"Peano’s String; A History of Spiritual Stories" has been accepted into the Bridges show in Leeuwarden Netherlands Aug 2008.
For the theory on this piece please check out "Axiomatic Poems"Pablo Kagioglu sent me a few slides showing some Mathematical Paradigm Poems. I am extremely limited in my understanding of Quantum Mechanics so I am sure there will be much reflexive didactic that I will miss however, I do find it interesting that he has substituted the idea of "quanta" for “self” in our human identity. I am interested in pondering these further I hope you do as well.
Here is an orthogonal space Poem submitted to us by Thierry Brunet via France.
In Delancyplace's excerpt for 4/16/08 --as discussed by political advisor Frank Luntz, the sequential arrangement of information often creates the very meaning of that information:
"[In film, when] two unrelated images are presented, one after the other, the audience infers a causal or substantive link between them. A shot of a masked killer raising a butcher knife, followed by a shot of a woman opening her mouth, tells us that the woman is scared. But if that same image of a woman opening her mouth is preceded by a shot of a clock showing that it's 3 a.m., the woman may seem not to be screaming, but yawning. The mind takes the information it receives and synthesizes it to create a third idea, a new whole. ...
"The essential importance of the order in which information is presented first hit home for me early in my career when I was working for Ross Perot during the 1992 presidential campaign. I had three videos to test: a) a Perot biography, b) testimonials of various people praising Perot, and c) Perot himself delivering a speech. Without giving it much thought, I'd been showing the videos to various focus groups of independent voters in that order--until, at the beginning of one session, I realized to my horror that I'd failed to rewind the first two videotapes. So I was forced to begin the focus group with the tape of Perot himself talking.
"The results were stunning.
"In every previous focus group, the participants had fallen in love with Perot by the time they'd seen all three tapes in their particular order. No matter what the negative information I threw at them, they could not be moved off their support. But now, when people were seeing the tapes in the opposite order, they were immediately skeptical of Perot's capabilities and claims, and abandoned him at the first negative information they heard. ... I repeated this experiment several times, reversing the order, and watched as the same phenomenon took place. Demographically identical focus groups in the same cities had radically different reactions--all based on whether or not they saw Perot's biographical video and the third-party testimonials (and were therefore predisposed and conditioned to like him) before or after the candidate spoke for himself.
"The language lesson: A+B+C does not necessarily equal C+B+A. The order of presentation determines the reaction."
Dr. Frank Luntz, Words that Work, Hyperion, Copyright 2007 by Dr. Frank Luntz, pp. 40-41
Delineation #11
Mathematics, among other things, is a language.
Art, among other things, uses language.
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I feel that one of the most dangerous areas of contemporary art comes when the artist makes him/herself a target by embracing spiritual concerns. Our society enjoys pointing fingers at the inadequacies of institutionalized religion (there are many) and ignoring the archetypical ideas of the spirit that have brought us the wonderful icons of the past. These spiritual metaphors have manifested themselves throughout history in many forms always relating to the culture of the artist. Many of the ideas of religions are obsolete and don’t function well in societies as diverse and ever-changing as ours. The artistic challenge of spirit is an extremely difficult task especially when trying to use historically loaded iconography of current dominant religions. I think many of the artistic phobias associated with the spirit are due to our experience of so many ‘so-called’ spiritual artists, who have created cliché kitsch or dogmatic concepts that accent the hypocritical ideas of the church or yet have forged an audaciously different direction aligning themselves with likes of aliens from other planets. Also to note, there seems to be a direct conflict between science and the spirit, which is anxiously evident when scientific minds address spiritual matters. I believe the problem is based in the illusiveness of Truth in both arenas. There are many that think that Truth is defined by science using the language of mathematics. Others believe Truth is beyond logic and only evoked through the metaphoric language of a spiritual ritual. Then there is my personally distasteful category of those who think that Truth is defined by their particular religion or should I say defined by their particular church. Focusing on the later idea we see that human nature tends to have many conflicts and from a historical perspective, one of the most destructive is the religious “us versus them conflict.” I see churches tending to promote this kind of behavior due to its doctrine being fed through so many egos. On the same line of thinking, the testosterone of the self righteous seems to have made its way into religion and spiritual matters to set up so many of the conflicts that we humans engage in. Many have died and continue to die in spiritual wars created by the religious intolerant.
The fact that the conflicts exist, illustrate how illusive Truth is. It seems to me that science is no better when it comes to Truth. The eminent scientist David Boehm points out that science does not find Truth, its purpose is to correlate experience. Also in this vein, we can see that there are those who provide great arguments against the platonic nature of mathematics pointing out numerous problems with using mathematics as a true model for reality. I see the bottom line being that the terra firma of veracity is constantly shifting; therefore, we must accept this fact and move on. The eastern mystics use the metaphor, “form is emptiness and emptiness is form”.
I believe it is the function of special artists to assimilate as much information as possible from the diverse cross-planet cultural ideas not limited to including the concepts of science so that they can re-contextualize, synthesize and synergize their metaphors to be acute and pertinent to the global culture today. They must fully embody the ideas of love and tolerance as if the ideas were new so as to debride the cliché skins attached to them. As impossible this task seems, it is the challenge of those artists to reconnect the loose strands of past archetypical works and re-contextualize them to breathe new life in today’s world. Their job is not to run from the spiritual confusion that permeates the ever-changing cultures on this globe by hiding in some self-conceived scientific illusion of truth without spirit. That is not to say that science cannot be the new religion … it can. However, the spiritual scientist must connect the magical and irrational mind to scientific metaphors so that our spiritual understanding can be flexible as science metamorphoses. The past mytho-spiritual ideas were always based in the science of the times. It takes courage to navigate through the mental minefield of past ‘truths’ finding new veracity that resonates in ones psyche as they express it and expose themselves to the ridicule of being an irrational kook.
I believe the special artist/poets should focus their efforts to make metaphors current to our historical and sociological condition. The purpose of a metaphor is to bridge the infinite to the concrete. Many people feel that past mytho-spiritual/religious metaphors are absolute in the notion that they permanently point to the infinite. Personally speaking, I see the veracity of metaphors being temporal with their cultural relevance having different half-lives. What can confuse matters is that the half-life in some metaphors have existed for such a long time that they seem absolute. There is an argument that the Bastian elemental ideas and Jungian archetypes are absolute. Even if this is true, the metaphors employing those elemental ideas always need recontextualizing to be relevant to the current cultural thought. The frustrating aspect for the artist is having so little control over the fertility of the inspiration process. I wish I could say that artists had full control over the source and production of their metaphors. However, it seems to me that their strength, viability and temporality are a function of graciousness, imparted from the muses. I believe it is though the struggle and success with life that these special artists acquire the molecular building blocks of a vocabulary that becomes the means of their expressions. These ideas logically coagulate around an infinite idea provided to them by the unknown.
Delineation #10
Mathematical creations are not unique in the sense that they could be discovered by anyone.
Artistic creations are uniquely invented by individuals.
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I receive a very important comment the other day from Jonathan who uses the JD2718 to identify himself on his blog. His comment was in reference to axiomatic mathematical poetry. However, I think his question is much broader.
Jonathan expressed the following:
Abraham, cats, Gods.
One, numbers, successors.
Which is really more poetic?
This is a sticky question because I want to avoid slipping into the bottomless void of the “What is poetry? What is art?” question However; I can discuss elements of poetry from which my idea of poetics is derived. I also want to add the following statements are not a value judgment on the aesthetics of mathematics. The mathematical aesthetic is one of the most wonderful experiences one may realize.
To answer Jonathans question; I am assuming that his question implies that pure mathematics is poetic. It is my view that pure mathematics is not poetic. Furthermore, the quick and dirty response to this question is that pure mathematics is different from poetics the same as pure mathematics is different from physics. Physics and Mathematical poetry, although vastly different, live in the realm of applied mathematics. Even when we ‘feel’ that pure mathematics is poetic, we are applying mathematics to some preconceived notion of what we believe poetry is without actually applying it. We may choose to argue that mathematics contains elements of poetry such as rhythm and pattern. Yet one may argue that it is not maths that has poetic elements but poetry that has mathematical elements. For the sake of argument, let us say that poetry possesses the mathematical element of pattern. I would like to make the point that it is difficult to get excited about these metric patterns when taken out of the context of poetry and view in only the light of mathematics. I know we are starting to get away from the intention of our question however, the point I want to make is that the aesthetics of mathematics is much different from the aesthetics of poetry and poetics. The ‘polyaesthetic experience’ that we are discussing is a vector sum experience of the aesthetics of art/language poetry and the aesthetic of mathematics. (They are different aesthetics) If we were to separate the mathematical aesthetic from a language poem how beautiful is it? Now let us look at the aesthetics of mathematical pattern by comparing the beauty of the pattern in iambic pentameter (or any other meter for that matter) to the beauty of self-similar patterns in a mathematically generated fractal. Which is more beautiful? Is the ‘isolated’ metric pattern in poetry more beautiful than a fractal? How about asking, “Is the fractal poetic?” If so what are the elements of poetry in the fractal. Is it the concept of rhythm that makes maths poetic? Are all things displaying rhythm poetic? The point I am trying to produce is that mathematical poetry, makes the structure of mathematics poetic only by application of poetics within that structure. Pure mathematics is not poetic by itself.
When addressing the metric beauty in language poetry; the metric beauty is not relevant to the mathematical pattern per se. It is relevant to the aesthetics involved in the relationship of the pattern to the words and the sounds of the words with its synesthetic energy igniting the meaning of the words as they point further to the cultural and historical relationships within the poem. The mathematical aesthetic devoid of the poetic aesthetic plays an extremely limited role in the aesthetics of language poetry. Yes, there is maths in the poetry however, break it out of the poetry, isolate it and I believe it becomes aesthetically trivial.
Let us look at metaphor – Does pure mathematics express metaphor? How could it? for pure mathematics is more about illuminating the logical structure of thinking. The key word that I want to stress is “logical”. Metaphor requires logical tension if not paradox to function as a concept to bridge the infinite to the concrete. However, I must say that mathematics does provide us with the linguistic structure to express metaphor. Again, this is the issue of pure mathematics relative to applied mathematics. To express metaphor you have to have an application of poetic concepts. You need a source domain and a target domain. (see the section on metaphor structure at Wikipedia) Pure mathematics does not have these metaphoric domains until we apply the poetic idea to the structure of maths as we do in mathematical poetry. The essay “Polyaesthetics and mathematical poetry” goes into more detail on this matter as well as an interview conducted by poetic aesthetician Gregory Vincent St. Thomasino. The interview will soon be published at “word for/word” an online journal of new poetry. I hope to announce the interview soon at this blog.
This is a page devoted to collect information on axiomatic poems.
Introduction to axiomatic poems -- Peano’s string; a history of spiritual stories.
Axiomatic Poems part two -- More structure added to Peano’s string; a history of spiritual stories.
The addition of another stanza and creating a metamorphic poem.
Proof that no cat is the God of itself (Peano’s proof by Professor Ray Balbes)
In March of 2007 I announced “Polyaesthetics and Mathematical Poetry.” published by
The contents of the paper are available for downloaded free at this link.
Journal of Mathematics and the Arts published “Polyaesthetics and Mathematical Poetry” March 2007 Volume 1 Number 1 ISSN 1751-3472
The published paper can be purchased at this link.
The Mathematician, Professor Ray Balbes will prove to you that “No Cat Is The God Of Itself”.
At the end of my blog entry for Axiomatic poems dated
Kaz
The Professor Ray Balbes wrote the following text:
For reference, here is what we have so far.
The Peano Axioms
Let us replace “number” with “cat” and let us also replace “successor” with “God”. Lastly, I am going to replace “One” with “Abraham”.
The Poetic Peano Axioms
OK, now to make the theorems more succinct, lets set up some conventions. With regard to the Peano Axioms, let us call the set of all numbers N and let us denote by n’, the successor of n
Also let:
11 = 1’
12 = 1’’
13 = 1’’’
etc.
We will refer to 1’ by the name of 2, 2’ will be called 3, etc.
Axiom 3 says that there is no n such that n’=1.
Axiom 4 says that if m’ = n’ then m=n.
Axiom 5 says that if S is a non-empty subset of N with these 2 properties:
i) 1 is in S
ii) If n is in S then n' is in S.
Then S = N.
Here are three theorems that lead up to the Well Ordering Principle. First, I will state them in terms of the Peano Axioms, next in terms of the Poetic Peano Axioms and finally I will prove something.
Theorem 1. For every n in N, n’≠n.
Theorem 2. If n ≠ 1 then n=m’ for some m.
We will say that m ≤ n provided that m = n or mp = n, for some p
Theorem 3. For every n in N, 1 ≤ n
Theorem 4 (The Well Ordering Principle) If S is any non empty subset of N then there is a number m in S such that m ≤ n for all n in S.
Here are the theorems in terms of the Poetic Peano Axioms. We will say that m is the source of n provided that m ≤ n. In other words, a finite number of Gods of m, yields n.
Theorem 1 No cat is the God of itself.
Theorem 2. Every cat, other than Abraham is the God of some other cat.
Theorem 3. Every cat has Abraham as a source.
Theorem 4 (The Well Ordering Principle) In any (non-empty) set of cats, there is one that is the source of all the others.
Here is the proof of Theorem 1 in terms of the Peano Axioms
Let S = {n| n’ ≠ n}. We will show that S satisfies the conditions i) and ii) of Axiom 5. By Axiom 3, 1 is in S so i) is true. To prove ii), suppose that n is in S then n’≠n. But if n’’=n’ then, by Axiom 4, we would have n’=n, a contradiction, so n’’<>n’. Hence n’ is in S. This means that S satisfies the conditions of Axion 5 and therefore S= N. So that n’<>n for all n in N.
Here’s the proof of Theorem 1 in terms of the Poetic Peano Axioms. Note that in the proof, I am referring to the Poetic Peano Axioms, not the Peano Axioms.
Consider the set S of all cats that are not Gods of themselves. We will show that S satisfies the conditions i) and ii) of Axiom 5. By axiom 3, Abraham is a member of S so i) is true. To prove ii), suppose that Isaac is a cat in S then Isaac is not the God of Isaac. Suppose the God of Isaac is Moishe. Now if the God of Moishe is Moishe then by Axiom 4, Moishe would be Isaac; that is the God of Isaac would be Isaac, a contradiction. Hence Moishe is in S. Since Moishe is the God of Isaac, we have shown that the God of Isaac is in S; in other words, the condition ii) of Axiom 5 is satisfied and thus S is the set of all cats. This means that all cats satisfy the property that they are not Gods of themselves.
The proofs of the other theorems are similar to this.
Ray
I have been having some wonderful conversations with the mathematician Ray Balbes. Ray has been asking some very important questions concerning the axiomatic poem. Ray has also helped me by correcting mathematical errors in my nomenclature.
Ray also has had concerns with the idea of God being a viable substitute for successor within the Peano axioms. For God in this sense must be comparable to a mathematical function. I personally have no problem with this idea for my understanding of the word God is metaphorical anyway. Therefore, I can see this metaphoric structure of “God IS mathematical function” as being nested e.g. metaphors within metaphors. The question then would be is God a mathematical function? Alternatively, can we say God functions mathematically? Historically God is described beyond language so I would not try to convince anyone otherwise. I personally do not see God functioning mathematically as a mathematical Platonist would however, I do see the accessibility of ideas mathematically expressed as phenomena attributed to a deity. I believe if you denote phenomena with words, you can do the same with math. Furthermore, I would go on to say that if you can be inspired to connote it with words you can do the same with math for those type of inspirations fuel mathematical poetry.
Therefore, the poem addresses the dichotomy of God being created by men or men being created by God.
To help anyone see how the logic in Peano’s axioms is functioning correctly in the Blog entry of January 29th, I created another axiomatic poem to show some more structure. The disadvantage to creating another ‘equal’ poem is that the new poem focuses the semantics in such a way that limits the metaphorical content. The advantage is that it gives more semantic structure, which enables one to see the Peano logic with ease. So in essence, we now have an axiomatic poem, which has metamorphic qualities. We see that the Peano axioms function as the underlying paradigm for the poem however, it could be viewed as the source domain with the other two ‘axiomatic stanzas’ as the target domains for the ‘overall metaphor’. In this case, we have three structures separated by two equal signs.
Poem #2 -- Peano’s string; a history of spiritual stories
From Poems 1972-1997 Copyright © 1997 by Scott Helmes
Can these axioms create interesting theorems?
View the show here
The AMS show is now visible in it physical construction in Exibit Hall B at the San Diego Convention Center. The good news is that you don’t have to be in San Diego to view it you can go to the link here. The bad news is that the internet destroys some of the subtleties in the images. For example, the image by Andy Lomas (above) has beautiful delicateness that cannot be imagined here on the internet.
Andy’s image is composed of layered trajectories followed by millions of particles. Each individual trajectory is essentially an independent random process, with the trail terminating when it reaches a deposition zone. Collectively the paths combine to form delicate complex shapes of filigree and shadow in the areas of negative space that the paths don't reach. Over time, as particles deposit they create a growing region that future particles will not be able to enter. There are no actual defined boundaries, simply intricately structured gradients of tone formed by the end points of trajectories.
Andy Lomas, Digital Artist, London "These pieces are part of a study into how complex organic forms can be created from simple mathematical rules.
The base algorithms used to generate the forms are variations on Diffusion Limited Aggregation. Different structures are produced by introducing small biases and changes to the rules for particle motion and deposition. The growth like nature of the process, repeatedly aggregating on top of the currently deposited system, produces reinforcement of deviations caused by forces applied to the undeposited particles as they randomly move. This means that small biases to the rules and conditions for growth can produce great changes to the finally created form. All the software used to simulate the structures and render the final images was written by the artist in Visual C++."
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The image above is of a three dimensional piece by Carlo Sequin in which he explores the geometrical relationships of a hole to a surface moving through a tube-like structure.
" Scherk's 2nd Minimal Surface" is a way to weave together two intersecting planes so that an infinitely long chain of holes and saddles replaces the intersection zone; it is possible to do that so that the resulting single surface has everywhere zero Gaussian curvature. The same basic scheme can be used to also blend together three planes that share a single intersection line. A small region, comprising just 5 monkey saddles and 4 Y-shaped holes, has been cut out of such a minimal surface; it has been artistically stretched and twisted to make a towering sculpture. Carlo H. Séquin, Professor of Computer Science, EECS Computer Science Division, University of California, Berkeley
Mathartist statement:
"My professional work in computer graphics and geometric design has also provided a bridge to the world of art. In 1994 I started to collaborate with Brent Collins, a wood sculptor, who has been creating abstract geometrical art since the early 1980s. Our teamwork has resulted in a program called "Sculpture Generator 1" which allows me to explore many more complex ideas inspired by Collins' work, and to design and execute such geometries with higher precision. Since 1994, I have constructed several computer-aided tools that allow me to explore and expand upon many great inspirations that I have received from several other artists. It also has resulted in many beautiful mathematical models that I have built for my classes at UC Berkeley, often using the latest computer-driven, layered-manufacturing machines. My profession and my hobby interests merge seamlessly when I explore ever new realms of 'Artistic Geometry'."
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January. The following image is by Reza Sarhangi and Robert Fathauer (see Robert's work)
Reza is one of, if not the most important person in the vismath genre for Reza is the nucleus of the Bridges conference on mathematical connections in art music and science. He is a very special man and I really appreciate everything he has done and continues to do for the genre.
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January. The following image is by Gary Greenfield who also happens to be the chief editor of the Journal of mathematics and the arts. I really enjoy Professor Greenfield no nonsense approach concerning this genre.
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January.
The following Image is by the mathartist Michael Field. Looking at these small jpegs is quite an injustice to these works. If you see them in person, you will be amazed at the complexity of texture. I have loved professor Field’s work since the first time I saw it at Bridges.
Part of a repeating pattern of type pmg. The pattern was generated using a smooth symmetric torus mapping and then lifted to the plane. The colors reflect the density of an associated absolutely continuous invariant measure.
Michael Field, Professor of Mathematics, Department of Mathematics, University of Houston
Mathartist Statement:
"All of my art work is based on ideas rooted in dynamical systems, chaotic dynamics and invariant measures (part of my field of research). I developed all the software, algorithms and coloring used for these images. I also built the computers used to generate the images and printed these images myself.
My interest primarily lies in the ways in which one can achieve certain desired artistic effects using a "mathematical palette" (as opposed to using images toilluminate the mathematics)."
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January. The following image is an M.C. Escher homage by Robert Fathauer. Also to note that Robert has curated many mathart exhibitions around the world.
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January. The following image is by Anne Burns who is a prominent figure in the vismath world and I continue to find her fractal imagery extremely fascinating.
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January.
While we are looking at space filling curves … The following piece is another space filling curve image similar to the last blog entry yet this one has a different premise.
"Outside Ring" is a continuous line drawing constructed from a 3000-city instance of the Traveling Salesman Problem. The line is a simple closed curve drawn with white ink. It divides the plane into two regions: in (drawn with red ink) and out (drawn with black). From afar, the piece looks like an alternating link, a knot formed from two interlaced loops, one red and one black. Robert Bosch, Professor of Mathematics, Robert and Eleanor Biggs Professor of Natural Science, Department of Mathematics, Oberlin College, Founder of http://www.dominoartwork.com/
Mathartist Statement:
"I specialize in "Opt Art", the use of mathematical optimization techniques to create pictures, portraits, and sculpture. I have used integer programming to create portraits out of complete sets of dominoes, linear programming to create pointillistic pieces, and instances of the Traveling Salesman Problem to create continuous line drawings. What all my pieces have in common---aside from how they were constructed---is that they look very different up close than they do from afar. I create my artwork out of a love of optimization---the theory, the algorithms, its numerous applications. I believe that optimization can be applied to virtually every imaginable field, and I believe that my artwork does a good job of helping me make that point!"
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January.
The following vismath piece is by Douglas McKenna who was kind enough to invite me to visit his studio a couple of years ago. Since then I have been a fan of his 'space filling curves'.
Graphical work based on links and interlaced structures.
The next few days I am going to diverge from mathematical poetry and display some of the visual mathematics work done by many talented people who have their work admitted to the American Mathematical Society mathart show coming up in January. The beautiful piece below is done by Slavik Jablan
Delineation #9
I am grateful and honored that the American Mathematical Society has accepted “DOG DREAM” (above) and “TEMPTATION” (below) to the 2008 art exhibition, which is concurrent with the 2008 AMS conference which takes place in January of 2008.
Both poems are in the form of an orthogonal space poem.
Double Derivative Poem
The poem above serves as an example of what I am calling a double derivative poem. In the study of mathematics a derivative is the rate of change of one thing as measured by the change in another. The double derivative poem addresses the first and second derivatives of a “phantom function”. The “mathematical function” that I am speaking of is to be experienced with ones intuition as opposed to being explicitly described as a single curve on a Cartesian coordinate system.
The idea for this poetic structure occurred to me while reflecting on the equations from physics that describe the relationship between position velocity and acceleration. Its similar triangles structure dawned on me because physics teaches the position of a moving object is to velocity as velocity is to acceleration.
The derivative:
Animation showing that when delta x approaches zero the secant line becomes tangent. (click on the image to see the animation)
