The Paradigm Poem

The Paradigm Poem

Unlike a simple mathematical structure as in the “Similar Triangles Poem”, the “Paradigm Poem” is a mathematical poetry technique that borrows its structure from an existing equation from applied mathematics of scientific or cultural significance. The “Paradigm Poem has many sub-categories which are as numerous as there are categories for applied mathematics. Examples that we could consider would be: “Physics Paradigm Poem”, “Chemistry Paradigm Poem”, “Business Accounting Paradigm Poem”, “ Psychophysiological Paradigm Poem” etc.
If we think in terms of metaphor using the cognitive scientific language of George Lakoff then we would classify the language of the variables “inside the equation structure” as the ‘target domain’ and the context or traditional meaning of the equation as the “source domain’
Let me show an example of a “Physics Paradigm Poem” using Newton’s second law. For this example I am going to use an excerpt from my essay on “Polyaesthetics and Mathematical Poetry” Journal of Mathematics and the Arts, Volume 1, Issue 1 March 2007 , pages 35 - 40 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954

“My personal view is that almost all mathematic applications rely on using equations with the intent similar to simile. The variables in the equation are compared explicitly with the result for uses in denotation. For example, in the case of an application of
Newton’s second law F = ma, or Force equals Mass times Acceleration, we are comparing the variables m (Mass) and a (Acceleration) explicitly to F (Force).
I can now make a mathematical poem based on the latter example by expressing the Force of ‘yesterday’s freedom’ as being equal to ‘a lush clover patch’
multiplied by ‘the swelling sweet summer breeze traversing the morning’. We can then put this in the form of a mathematical equation as; Yesterday’s freedom = (a lush clover patch) (the swelling sweet summer breeze traversing the morning). In other words, I set the Force to ‘Yesterday’s freedom’, the mass to ‘a lush clover patch’, and I accelerated the mass by ‘the swelling sweet summer breeze traversing the morning’. All of these
phrases relate back to the original equation from physics F = ma. I want to emphasize that I was very careful when I made my choice for acceleration so that the phrase is evocative of the mathematical description of acceleration as defined by physics. Acceleration is the
change in velocity of an object per unit of elapsed time during that acceleration. Here, the change in velocity is implied by ‘swelling’ and the change in time is implied by ‘traversing’.”

In the above example we are using the words “Yesterday’s freedom; a lush clover patch; the swelling sweet summer breeze traversing the morning” as the variables of the equations which supply us with the target domain and then we are using the equation from physics F = ma, its scientific meaning and historical significance as the source domain.

The bottom line concerning the ‘paradigm poem’ is that we borrow an equation from the past which inherently contains historical significance and serves as an paradigm or mathematical model that seems almost “a vessel” to carry the mathematical poem. The paradigm poem always borrows an existing mathematical structure to serve as a source domain in our metaphor.

Paul Klee Quote

To emphasize only the beautiful seems to me to be like a mathematical system that only concerns itself with positive numbers. — Paul Klee (1879-1940).

Thierry Brunet Mathematical Poetry Links

This page is reserved for all of Thierry Brunet's Mathematical Poems

No Boundaries

Philosophic Cocktails

Rhesus

Japanese Elvis

Toaster

A Math Art Moment #5

Delineation#5

Pure Mathematics has no expression for metaphor however; it does provide us a structure that can be used for it.

To see more math art delineations click here

RHESUS By Thierry Brunet

Here is another orthogonal space poem via France by Thierry Brunet. Titled “RHESUS”

I have noticed something that I find interesting about Thierry's technique when approaching an orthogonal space poem. It seems he acquires three ideas that can be viewed as being near synonymous. He contextualizes them in an orthogonal space which seems to me to work very well together. If one remembers the prismatic structures in my verbogeometry paper then you may also remember that synonyms are viewed existing on a plane. The arrangement of the synonyms on that plane could easily be viewed orthogonally. Furthermore, I think this is why Thierry’s technique works so well.

L'anxieux

Thierry Brunet has kindly translated “The Insecure” into French. This is an expanded similar triangles poem. Flavor three: h = ((e(a+g))/d)-b

Japanese Elvis by Thierry Brunet

Here is a orthogonal space poem by the French poet Thierry Brunet

Foul Of Pray

The mathematical poem above is an example of a expanded similar triangles poem.

Flavor two: a = (g(b+e)/h)-d

The Orthogonal Space Poem

The orthogonal space poem is one of the simplest mathematical structures one can use for mathematical poetry. The structure can be seen in numerous contexts in the discipline of the sciences. Examples in physics would include Newton’s second law “F = ma”, Ohms Law “E = IR”, the kinematical properties of “d = vt”, “p=mv” and E = Fd. Please notice all of the equations are in the form of ‘a’ equals ‘b’ multiplied by ‘c’ or “a = (b)(c)”. This wonderful equation states that the value of one particular concept is equal to the product of two values held by two other concepts. When this equation is depicted in a Cartesian coordinate system you can see that the latter two concepts exist in an orthogonal or perpendicular space.

Before I explain the “orthogonal space poem’s” use in mathematical poetry, let us look at this same mathematical structure in the context of science. Furthermore, before we look at a scientific example let is review a little mathematics. Let us first review the Cartesian coordinate system and its nomenclature.

When we look at the two-dimensional axis of a Cartesian coordinate system, we can see that by randomly picking a point somewhere on the Cartesian plane, then we see there exists is a relationship between this ‘chosen’ point and the point defining the origin of the coordinate system. This relationship is understood by the nomenclature of the coordinate pair (x,y) where x and y are distances along each axis from the origin. Furthermore, if we draw lines from a newly created point, orthogonally (perpendicular) to both the x-axis and the y-axis and taking into consideration the axis system in the background then we will make a rectangle.



The area of a rectangle is product of the lengths of its sides furthermore, in the upcoming example, it is the product of the values for the x and y coordinates of this “chosen” point. I assigned one corner of the rectangle at the origin point to make our example easier to see. Example: Let us pick a point defined by the x-y coordinates of (11,13) and draw lines perpendicular to the axes to illuminate the concept that I just stated. The area of any rectangle is equal to its height multiplied by its base (The product of the lengths of its sides). We have a green rectangle delineated on our axis system. (See figure. 1) The height of our rectangle is 13 units and its base is 11 units. The area of our rectangle is 143 square units … or 13 X 11 = 143 This later example is one of pure mathematics. However if we want to use math as a language then we will have to apply concepts or words to our axis system.

Let us look at a typical physics problem of distance, velocity and time displayed on a two dimensional axis system. Let us assign the y-axis to be levels of velocity in units of miles per hour and the x-axis to be amounts of time in units of hours. Furthermore, let us look at an example using the concept of the “distance an object has traveled is equal to the velocity of that object multiplied by the time the object has traveled.” or “d = vt” In this example let us look at the Cartesian coordinate system as well as its orthogonal construction.

We will use the same pure mathematical example as before but by our contextualizing the axis and assigning the y-axis to represent velocity and the x-axis to represent time, our original point from the last example (11,13) has a new meaning. To reiterate … the point before was in the realm of pure mathematics but now the point represents a moment in time of a speeding object. The object is traveling 13 miles per hour and has been traveling for 11 hours. So to calculate, (d = vt), the distance the object has traveled we must multiply the velocity by the time or 13 miles per hour times 11 hours which equals 143 miles.

In essence, what we have done has been to assign a concept by using words (velocity and time) to our axis system. Moving our attention up or down on the y-axis displays different values of velocity. Moving left and right on, the x-axis displays different amounts of duration or time. Physical experimentation can easily verify the veracity of this equation. In addition, the same experimentation verifies the verbal concepts and their relationship to each other that we have assigned to our axis system. We can see the relationship between the concepts of distance, velocity and time spread out on a two-dimensional plane via our axis system. It is important to note that these concepts occupy orthogonal spaces as well as all equations in the form of a = (b)(c).

The orthogonal space poem possesses the exact same form as our scientific equations however, our intention is poetic as opposed to science.

For our example lets look at the following orthogonal space poem which is titled “Prometheus’s Epistle To Job”

In this poem Prometheus expresses to Job that the suffering of pious people is equal to the arrogance of their God divided by the level of ostentatious generosity imparted by their God.

Lets see how this poem relates to an orthogonal space using a Cartesian coordinate system. We can see how it follows the same structure as the previous physics example.

Another important aspect of creating a orthogonal space poem is to examine all the syntactically different synonymous permutations. Let me reiterate, the structure for an orthogonal poem is a = b c which means that we can solve the equation three different ways. a = b c, b = a/c and c = a/b. When creating an orthogonal space poem you would want to solve and analyze your poem all three different ways to make sure that the poem makes ‘aesthetic sense’ in each different form. Lets look at the latter poem as an example.

The suffering of pious people = The arrogance of their God “divided by” The ostentatious generosity of their God

This poem should also make sense in its two other syntactically different however, synonymous structures.

The arrogance of their God = The suffering of pious people “multiplied by” The ostentatious generosity of their God

And:

The ostentatious generosity of their God = The arrogance of their God "divided by" The suffering of pious people

Toaster By Thierry Brunet

The following is a similar triangles poem sent to me by the French Poet Thierry Brunet.

Physics

Below is another similar triangles poem

Taxonomy Of Experience

Equations from physics are a taxonomy of correlated experiences.

Korean Dog Dream In Spain

I am grateful and honored to have “Dog Dream” accepted into the show Bridges Donostia in San Sebastian, Spain. July 24-27, 2007

The other artists in the bridges show are referenced here

Dog DreamThis is my first Korean artwork

The Korean text translates as:

Dog Dream = Irrationality / Importance

The equation is in the form of an orthogonal space poem

The Expanded Similar Triangles Poem

The similar triangles method for constructing mathematical poems easily enables us to manipulate a logical structure for metaphoric expression. What this pedagogical blog entry is trying to accomplish is the introduction and showing of the expanded version of the similar triangles poem. This poetic structure is logically more complex however, still easy to manipulate.

Before you read further, please be familiar with the similar triangles poem. You may read about it here.

The 'regular' similar triangles poem only utilizes two similar triangles and is defined as an expression of two ratios. The expanded similar triangles poem utilizes ‘n’ number of ratios and is able to create very large (infinite) complex mathematical poems and still have a logical structure to act as a source domain for our metaphoric expression.



I am going to create an expanded similar triangle poem made of three similar triangles to serve as our example.
The following image shows three similar triangles with three different sets of relationships:‘a is to b’ as ‘c is to d’ and ‘e is to f’

The trick in the construction of the poem is to add or subtract the numerators of two of the relationships and also add or subtract the two corresponding denominators of the same two relationships. The next image shows us how we construct our three ratios and associated them differently with subtraction and addition to form six flavors. There are actually 12 flavors possible however; I wish to focus on these six, which I consider the best ones to give us a good understanding of what is occurring in this method.

Each flavor yields a group of six equations after the flavor is solved for each variable. The next six images show the six syntactical arrangements for each flavor. There are at least seventy-two different syntactical arrangements however; I wish to focus on what I feel to be the thirty-six most important ones. The next six images show each of the six groups created by each flavor.

FLAVOR ONE YIELDS:
FLAVOR TWO YIELDS:

FLAVOR THREE YIELDS:

FLAVOR FOUR YIELDS:

FLAVOR FIVE YIELDS:

FLAVOR SIX YIELDS:

Lets now create a poem using the following text.

a = Love Lies
b = The Lonely
d = Sugar
e = The Starving
g = Sexual Conquests
h = The Insecure

The structure follows as thus: ‘Love Lies’ is to ‘The Lonely’ as ‘Sugar’ is to ‘The Starving’ as ‘Sexual Conquests’ are to ‘The Insecure’

We then substitute the variables with the poetic phrases to yield thirty-six poems in six groups relating back to the flavors shown above.

I find it interesting to savor each syntactical permutation to get a ‘feel’ for each poetic expression.

GROUP ONE CONSTRUCTED FROM FLAVOR ONE:

GROUP TWO CONSTRUCTED FROM FLAVOR TWO:

GROUP THREE CONSTRUCTED FROM FLAVOR THREE:

GROUP FOUR CONSTRUCTED FROM FLAVOR FOUR:

GROUP FIVE CONSTRUCTED FROM FLAVOR FIVE:

GROUP SIX CONSTRUCTED FROM FLAVOR SIX:


Many of the links below are examples of the expanded similar triangles poem.

Love And Ego

Today, I would like to express my gratitude to JoAnne Growney and Sarah Glaz for accepting one of my works (above) into their collection of mathematical love poems that they are editing for their upcoming book.

Another From Gregory Vincent St. Thomasino

The NYC philosopher/poet Gregory Vincent St. Thomasino would like to see a math poem of his cartoon shown above. The result is a similar triangles poem shown below.

Two Emblem Poems From Thierry Brunet

I just received two images from Thierry Brunet and would like to discuss them. Both of these images I would consider emblem poems with a mathematical emphasis. Thierry has made some interesting connections in these poems. The first one I wish to point out is in his piece titled "HeXaedron". A Hexahedron is commonly known as a cube as well as being a regular polyhedral. Thierry asks “how many platonic solids do we need to dream TOMORROW? The answer depends on many things but my first thought is the question of what dimensions are we limited to? There are 5 platonic solids in the third dimension 6 in the fourth dimension and 3 in every dimension above the fifth. So if we include all dimensions we have an infinite choice. However, he is pointing at time by mentioning the PRESENT as well as TOMORROW so this also makes me entertain the idea of relativity.

Another thing I find interesting is the statement “Don’t forget the empirical BEAUTY of experience. Thierry now connects by capital letters time with beauty however, more important to me is the idea of an aesthetic of beauty tied directly to logic through an empirical process. Some artists would argue that beauty cannot be expressed with a logical statement. I feel they should be nauseous from eating only the icing and none of the cake.

The second piece is titled “mouTH THeory” I like this piece as well because I find my reality can be seen in terms of topology and beliefs are the forces that distort that topology. And here we have a lady whose scream swallows it.

Two Mathematical Poems For Gregory Vincent St. Thomasino

“Profundity” is the title of the first mathematical poem and the second “Logoclasody”.

The drawing above is by Gregory Vincent St. Thomasino

If discourse is a river then what is a lake?
Is not the Philosopher a dam?

Logoclasody


The two mathpoems above are similar triangles poems
Also related is the Avrin proposition

The Account Of Oscar

The following is a mathematical adaptation and edition of an Oscar Wilde quote.

The later poem is in the form of an orthogonal space poem

Fall Well

I forgot to post this two weeks ago.

also see Orthogonal Space Poem