The Metamorphic Mathematical Poem

From Poems 1972-1997 Copyright © 1997 by Scott Helmes

"Philosophic cocktails" by Thierry Brunet 2007

I would like to introduce a new term for a technique used in mathematical poetry. The first person I know to have used this technique is Scott Helmes. His poem from 1997 (upper image) illustrates the technique well. One can see that it has five structures separated by four equal signs. What occurs is that the mathematical poem contains several structures (equations) all set equal to each other. In effect, the poem reads as a series of statements that metamorphose into each other through the duration while reading the poem.

The lower image, by Thierry Brunet, titled “Philosophic cocktails” is also a metamorphic mathematical poem as you can see three structures separated by two equal signs.

A metamorphic mathematical poem could possess unlimited structures and equal signs however; it must contain at least three structures separated by two equal signs to be considered metamorphic.

The aesthetically interesting thing about these poems is that the target domain and the source domain for the ‘overall whole’ metaphor bounces and shimmers in ones mind as you swap or rotate the domains around each other. This is due to there being multiple domains for the target and source. **

**The metaphor nomenclature borrowed from the cognitive scientist George Lakoff can be viewed in more detail at this link.

Axiomatic Poems

Peano’s string; a history of spiritual stories (Image above)


Axiomatic Poems

I would like to introduce a new mathematical structure to be used with mathematical poetry.

I understand that for two thousand years Euclid’s axioms stood alone as a meaningful axiomatic system. However, in 1889 Italian mathematician Giuseppe Peano created a new axiomatic system based on two primitive notions and the five following statements:

1. One is a number
2. If x is a number, the successor of x is also a number.
3. One is not the successor of any number.
4. If two numbers have equal successors, they are equal.
5. If a set of numbers contains the number one and it contains all the successors of its members then the set contains all the numbers.

What is interesting is that this system does not have to be limited to number. Calvin C. Clawson in his book “Mathematical Sorcery: Revealing the Secrets of Numbers” gives us the same five statements in the following form:

1. Heinsforth is a gelb
2. If x is a gelb, the ranker of x is also a gelb.
3. Heinsforth is not the ranker of any gelb.
4. If two gelbs have equal rankers, they are equal.
5. If a set of gelbs contains the gelb Heinsforth and it contains all the rankers of its members then the set contains all the gelbs.

Clawson has substituted the number “one” with Heinsforth, the term “number” with “gelb” and used “ranker” in place of successor. The point that Clawson is trying to make is that we need not be concerned with the primitive notions per se. What we need to be concerned with is the relationship of these notions within the axiomatic structure. From what I understand there could be incalculable different ways to describe the primitive notions however, only one way to logically relate them to each other. After reading Clawson’s axioms, I became aware of the ability of this structure to create metaphor. The source domain of the metaphor is the Peano axioms. The target domain is the same set of axioms with poetic substitutions placed inside the axioms. Therefore, I have created the axiomatic poem shown below:

Let us replace “number” with “cat” let us also replace “successor” with “God”. Lastly, I am going to replace “One” with “Abraham”.

1. Abraham is a cat
2. If x is a cat, the God of x is also a cat.
3. Abraham is not the God of any cat.
4. If two cats have equal Gods, they are equal.
5. If a set of cats contains the cat Abraham and it contains all the Gods of its members then the set contains all the cats.

Now the next interesting idea is:

Can these axioms create interesting theorems?

Blog Update

I obviously have not been working on my blog lately. My time has been consumed being interviewed by the Poet/Philosopher Gregory Vincent St. Thomasino. I am very happy with the interview for Gregory has asked some very interesting questions, which has inspired me into better defining the aesthetics of mathematical poetry. I hope to see it published next month on Jonathan Minton's “Word for/Word”.

Although the last few blog entries have interesting, they have not had any direct relationship with mathematical poetry. I am now looking forward to getting back to posting issues of mathematical poetry.

The American Mathematical Society Show Is Up And Running

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'."

Reza Sarhangi - At The AMS Show in San Diego - January

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.

This print was inspired by Abu’l-Wefa Buzjani's (10th Century Persian mathematician) construction of a regular heptagon contained in his treatise, “On Those Parts of Geometry Needed by Craftsmen”. His construction is illustrated by the linework in the center portion of the print. The characters around the perimeter of the design repeat Buzjani’s name in Farsi.

Reza Sarhangi, Professor of Mathematics, Towson University and Robert Fathauer, Small business owner, puzzle designer, and artist, Tessellations Company

Reza Sarhangi is interested in Persian geometric art and its historical methods of construction, which he explores using the computer software Geometer's Sketchpad. Robert Fathauer creates digital artworks on a Macintosh computer, primarily using the commercial programs FreeHand and Photoshop.

Gary R. Greenfield - At The AMS Show in San Diego - January

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.

Virtual interacting particles are realized as small paint droplets encased in hard shells. Particles move under the influence of artificial gravity. When a particle touches the canvas it adheres, its shell disintegrates, and the particle's footprint becomes visible. Particles stream from fountains located slightly above the canvas. Back scattering and dispersion occur when particles from two or more intersecting streams collide. This series of images was made by sequentially turning on and off 120 pairs of streams where some collision potential existed. Each stream contained 400 particles, all particles in a stream were of the same color, and four colors were available. The resulting "fountain paintings" lie on a spectrum somewhere between simulated Pollock style drip paintings and simulated air brush paintings.Gary R. Greenfield, Associate Professor of Mathematics and Computer Science Mathematics & Computer Science Department, University of Richmond, Richmond, VA 23173

Mathartist Statement:

"Many of my computer generated algorithmic art works are based on visualizations from simulations that are inspired by mathematcal models of physical and biological processes. Examples include cell morphogenesis, swarm behavior, diffusion limited aggregation, and interacting particles. By experimenting with the parameters affecting simulation settings and drawing attributes, I try to focus the viewer's attention on the complexity underlying such processes. "

Michael Field - At The AMS Show in San Diego - January

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)."

Robert Fathauer At The AMS Show in San Diego - January

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.

"Angels and Devils" is a digital artwork based on a fractal arrangement of circles within circles. Two half-scale circles are placed within the starting circle and rotated by an angle of π/4 in opposite directions. These steps are then repeated in the smaller circles, etc. The motifs pay homage to one of M.C. Escher's most famous prints, "Circle Limit IV", which also contains angel and devil motifs. Escher's print is based on hyperbolic geometry, which distorts the motifs as they get smaller. All of the tiles in "Angels and Devils" are similar in the Euclidean plane.

Robert Fathauer, Small business owner, puzzle designer, and artist, Tessellations Company

Robert Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity. His artworks are created on a Macintosh computer, primarily using the commercial programs FreeHand and Photoshop. More recently, he has been exploring fractal arrangements of polyhedra

Anne Burns At The AMS Show in San Diego - January

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.

I use mathematics to invent algorithms and recursive subroutines that model structures found in nature such as clouds, trees and flowers. The program that generated this picture was written in Visual Basic.

Anne Burns, Professor of Mathematics, Long Island University.Mathartist statement:

"Just out of high school I entered Pratt Institute to major in art. For a number of years I painted in oils and water colors. I returned to college in my thirties and found that I loved mathematics. I never realized the connections between math and art until I bought my first computer and began writing computer programs. This enabled me to combine my love of art with my love of mathematics. Another of my interests is identifying wildflowers; this led to writing programs trying to imitate the structure of plants and other forms found in nature. I love programming and I am fascinated by the process of recursion and how it can be used to create pictures of astonishing complexity with very little code."

Robert Bosch - Outside Ring - At The AMS Show in San Diego - January

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!"

Douglas McKenna - Thirteenski - At The AMS Show in San Diego - January

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'.

Mathartist statement:

The original Peano and Hilbert Curves represent two out of three techniques for "threading a square". The generalized third technique I recently conceived connects square corners with side centers. "Thirteenski" is an asymmetric, recursive traversal of 13 symmetrically arranged sub-squares that eventually converges as a space-filling curve at different geometric scales according to a Sierpinski Carpet-like pattern. The resulting pattern wonderfully illustrates a struggle between symmetry and asymmetry, arising from the underlying combinatoric constraints governing the solution space.

Slavik Jablan At The AMS Show in San Diego - January

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

Graphical work based on links and interlaced structures.Slavik Jablan, Professor of Mathematics, The Mathematical Institute, Belgrade, Serbia.

Mathartist Statement:

"For many years I used almost all painting techniques (oil, watercolor...), painting in a color-expressionist manner. Later I transferred to computer graphic and mathematical art, trying to preserve the individuality and originality of math-art works, so my math-art works are not computer-generated. In fact, I am using a computer only as a tool for producing artworks."

A Math Art Moment #9

Delineation #9

A mathematical theory seems to come in a flash of intuition before the final product is rigorously constructed.

An artistic theory seems to come much after the artwork that has been constructed in a flash of intuition.
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To see more math art delineations click here

Caring Metered

Here is a double derivative poem (above) based on the previous poem (below). I made substitutions in the poem as stated below and the new poem has elements of iambic trimeter. If you are not familiar with the double derivative poem then please click here.


What I have done is substituted the word caring with:

I am stunned in how her
soft spiritual wisdom
confiscated my care
and delivered it in
summer shining prairie
windswept floral baskets
… of love

Liking is substituted with the following:

Cant help but be seduced
by her chocolate locks
and eyes shining a black
night horizon much less
the subtle curve in her
… waist

Lastly daily rituals of love is substituted with:

I breathe my care’s expanse
Cultivating my love
Tilling the space spreading
Compassion’s grace always
Practicing awareness
as she blankets all my
new experiences
teaching me to focus
… care

개꿈 The American Mathematical Society Has Accepted “DOG DREAM” And “TEMPTATION”

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.

The Double Derivative 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.

In the study of physics, calculus tells us that the first derivative of position is velocity and the second derivative of position is acceleration. Furthermore, it just so happens that this ‘ratio form’ is the same structure used for a similar triangles poem. (You need to be familiar with the similar triangles poem and mathematical differentiation for this to make sense)
Shown below are visual aids to understand the derivative.

The derivative:

Animation showing that when delta x approaches zero the secant line becomes tangent. (click on the image to see the animation)

Each mathematical term “a” “b” “d” or “e”(see similar triangles poem) is a word or phrase. In the derivative poem each mathematical term is a metaphor relating to the derivative paradigm which is an expression relating the first derivative to the second derivative. (the change and the change in the change)

The first word is mapped into the source domain for the first metaphor in our paradigm serving the target metaphor of ‘position’.

Liking IS “position” (I use George Lakoff’s nomenclature for metaphor see "parts of a metaphor" at this link)

The second word is mapped into the source domain for the second metaphor in our paradigm serving the target metaphor of ‘velocity’.

Caring IS “velocity”

Lastly, the third phrase is mapped into the source domain for the third metaphor in our paradigm serving the target metaphor of ‘acceleration’.
Dailey rituals of love IS “acceleration”

Here is the poem shown in the similar triangles form

Here is the poem solved for “Liking”


I find the poem more interesting if solved for the word “Caring”

Visualizing the exact mathematical function of “Caring as a function of time” would be impossible with the small amount of information given however; a single point on that function is present in the poem even if it is nebulous in form. Furthermore, other poems could be written to describe other points along the phantom curve. I find it very interesting to ponder the shape of the "phantom curve"

The challenge in making a successful derivative poem is in the ability of the poet to express the proper metaphors to fit the paradigm of first and second derivatives. That is to say that the poem expresses a change relative to an idea and then a change in the change relative to that same idea. If all of these requirements are met then one may be able to visualize a point and part of the nebulous function graphically in their mind.

Now to take this idea a little further lets think about this. As I had mentioned before; a derivative is just the rate of change of one thing as measured by the change in another. Furthermore you can have a third derivative, a fourth derivative etc. So in effect we could create a mathematical poem that has its structure as a ‘string or sequence’ of derivatives expressed as an expanded similar triangles poem. Now that sounds like an interesting challenge! Who is going to be the first to do it and send it to me?



also see this

Jennifer Karmin And The Monocular Mistress

During the 2nd Annual Chicago Calling Arts Festival, Chicago-based artist, Jennifer Karmin will be reading Beast Poetry that she has assembled from sources outside of Chicago. One of the poems she will be reading is the similar triangles poem “The Monocular Mistress” shown above.

Death by Thierry Brunet

The mathematical poem above is a similar triangles poem titled "Death" Furthermore it was sent to me by the French Poet Thierry Brunet.

Mathpo Eclipsed By Fire

This is the only thing on my mind tonight ... This is a photo shot from space of southern California. I am between the first and second fire from the bottom of the picture. (San Diego)

A Math Art Moment #8

Delineation#8.

Artists have an insouciant tendency to get lost in their imagination.
Mathematicians have an attentive tendency to map their imagination.

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To see more math art delineations click here