Tuesday, January 29, 2008

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.

Sunday, January 06, 2008

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++."

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'."

Visit the National Gallery of Writing