of those remarkable results you found in MathWorld. See the attached image from

M. Kirsh, Where the shape of an egg comes from, page 26 (downloadable from your RG page).

James: Thanks for the comment. The figure caption is written incorrectly. The egg shape and hyperbolic cone is from Schauberger. The equation and graphing parameters for the figure are from math world. Math world at one time had a more complete discussion of the egg shape as a mathematical object, is now reduced for some reason to exclude old (15th century) egg versions made from pentagon overlays-I think most important, rather than the specifics of shape (e.g. egg) is the potential description of nature as open but closed looking surfaces.

Dear James,

I think, the article from "Arts Theory and History"2013, which is called "The Being of a Thing and its Meaning in Social Communication" by Q.Nguyen will be interesting.The author proves that being of Maths (numbers), language (sentences) is not only logical patterns, but also beautiful ideas. They must gear to humanism, a noble ideal. The beautiful of human intellect is eternal. Painting is the most powerful language, it's a human language at its best. It has the power to move, the power to transcend all boundaries, the power to express the depth of feelings of one's heart.

Marvin,

Helpful post! Many thanks for the corrections. The implications in what you have written in your article are important. Have you considered introducing axioms and theorems connected with your work?

Dear Irina Pechonkina,

What you have found is important. Q. Nguyen writes: Hardy wisely concludes, “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations”, are simply the notes of our observations.” (Hardy, 1993). This may spurs our attention to Heidegger’s central idea that all things must gear to humanism, a noble ideal, not idea. Hardy may have no problem with Heidegger on humanism he sees art and science different in substance but common

in terms of the beautiful of human intellect, that like a painter and a poet, a mathematician is a maker of patterns (Hardy, 1993). We tend to think of art as something beautiful, do we think theorems are beautiful or ugly? While the functions and idea of making patterns greatly differ from mathematics to poetry and painting, Hardy enjoys the fact that “the mathematician’s patterns must be beautiful. (Hardy, 1993). I agree. Concerning the perseverance of the beauty of idea, Hardy finds that while the beauty of the idea of verbal pattern seems “hardly” affected by “poverty”, that of mathematics “lasts longer” (Hardy, 1993), for instance the Pythagoras’s theorem or the ratio 60 in Babylon’s culture.

An observed pattern (represented in a theorem) must have some form of beauty.

Thank you, dear James, for your useful and original comment. My students adore Maths and Physics.I'll share these ideas with them."In Geometry we need inspiration as well as we need it in poetry"A.S.Pushkin.V.G.Boltyansky created a formula of mathematical aesthetics: "beauty= visualization+surprise+simplicity."..

thJames: Thank you for your comment. I have begun to consider the prospect of defining axioms. To begin I consider that if there is space with shape there is always and exclusively space with shape. I also consider the following to distinguish the real from the imagined:

A + B = B + A to have no coherent real meaning

but A(t) + B (t) = B(t) +A(t) to be exclusively false

Entailed is a directionally oriented, or inclined, world defined by order of occurance, uniqueness of all spaces.

Scaubergers' vision demonstrated in graph is exceptionally intuitive rather than rigorous mathematically. He sees an always open world he represents with a (closed) 2-D shape; though in reality a universal shape and world must both be 3-D and open. I think my egg depiction captures an eccentricity to two planes in which both axises, the long axis with the prominent blunt pointed ends and the shorter radial dimension each, are each eccentric and open with respect to temporal ordering along the surface: the egg as if constructed of intersecting perpendicularly oriented hyperbolic cones in which the final shape represents an intersection of (like shaped) causes that are maximally distal from one another yet 'proximal' and witnessible to the linearly progressing first perspective in the sense that regardless of attributed physical dimension to causes and perspectives interactions involve comparable rates of motion, the form-bearing past occupying 1/2 of the time embodied to continually progressing present form. A "homeostatis" resulting in eternal existence of like shapes in which rates are linear (e.g.fixed mutation rate) as if in a constant velocity free fall effected from the engagement with causes in which forces are tangential, e.g. centripital, centrifigal. Homeostatis is absent in cases like Uranium where 1/2 life refers in a predictable manner to the future, the memory state is unstable possessing an accelerating rate...one can also predicte the existance of de-accelerating rates (alike to loss of orbital velocity), death. The world can be described this way in terms of mass(=number)/volume containment by causes and the existence of an of-itself shape= universe contained to a 'world' of diverse world forms. The existence of an absolutely "unwitnessible" 'black hole' (a phenomenon of deaccelleration, death) containing the familiar world is imaginable, the processes towards death universally preceeding birth. Einstein failed to capture this to mathematics but dissented strongly to what I interpret as a propensity for escape, to jump, de-accelerate to accelerate prematurely from great heights in descriptions attempting to capture from imagination what absolutely also contains it...absolutely fixed on the axiom "A + B = B + A"

Chemists and biologists are not so succeptable, but physicists have become fixed on dualisms that can deleteriously affect imagination, to affect redundancy in conceptualization and language facilities. Consider the working of the modern cell phone system....it is proclaimed that it is impossible for a single mind to trace through the entire program steps of functioning, asembled by diverse programmers from different cultures with different languages; known to have potential hardware bugs involving plural access codes. Is this beauty, or just evolved structure looking through a tunnel? Who can say we dont already understand the issues but are captured and addicted to the pleasure of the fall. Terrible paradox, strigent need for reordering arises when social scientists enter the picture, especially when it is considered that some tools of the trade are dependent on such technologies. Dependence and addiction are terrible social problems, and the recipients of major governmental attention, but does an industrial economy realize how deeply it is infested with addiction like. I do not think the two mix so beautifully together; industry is self-organized with a tunnel like scientific window, doesnt perceive the concept of 'containment' clearly: there is such a thing as absolute containment; while government seeks to 'contain' according to necessity, but seems to becoming blinder and blinder. Look how I can order parts for my computer from mainland China at whim at the same time as there are active differences and inflamations referring to military contest. We cannot have world governments driven majorily by a philosophy of 'structure without borders' simultaneously with existing beautifull patterns to recognize. Is obvious that proximality of many kinds of structures can exist, but some should be necessarily avoided for focus as all we have so far accomplished is tangentially intersecting 2-D closed forms made into fixed ideas involving the motion of perfect roundness, but no whole conceptualization of eccentricity and real sustained shape.

Marvin,

Great post! Here are axioms to consider, branching of what you wrote. First,

let X be a set of 3D shapes endowed with a descriptive nearness (proximity relation).

By allowing for the description of 3D shapes, we give ourselves handles (shape descriptions) that can be compared. I am hoping that you have dabbled in or know LaTeX (the following notation is written in LaTeX).

Let $\Phi$ be a set of probe functions of the form $\phi: X\rightarrow \mathbb{R}$.

That is, each probe $\phi$ in $\Phi$ maps element of X to real number that represents a feature value of x. Then a description of an x in X is a feature vector of the form

\[

\Phi(x) = \{\phi_1(x), ..., \phi_n(x)\},

\]

for an element (point) in x in X that has n features. Then we write

\[

A \delta_{\Phi} B, \mbox{provided}\ \Phi(a) = \Phi(b),

\]

for at least one pair points $a\in A, b\in B$. Now imagine that each shape is a set of points $A$ sliced by a horizontal plane. Then a good probe function to consider is

\[

h(x) = \mbox{height of a surface point above the horizontal plane}.

\]

Then try out

Axiom 1 3D volumes A, B in X are descriptively near, provided h(a) = h(b) for a surface point $a\in A$ and a surface point $b\in B$.

Axiom 2 A 3D shape C in X has zero volume, provided h(a) = 0 for every $a\in A$.

Irina ... re "A.S.Pushkin.V.G.Boltyansky created a formula of mathematical aesthetics: "beauty= visualization+surprise+simplicity."

It is when there is an event horizon that we are surprised with meaning, comes from within to the understanding. Einstein felt that nothing can be conjectured to exist that does not have an event horizon, yet a new idea about black holes from Steven Hawkings says that maybe blackholes (spaces so dense they collapse under their own gravity and no light can escape them) exist but do not have event horizons, in Einsteins interpretation cannot exist. Maybe are but nothing but a ugliness(Einstein once claimed "he thought he understood mathematics")(maybe a "prepared with" mathematics that had become transformed unawarely by a fatal attraction of some kind to a pushy " preparing for" mathematics.

Though agreed that blackholes are nothing but carcasses, does the sum of all numbers exist? I'd not be 'surprised' if the next step isnt to postulate that all numbers exist both within and without black holes. Are we learning or seeking?.. immortality?...a condition where we are dressed uniformly like soldiers and suprised by nothing, even death or doom?

James: The problem confronting me, though I have a way to generate the egg, to understand it better would be to find a possible series and recurring element to generate data that parallels that from the graphing function I use in which data points are determined by rounding and division of pi by whole numbers. The graphing function involves algorythums used to render trig values, rounding paramters and maybe even common eccentricities of computer processors, the resulting 3D shapes are an "everything but" what usually comes from the even lines of engineering method. Such a series might be very complex. The simplicity in the method involves an opposition of kinetic to potential energy that entails a whole view=universe and not laws that are valid for each and every intersection, and an associated philosophy and cosmology that does not involve the same treatments of matter and energy as in modern physics. I think fuzziness has a role here within a system homogeneity of shape type with respect to diection of pointing, the momentum vector, age. It does not involve pattern recognition as one might sort a face from a moving background but identification of plausible prominent historical patterns as pieces from which they are evolved. The pattern recognition you discuss and to what I am referring maybe quite distinct , entailing an added dimension that maybe or may not be handleable....I think, regardless of computer capability, a studied historical imagination is first required to produce results that are selective and unique for pursuit ...my premise is that a long term chronic condition exists upon which pathological behavior leans throughout history. e.g an invisible and extraneous, but prominent influencing pattern.

I do not have experience with LaTex, and am only capable to work with 3D vector =radius, theta and phi, point data that is selectively obtained.

Marvin,

Very interesting post! Okay, for a series with a recurring element, one of the things you may want to consider is how to identify a recurring element, especially in a case where the recurring terms are widely separated (perhaps with an infinite number of terms in between). For example, consider the series

...1, 1, 2, 3, 5, x, 5+x, 5+2x, ... , y, 5+x, 5+x+y, 10+2x+y, ...

Assume the recurring term is 5+x. If we have a description of the recurring term, we can then determine those terms that are descriptively close to the recurring term of interest, even if the terms are an infinite distance apart. The description would be something like a serial number that reflects our knowledge of the parts of a recurring term.

For followers of this thread, I have just found something remarkable.

Brains scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers.

Mathematicians were shown "ugly" and "beautiful" equations while in a brain scanner at University College London.

The same emotional brain centres used to appreciate art were being activated by "beautiful" maths.

One of the researchers, Prof Semir Zeki, told the BBC: "A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain - the medial orbito-frontal cortex - like looking at a great painting or listening to a piece of music."

The more beautiful they rated the formula, the greater the surge in activity detected during the fMRI (functional magnetic resonance imaging) scans.

...He said beauty was a source of "inspiration and gives you the enthusiasm to find out about things".

The hugely influential theoretical physicist Paul Dirac said: "What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating." (see attached image)

Mathematician and professor for the public understanding of science, Marcus du Sautoy, said he "absolutely" found beauty in maths and it "motivates every mathematician".

For more about this, see

http://www.bbc.co.uk/news/science-environment-26151062

I think we do get a certain high from perfection reflected in mathematical expression because it unites to circumvent the sense of chaos and tedium associated with measurement; but nature working as a concerted whole only produces witnessible dimension at time bearing intersections seen amenible to mathematical description. Mathematical description can also be a means of corner cutting in the same sense that criminals cut corners, over time can lead to habituation, seeing beauty in criminal methods. The atomic bomb is clearly a case where the sense of beauty is lost, beautiful designs, though, can lead to applications that subvert the senses in recognizing intersections that are only parts of a whole beauty bearing unity and that are strictly constructed of time bearing dimension. It is possible to first recognize beauty in design that subsequently can hone the ordinary street corner from its natural place in the forest. In comical view, under the stress of an exceptional weather, nature seen as an adversary akin to the threat of a criminal, collective thought and behavior might resemble a long distance telephone call deep in space to resemble its' source in actions that with it. Viewing modern society there are two problems:

1) identifying the characteristics of a poor (information bearing) weather that also seems to elicit eccentric behavior in both scientific (e.g.the overuse of resources to contact and explore very distant space targets, engineeri ng towards the mechanization of natural processes).and sociological pursuits(e.g globalization efforts reflecting ideas of unification). The fetures if the poor weather become simple to elucidate from just an overview.

2) more difficult is the production of teaching and wisdom to inhibit behavior becoming dominated by mass communication devices (as described previously operate also by corner cutting methods in communication apparatus desigm that nearly duplicate the etiology of rising immune disease problems ..evidenced to surface in cultures among individuals with a similar sense of ingrained perfection in principle belief are ideas of secret government activities that are absurd to common sense. ...deliberate invention of aids, secret underground dens and plots of a science fiction nature, loss of respect for government and authority, the world as a game machine). I do not think it is recognized that the very wor

ds and principles used to combat such troubles are awarely known to contradict to align more deeply within problems solving techniques of organizations in approach to sociological problems, is filtering into the home, mathematical like symmetry as a guiding theme, has beauty only on paper, cant be realized in application.

The pattern I am describing becomes distant to guide attention apparatuses rather than attracting active focus itself...it is possible, as nature itself possesses templates that are also a major topic in education and mass media, for the real and important patterns to emerge likewise to be applied reflexively for identification of those with infinitely lesser practical importance. It is also clear that nature is not understood very well...as quoted previously by Irina "our most complex theories maybe but observational notes".

to add: If interested the link to "A Live Wire : Machismo of a Distant Surface" is included; I would appreciate any comments or inputs.

Article A Live Wire: Machismo of a Distant Surface

Irina and Marvin,

I remember reading in Bertrand Russell's ABC of Relativity that it is amazing how much we have done with the little that we know. In his last chapter on the philosophical implications of science, Russell writes:

One thing which emerges is that physics tells us much less about the physical world than we thought it did. Almost all the 'great principles' of traditional physics turn out to be like the 'great law' that there are always three feet to a yard; others turn out to be downright false. The conservation of mass may serve to illustrate both these misfortunes to which a 'law' is liable. Mass used to be defined as 'quantity of matter',

and as far as experiment showed it was never increased or diminished. But with the greater accuracy of modern measurements, curious things were found to happen. In the first place, the mass as measured was found to increase with the velocity; this kind of mass was found to be really the same thing as energy. This kind of mass is not constant for a given body. The law itself, however, is to be regarded as a truism,

of the nature of the 'law' that there are three feet to a yard; it results from our methods of measurement, and does not express a genuine property of matter.

...The world which the theory of relativity presents to our imagination is not so much a world of 'things' in 'motion' as a world of events. It is true that there are still particles which seem to persist, but these (as we saw in the preceding chapter) are really to be conceived as strings of connected events, like the successive notes of a song. It is events that are the stuff of relativity physics. Between two events which are not too remote from each other there is, in the general

theory as in the special theory, a measurable relation called 'interval', which appears to be the physical reality of which lapse of time and distance in space are two more or less confused representations. Between two distant events, there is not any one definite interval. But there is one way of moving from one event to another which makes the sum of all the little intervals along the route greater than by any other

route. This route is called a 'geodesic', and it is the route which a body will choose if left to itself.

The attached image shows the cover of Russell's book, which can be downloaded from

http://www.sadena.com/Books-Texts/Bertrand%20Russell%20-%20ABCs%20of%20Relativity.pdf

James : In better words it seems to be the recurring element itself that is alluring; does not always result in beautiful structure and can result in behavorial addiction. This possibly indicates a deficiency of naturally occuring recurrance, an prominet guiding sense of uncertainty. Recurrance is a guiding feature of nature (e.g. genetic elements) but it can produce success or failure in a bed guided by possibility rather than governed by proximal transmission. I have attached a link from another RG thread (Is internet making us lose our skills to exercise our brain?) on super normal stimuli. I think the recurring element in nature in my description is a radiation that I also believe is known to exist, though not realized how chronically close it is held; leads to super normal behavior like colonizing and constructing umbrella states followed with remedies to build a gigantic umbrella to block excessive radiation. Scientific umbrellas tied to mythological belief centered around educational and medical facilities are normal occurance block to block amidst city streets in American cities, a large science fiction and "actual fact" media market surrounds them. I do not think "umbrella" is the answer and could prove as deadly as other behaviors, but the egg figure resembles an umbrella with a line rather than point as the center....I think to try to sort out two surfaces perceived as one (within a bed constructed singularly of intercourses of the same type of surface) to improve perceptual and conceptual clarity.

In the etiology of both cancer and aids genes discovered to be associated or responsible are also normal genetic components of cells.

http://www.sparringmind.com/supernormal-stimuli/

James

"Beauty depends on size as well as symmetry."--Aristotle

Size and symmetry tacitly imply structure, I am of the opinion that the greatest source of mathematics is the beauty of patterns and that beauty of patterns and structure of patterns are not unrelated.

You ask "is it the painting Mona Lisa or the golden ratios in the painting that first attracts us?" My reply is that people were attracted to the Mona Lisa not knowing anything about the golden ratio! The question is did the golden ratio gave beauty to the Mona Lisa? IMHO the answer is yes; this clearly demonstrate that beauty of patterns and structure of patterns are not unrelated.

Issam, James : I think there is a temporality factor involved. If first we see the Mona Lisa and that is beautiful, and we see that it is constructed of golden ratios then golden ratios become beautiful. Beautifulness maybe altogether something else, the mind pursues it because it knows ugliness and beauty is the opposite: one does not exist without the other. Perhaps, as the mind is able to predict and project to navigate using causality learned from experience, ugliness and uncertainty are associated as one, the discovered pattern a potential path towards beauty. The failure to find beauty then results from the failure to understand the association of recurance and nature; recurance does not necessarily entail predictability.

to add: then again the mind is able to suffer ugliness delivered from the hands of beauty...e.g. Samson and Deliah..or a violent storm that can be both beautiful to the senses and destructive to untasteful to the senses and destructive to the body. At this juncture body/mind nature in proximal range looks more like a contrivance than an all time friend. Man can be his own enemy when he digs with prosthetics (e.g microscopes,telescopes) mixing what is in the world of his own dimensions with .the vast, microscopic, to cut corners.

I think the association of language and thought has a temporality factor involved that is similar to the idea of beauty that is transferable from a whole painting to the golden ratio discovered buried within it. Many would agree that thought can be languageless and composed of only images, yet words can become substitued to describe after first exposure It can become difficult to distinguish the "this x" and "that y" from whole scenes. Perhaps it is real that social patterns are the consequence of conceptual repression in a field of defining according to thisks and thats and broadened physical ranges. Looking at the natural history of humans it looks like the seeking to broaden the physical range is instinctual, humans are the most successful at it: but it is also possible to ask about nature, contrivance and recurrance whether first steps towards seeking and an a resulting eccentrically excessively ordered world involve tricks to the instinct.

Marvin, you write: I think the association of language and thought has a temporality factor involved that is similar to the idea of beauty that is transferable from a whole painting to the golden ratio discovered buried within it.

Good point. It takes time to make discoveries after on is attracted to an object with beauty such as the Mona Lisa.

Many thanks to Professor Costas Drossas, University of Patras, who made the following observations about patterns, beauty and structures:

The basic element which give us a sense of beauty is the “patterns”.

I think that “patterns” owe their beauty to the underline mathematical structure. This structure is group structure which express the concept of symmetry. The fascinating book of H. Weyl is a good source of examples. I should add the designs of the Dutch artist Escher. For patterns see also the nice book: F.J. Budden The Fascination of Groups 1972, Chapter 26. Relevant to the question is the paper: GIAN-CARLO ROTA THE PHENOMENOLOGY OF MATHEMATICAL BEAUTY, Synthese 111: 171–182, 1997 (see attached pdf file).

Gian-Carlo Rota observes that instances of theorems that are both beautiful and surprising abound. Often, such surprise results from a proof that borrows ideas from another branch of mathematics.

An example of mathematical beauty upon which all mathematicians agree is Picard's

theorem, asserting that an entire function of a complex variable takes all values with at most two exceptions. The limid statement of this theorem is fully matched by the beauty of the five-line proof provided by Picard himself (p. 173).

@Issam Sinjab: You write: "Beauty depends on size as well as symmetry."--Aristotle

Size and symmetry tacitly imply structure, I am of the opinion that the greatest source of mathematics is the beauty of patterns and that beauty of patterns and structure of patterns are not unrelated.

Yes, I agree with you. Apparently, in doing mathematics, we invariably start by structuring some form of a set of points such as the set of points in a picture. By "structure", I mean the arrangement of the parts and relations between the parts of a set. There are different ways to obtain a structure on a set. Here are two examples.

1. An ordering of the parts of a set. Example: painting by DaVinci, where the picture points are ordered in terms of the colour and geometry of the points and lines in depicting the Mona Lisa. Another example: the set of all points that are close to a given point.

2. Any set endowed with a relation is a structure. Example: A set X endowed with a nearness relation so that, for each given subset A in X, one can find all subsets that are near A. Another example: binary relation on a set.

Dear James, thanks a lot for your publication by Gian-Carlo Rota- so aesthetic! Dear Marvin, "Are we learning or seeking? immortality?" An ordinary situation- "a condition where we are dressed uniformly like soldiers and surprised by nothing, ever death or doom?"To Brentano, "Reality is individual, but, learning it- we generalize it". To Husserl, difference between a fact- entity- and a thing- isness is becoming a means of substantiation of logic and math. Mathematical judgments are necessary and universal, because they fix relations between things. They needn't in piece of evidence of their value. "The sum of interior angles of Euclidean triangle is 180*" needn't in a such kind of verification.

I think it is the beauty of patterns that is the starting point that attracts everybody's attention. However, when others stop with just appreciation of the beauty, a mathematician starts looking for the underlying structures that lead to the perception of beauty by the human brain. And that has the potential to become the source of next level of unraveling in mathematics.

I refer you to our mathematical art gallery on yutube to see the beauty of mathematics. Use the link:

https://www.youtube.com/watch?v=nd6Dx9c1txw

This link was established on April 28, 2009 to show the result of a mathematical art algorithm.

This algorithm has two stages. The first is to produce all the curves 3D representing the shape. It does not require much calculation time. The second step will cover all the curves produced by the first stage, to give them a shadow and color. It takes a lot longer than the first using the software Matlab.

To make the processing of 3D shapes, we vary a parameter. At each value of the latter is a form. In most cases, a small change in the parameter induces a small change in the 3D shape. .

Without involving the transformation symmetry eliminates all forms synthesized have some symmetry. More symmetrical shape can be eliminated.

Despite the variety produced forms, they are generated by the same algorithm.

Although work on this algorithm is far from over, we believe that this algorithm has two interesting points:

a) his approach is totally new

b) it generates a wide variety of forms.

Authors:

Jelloul ELMESBAHI; Ahmed ERRAMI; Mohamed KHALDOUN; Omar BOUATTANE

Dear James Peter;

I share the idea behind your following question :

The question here is not whether mathematics is beautiful but rather whether beautiful patterns are the starting point of mathematics. ?

Our modest gallery, of 1076 synthetic videos,  and the interpretation of the equation each 3D image video  can comfort mathematicians and give them some response elements to  this question.

the video files are zipped to fit yutube image resolution ! 

https://www.youtube.com/channel/UCjnyt-WYnqb6nl-Ka4B2rNw/videos?feature=c4-feed-u

Good show !!

All of you, ingenious souls, may be happy to read this, extracted from the book, “The Evolution of Physics”, by Albert Einstein & Leopold Infeld, published in 1938:

“Physical concepts are free creations of the human mind and are NOT, however it may seem, uniquely determined by the external world. In our endeavor to understand reality, we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious, he may form some picture of a mechanism which could be responsible for all the things he observes. But he may never be quite sure his picture is the only one which could explain the observations. He will never be able to compare his picture with the real mechanism & he cannot even imagine the possibility or the meaning of such a comparison.”

If the motions of nature, its unity and its surprises. are not really captureable,  I think mathematics finds beauty when it creates motion and that looks, as nature, to recurr from within.  Beauty is attached to recursives that function to paint larger perspectives.

I completely agree with James Peters that the beauty of mathematics is similar to Mona Lisa Giaconda, but my guys take offense at it. Because cockroaches consider that the beauty of mathematics is similar to a female of a cockroach, snakes consider that the beauty of mathematics is similar to a viper female, and only the Ebola viruses consider that the mathematics is similar to the Ebola viruses because viruses have no females.

Recursivity, Fractal, Modulo, symmetry; are the impressive tools to show few aspects of the beauty of mathematics. naturally and biologically we are symmetric our life is cyclic and recursive. some of our organ are fractal. etc.

By mathematics we are discovering just what we are; so the humain map is so beautiful that mathematics can do. Our inspiring behaviour shows us that, by mathematics we represent closely the secret of the nature.

   Though science and math can conceive of many possibilities for things that have beauty, elicit what philsophers of aesthnetic call an "aesthetic emotion" the world of nature or real identities might be different from when it is considered that real nature and the disposition of "the whole" is what matters for the eye in terms of survival. If the world can be defined exclusively just to consist of unique identities rather of groups of things organized statistically, the set of prime numbers, random and infinite, might fit to contain it. The set of primes is random and infinite, primes are each unique, divisible by themselves and one only 

     When one looks at the night sky with unaided vision he might say to have observed "primes"...surfaces of many points each that are divided from those of an infinite variety of surface with points, by prime gaps to what ever points on what ever surfaces lead to the existence of the set of points to a surface.

    I think.it is possible to build bubbles that are displaced from the consensus, from past progresses that are based on the directly experienced universe, when scientists make models based on second hand observations using prosthetics like radio waves, radar, in which emissions and detection are secondary to those of unaided witness in which the first-hand, person itself is both emitter and receiver, the test equipment.  

Such method can be visualized to produce changes in surfaces, to cause displacements in which some of the surface points of surcaces are no longer "prime gap" to orininal defining identities, to cause a redundancy of identity (to be come non prime or even/divisible by other than the self), loss of identity in a drift towards that which is closed, towards loss of uniqueness/identity.

   Handling primes is difficult mathematically and refers to the wierd rational facility that I think exists to witness by other than distance and time, but by first hand experienced form/shape that gives meaning. I think the cosmos is not safely presented to the rationale by second hand measurments that give scientifically "truthfull" meaning to observed geometrical form. Thinking of mythology and the constellations, mankind had substantially drifted, dangerously?, with respect to the way meaning is given to life experience.

http://www.wired.com/2014/12/mathematicians-make-major-discovery-prime-numbers/

I consider. that Omar mixed sequence of the concepts "beauty" and "secrets". The beauty is glossy magazines, and secrets are CIA and KGB. Therefore in the correct sequence at first there is a disclosure of secrets of intelligence services by means of mathematical algorithms, and the beauty which was stolen from archives of intelligence services opens then. After secrets learned everything, and confidential agents discharged from office and they went to draw beautiful pictures to Putin as Mr. Snoudan.