Terminology for Mathematical Poetry and Related Endeavors*

‘ Visual Art Aesthetic’ is the aesthetic that concerns itself primarily with the beauty or horror expressed in direct sensory experience, whereas the ‘ Mathematics Aesthetic’ concerns itself with the beauty in the structures of logic and thinking.

‘Mathematic Aesthetic’ is the aesthetic that concerns itself with the beauty in the structures of logic and thinking, whereas the ‘Artistic aesthetic’ is concerned primarily with the beauty expressed in direct sensory experience.

'Mathematical Conceptual Art’ This form of Art focuses on the Math aesthetic and re-contextualizes it as Art personally I feel conceptual art is not art however, it is aesthetic but. That does not necessarily mean that Math is art. The main difference between ‘Mathematical Conceptual Art’ and ‘Visual Mathematics’ is that in the former the artist presents their the work as Math, where as in the later they display the mathematical object as Art. In both types, they display the object in the context of an Artistic space. A good example of “Mathematical Conceptual Art’ would be the work in the late 1960’s of Benar Venet in which he would study Math and Physics and present what he had learned purely for the aesthetic of the topic involved. There are many works of Sol Lewitt that could be considered “Mathematical Conceptual Art’ as well. A contemporary Artist who I would consider a ‘Mathematical Conceptual Artist’ is the British artist Justin Mullins although he does some work that could be considered as ‘Mathematical Visual Poetry’. The main difference between Mathematical Conceptual Art and Mathematical Poetry is that the Conceptual Art movement as a whole was not concerned with the intention of metaphor in any form and Mathematical Poetry relies mostly on metaphor to make its connection to poetry in general

‘Mathematic Constructivism’ Is one of the most popular forms of Mathematically related Art. It is a term I will use to sum up a conceptual thread that started with the Russian constructivists and ended up in the modern movement of visual mathematics. The former started in the political and social upheaval of the 1920’s with the emergence of Artists such as Naum Gabo, Vladimir Tatlin and ended up in the latter movement with mathematicians such as Donald Coxeter who felt their mathematical work is a form of Art. Donald Coxeter imparted much mathematical assistance to M C Escher.
The conceptual idea of Cubism pushed visual Art into a process of abstraction whereby the artist removes unnecessary visual layers of an object in order to point to a metaphysical idea of the object. Art Constructivism moved to push the methodology of abstract Art more and more abstract to the point of the object being something not found in nature -- a “construction”. If we push this idea further we end up in realm of ‘Visual Mathematics’ where the object of Art is pure logic, a reflection of the logical structures of language in our mind. Today ‘Mathematical Constructivist’ work has moved more toward ‘Visual Mathematics’ and can be seen in the work of Max Bill, Helaman Ferguson, Rinus Roelofs, Robert Fathauer, Brent Collins and many others.

‘Mathematical Poetry’ – Mathematical Poetry is a umbrella term that covers any poetic expression involving Mathematics. An initial list of categories is as follows: Equational Poetry, Mathematical Visual Poetry, Visual Mathematical Poetry, Mathematics Poetry and Number Poetry


‘Equational Poetry’ – This is literally performing mathematical operations on concepts whether they are words or images. A good example would be my page at the following link: Mathematical Poetry

'Visual Mathematical Poetry' -- This is a mathematical poem where the elements in that poem are visual objects. The difference between mathematical poetry and visual mathematical poetry is that the former uses words and the later uses images. Visual mathematical poetry is more similar to mathematical poetry than it is to mathematical visual poetry. However, one could create a poem that has aspects of all three of these types. For an example check out "Americana Mathematics"
‘Mathematical Visual Poetry’ – This is more difficult to define because of the vast areas and the many competing definitions of visual poetry. However, I consider mathematical operations on text as well as mathematical textual information composed for aesthetic purposes to be ‘Mathematical Visual Poetry’ Also words, text or textual elements mixed with mathematical symbols or formulae that are not performing mathematical operations on the word meanings. Although Karl Kempton has worked in many categories, I feel the following is a good example of ‘Mathematical visual poetry’: Another good example is Marko Niemi’s fractal poem described in the following link: Midwinter nights dream Scott Helmes was one of the first visual poets that moved into mathematical motifs. Bob Gruman has probably been the most prolific in this catagory.

‘Mathematics Poetry’ -- This poetry is what I would call traditional language poetry about or inspired by or uses mathematical imagery. I also would consider this catagory to include language poetry that has an interaction of numbers with words. There are numerous examples all over the web but the most popular from google's perspective seems to be Marion Cohen: other sources would be JoAnne Growney and Katherine Stange:

'Polyaesthetics' is a word used in relation to aesthetic works which incorporate many diverse aesthetics. This is not limited to but includes the aesthetics of Mathematics, Art, Music, Science, Religion etc.

'Visual Mathematics' Is one of the most popular forms of mathematically related art. It sometimes has been called “Concrete Art” This is a form of Art that focuses on the Math aesthetic and re-contextualizes it as Art. The main difference between ‘Mathematical Conceptual Art’ and ‘Visual Mathematics’ is that in the former the artist presents his/her personal/emotional relationship with the aesthetic of Mathematics where as in the later the display is less personal and more cerebral. In both types the object of that presentation is displayed as a form of Art. The hero of visual mathematics is M C Escher whose work is so strong anything that resembles it looks cliché. Fortunately there are other arenas in Visual mathematics. A good example of contemporary Visual Mathematics is found in the work of George Hart, Paul Gailiunas, Carlo Sequin, Robert Krawczyk, Michael Sussna and many others. This type of work is primary interested in visualizing mathematic structures. These structures could be anything from computer algorithms not limited to fractal Art or polytopes to hand drawings, plastic sculpture or origami.


*Disclaimer: These are the views of Kaz Maslanka and are a rough attempt at trying to put mathematical poetry in context with most of the mathematical influences in visual Art of the last 100 years

Polyaesthetic example

Boeing 747 Landing gear in the process of being manufactured

Walt Gillette of Everett, the lead engineer on the 787 program and a man Business Week recently called a "plane genius," announced his retirement from the Boeing Co. on Wednesday.

You may ask what does this event have to do with mathematical poetry? … Not much however, the following quote of his is a perfect example of polyaesthetics whereby aesthetics comes in many forms even in engineering and technology.

Walt Gillette says,
"One of the most incredible experiences is to go out ... and stand in the middle of full landing gear of a Boeing 747," he said. "To stand there, right there under that big, fat, huge machine, and you think this thing goes 625 miles an hour and a little-bitty human brain ... tells it exactly what to do and where to go, and it follows just like a docile family pet."

There are many forms of beauty in concepts and some say that if the concept is beautiful then it must be considered as a form of Art. I personally don’t find beauty and Art to be synonymous but I will admit that it is a nebulous concept and very difficult to nail down. Art for some reason seems to be a ‘catch all’ for anything anyone wants to call Art. I wish there was a better ‘catch all’ term other than Art. If you can think of one then I invite you to comment.

More Math Humor

click on it to read

It seems I have found myself in a sexist corner again … I was feeling a bit guilty for posting "girls = evil" so I had to post the cartoon above to give equal voice to our lady friends.

smile! its not that bad :)

Fabulous Finnish Fractal Flooring

Nokturno.org has a new fractal poem titled Palasista!
by Saara Lehto. The piece is written in Finnish and is in the form of a Sierpinski Carpet. The words are all anagrams of each other as well as the title.

Thanks to Marko Niemi for bringing this to us!

Rescuing Metaphor

Gregory Vincent St. Thomasino has sent me a math poem that I would like to share with you.

to + to = too

This is a nice little poem that has three Important elements that strike me right off the bat:
1.) it plays on the words to and too … in other words we have too many to's
2.) is the clever addition of o's … o + o = oo
3.) the most important element of any mathematical poem is the equal sign.**

I really like the feeling in this piece … metaphor is so hard to describe without using another metaphor and if we do then we miss the point.

By the way … eratio seven is now available! Check it out!


** It is the equal sign that creates the metaphor in mathematical poetry. It is the fact that a poem of the latter form says a + b = c and we know that a + b IS NOT c

There exists a mathematical form that is the logical crux of all metaphors in all poetry mathematical or otherwise:
A = B
Given: We know for a fact that ‘A’ does not equal ‘B’

There is also one more key ingredient for metaphor to exist. That ingredient is connotative intention. In other words, the physics equation d = vt is not metaphorical because the intention of the equation is denotative.

‘A’ is similar to ‘B’ is not a metaphor
‘A’ is proportional to ‘B’ is not a metaphor
‘A’ looks like ‘B’ is not a metaphor
‘A’ is compared to ‘B’ is not a metaphor
Simile is not metaphor that is why they are different words

Marius de Zayas

Marius de Zayas and Francis Picabia, FEMME!

Karl Kempton shares some links with us:

Marius de Zayas, Agnes Meyer. Eye Contact: Modern American Portrait Drawings from the National Gallery, Nov 2005

Picabia. Between Music and the Machine: Francis Picabia and the End of Abstraction, fig 28 mathematical formulas. Nov 2005.



I really am not able to tell whether Zayas was trying to express something mathematically or not. I have seen an abundance of artists decorating their work with equations in order to express a math feeling or maybe add a cryptic quality to their works and that may be what Zayas is trying to do as well …These days the latter idea is a bit trivial however, the case with Zayas is probably one of the first times that equations are inscribed within visual work … (I wouldn’t put it past Hieronymus Bosch … but I don’t think he did it)

Thanks Karl for passing this on!

Three Different Kinds Of Mathematical Poetry

There are many who claim the endeavor of ‘mathematical poetry’ and we can see as many forms as there are claims. I personally see three different forms however, the second category I describe may be broke into many other forms. I will list them in order of popularity at this time in 2006. I also want to mention that I am not one to engage in taxonomy for taxonomy’s sake but delineate things only if I feel there is confusion. There also can be works that incorporate all three categories.

1. ‘Mathematics poetry’ -- This poetry is traditional language poetry about or inspired by mathematics. I also would consider poetry that plays with numbers and words in this category. There are numerous examples all over the web but the most popular from my perspective is Marion Cohen:
Another source would be Katherine Stange:
2. ‘Mathematical Visual Poetry’ – This is more difficult to define because of the vast areas and the many competing definitions of visual poetry. However, I consider mathematical operations on text as well as mathematical textual information composed for aesthetic purposes to be ‘Mathematical Visual Poetry’ Also words, text or textual elements mixed with mathematical symbols or formulae that are not performing mathematical operations on the words. Although Karl Kempton has worked in all three categories, I feel the following is a good example of ‘Mathematical visual poetry’: Another good example is Marko Niemi’s fractal poem described in the following link: Midwinter nights dream
3. ‘Mathematical Poetry’ – This is literally performing mathematical operations on concepts whether they are words or images. A good example would be my page at the following link: Mathematical Poetry
Also the following link has an example of Scott Helmes: And Bob Grumman at the following link is a good example of a mixture of ‘Mathematical Poetry and Mathematical Visual Poetry’:

The mathematical poetry blog links page for Marko Niemi

Marko Niemi's Mathematical Visual Poetry:

Midwinter nights dream
http://www.nokturno.org/marko/haynaku/midwinter.html


The link above is my personal favorite of his and you can read my little essay about it click here

Fibonacci visual poem
http://www.nokturno.org/marko/nurotus/fibonacci/

The following is a Finnish visual poem based on the Sierpinski triangle

Kolmioita
http://www.nokturno.org/marko/nurotus/kolmioita/kolmioita.gif

Brownian motion series
http://www.nokturno.org/marko/brown/

"Eloquent Fern"
http://www.nokturno.org/marko/fern.html

The following statement is Marko's preface for the "Eloquent Fern"

The "text" in this piece is the following paragraph from "Rhetoric" by Aristotle, "translated" into four-letter DNA code: (It is clear, then, that rhetoric is not bound up with a single definite class of subjects, but is as universal as dialectic; it is clear, also, that it is useful. It is clear, further, that its function is not simply to succeed in persuading, but rather to discover the means of coming as near such success as the circumstances of each particular case allow. In this it resembles all other arts.)

Narcissus The series consists of six pieces, and in each piece, locations of individual letters are defined by the same polar coordinate function:RADIUS = A * cos(0.5 * ANGLE), where A is constant, and ANGLE goes from 0 to 720.The direction of each letter is independent from its location, but it's a function of ANGLE, too. This direction function is different in each piece, and it's the reason why pieces differ from each other, even though they all use the same location function.

Animated mathematical/scientific visual poems:
Critical Mass
Divine Intervention
Eye of the beholder
Party NRJ

New Mexico State University has acquired "Golden Fear"

GOLDEN FEAR
by KAZ MASLANKA

I am pleased and grateful to announce that the “University Art Gallery” at New Mexico State University has acquired a limited Edition Lambda Duratran of “Golden Fear” for their permanent collection.

Blends & Bridges: A Survey of International Contemporary Visual Poetry & Related Art

I feel grateful to be a part of the vispo show in Cleveland Ohio this month the show is full of a lot of very talented artists. Check out the link below.

VISPO!

Blends & Bridges: A Survey of International Contemporary Visual Poetry & Related Art

Gallery 324 at the Galleria at Erieview is hosting a show of contemporary international visual poetry (Vispo for short).

The show is curated by Cleveland visual poet/publisher John Byrum, Florida visual poet/publisher Bob Grumman, and Cleveland artist Wendy Collin Sorin.

Gallery 324
1301 East Ninth Street
Cleveland, OH 44114

click here for show index

Fractal Vispo by Marko Niemi

Fractal Visual Poetry

I want to get back to the fractal poem that was created by Marko Niemi and the post of mine from March 25th Midwinter night's dream (click here)

To understand some of the ideas incorporated into Marko’s piece we need to understand a little about this idea of fractals. Fractals have many facets of interest however; I want to focus on two:
1.) Self-similarity
2.) Fractional dimension

You may ask yourself, “Have I experienced a fractal in nature?” The answer is yes! I believe the easiest way to recognize that you have experienced one is by asking the question: “Have I ever noticed something that looks the same viewed at different scales: In another words the look very much the same whether I am close to it or far away.

Here is a short list of fractals patterns found in nature

*the branching of tracheal tubes
*the leaves in trees
*the veins in a hand
*galaxies
*water swirling and twisting out of a tap
*a puffy cumulus cloud
*tiny oxygen molecule, or the DNA molecule
*the stock market
*sea shells

I believe out of this list the easiest one to comprehend is a cumulus cloud. These kinds of clouds look the same from 60 feet away or 6 miles away. From either distance clouds basically have the same shape.
The picture below looks very much like the runoff from my neighbor’s water sprinkler before he got his Arizona lawn seeded properly.






But the fact is that I shot this photo of the Grand Canyon from one of my airplane flights between Denver and San Diego. Here we can see another self-similar example in the fact that water erosion looks pretty much the same from different scales.

Mathematical Fractals
We have all probably seen some ‘self similar’ mathematically created fractal images on the Internet or at a bookstore. However, to get a good understanding of it I wish to focus on the image below. The figure below is called a Sierpinski triangle named after the 20th century polish mathematician.
I don’t think that I could explain self-similarity any better than the following link so I stole the text “shown in red” from that site and posted it below:

http://math.bu.edu/DYSYS/chaos-game/node5.html

Self-similarity
One of the basic properties of fractal images is the notion of self-similarity. This idea is easy to explain using the Sierpinski triangle. Note that S may be decomposed into 3 congruent figures, each of which is exactly 1/2 the size of S! See Figure 7. That is to say, if we magnify any of the 3 pieces of S shown in Figure 7 by a factor of 2, we obtain an exact replica of S. That is, S consists of 3 self-similar copies of itself, each with magnification factor 2.

Figure 7: Magnifying the Sierpinski triangle

We can look deeper into S and see further copies of S. For the Sierpinski triangle also consists of 9 self-similar copies of itself, each with magnification factor 4. Or we can chop S into 27 self-similar pieces, each with magnification factor 8. In general, we may divide S into 3^n self-similar pieces, each of which is congruent, and each of which may be maginified by a factor of 2^n to yield the entire figure. This type of self-similarity at all scales is a hallmark of the images known as fractals.

Fractional Dimension

The word fractal actually comes from the notion of an object having a fractional dimension. In other words we can have an object existing in a dimension between the first and second dimension or the second and third dimension etc. A point has dimension 0, a line has dimension 1, and a plane has dimension 2. So how is it we can have something that has a dimension of 1.6 or 2.34? Again I found a wonderful link that explains it good enough that even I (an artist) can understand it.

http://math.bu.edu/DYSYS/chaos-game/node6.html

Understanding Marko’s piece

Midwinter night's dream (click here)

Marko’s piece has 6 triangular shaped words stacked together to form a larger triangle each of the words in this structure have white letters except one that shines yellow-orange. When you first click on this piece six words appear and if you notice the yellow-orange letters also spell a word. When I first started clicking on the words I could not make out any pattern. However, once I started writing them down I noticed that the triangular word that I clicked upon was soon duplicated throughout the entire structure of the large triangle with the yellow-orange letters. So in effect the smaller triangle conceptually contained the larger structure within its smaller structure. Thus being self-similar! A fractal made of words!

If you don’t get this at first then just think of each letter in the smaller triangle as being the possibility of also being the yellow-orange letter in a larger triangle and when you click on the small triangle it pops into a large triangle made of 6 more words. The larger triangle is contained in the smaller one you just can not see it because the resolution of the screen gets in the way … there are countless levels you can go through with this poem (its not really countless there is a finite number) but the bottom being it is a very beautiful structure.

I asked Marko about his choices for the words in the piece and the following in blue is his response:

In the Midwinter piece, there's the "source text," consisting of all the six-letter words from "Midsummer Night's Dream" by Shakespeare, in their original order. When you click for instance the word "hermia," the program gathers all the words from the source text which include the first letter of the clicked word, in this case "h", the first letter of "hermia." Then the program chooses one of those words at random and, after that, the program chooses the next word after previously chosen word in the source, which includes the second letter of the clicked word, in this case "e", and so on, until six words are chosen. Those six words are then displayed on screen, and the letters "h", "e", "r", "m", "i", and "a" are higlighted. This process takes place again and again, after each clicking.

So my next thought what is the fractional dimension for Marko’s piece? Here is the link again to help you:

http://math.bu.edu/DYSYS/chaos-game/node6.html

Here is our equation:

So Marko has 6 self similar pieces with a magnification factor of three. Therefore midwinter nights dream has a fractal/fractional dimension of approximately 1.63

Now isn’t that cool!

A numerically significant point in time?

My mother sent this to me:

It is currious to note that on Wednesday of this week, at two minutes and three seconds after 1:00in the morning, the time and date will be

01:02:03 04/05/06.

This won't ever happen again

You may now return to your normal stuff

Mathematical Poetry from Finland

I received an interesting email from Finland yesterday. A mathematical poet there named Marko Niemi sent me some links to some mathematical vispo. One link in particular, directed me to one of his interactive math poems. I have to declare this is my favorite mathematical vispo piece to this date. It is a fractal poem and due to the fact, I have never seen one before, I find it quite exciting. Before I go into the details of how it works, I am going to leave it with you to figure out. I had a lot of fun poking around to see how it works and when the fractal element hit me, I was quite surprised. Marko’s hint was to click on one of the words to “zoom in”

Midwinter night's dream (click here)

I will be discussing the details in a future blog entry.

Scott Glassman on Verbogeometry

I wanted to share something that Scott Glassman wrote to me, which I find elegantly written. However, I don't want to appear pretentious and hope no one feels that my blog has advanced mathematics … all of what you find here, at this time, comes from the first and second semesters of algebra based physics. I am primarily an artist not a mathematician.

If there's anything verbogeometric / mathematical poetry can accomplish, at the very least, it can begin to help people understand that human qualities, spiritual qualities, operate in many dimensions-- they are the true conceptual equivalents (& metaphorical equivalents) of physical wonders, like say neutrinos. I see how the incorporation of these concepts with advanced mathematics begins to illustrate a depth and a myriad of functional "flourishes" (as say, experiences with music) that is closer to the truth. It begins to get at that "one hidden stuff" of which we are all made.

Scott Glassman
http://scottglassman.blogspot.com/


Thank you Scott for your kind words,
Kaz

5 out of 4 people have a problem with fractions

I was attending an opening for the Imperial Beach Art Guild show held March 4 in Imperial Beach, California and happened to run across an artist there who was wearing a t-shirt that said, “5 out of 4 people have a problem with fractions.

Lets look at it in equation form:



I find it to be a nice math poem with a very curious metaphoric image in it. Five fourths of the population is a party balloon slightly twisting your mind before it blows up in your face. This twist of denotation turning into connotation sets up great tension, then exploding into a joke, as you walk away … some of us savor this kind of nonsense.

Chauvinistic Mathpo

I have been hesitant to post this on my blog because I really don’t appreciate sexist humor. However, the text above has been sent to me numerous times from all types of people including my sister! However, I must admit that I do find this interesting because it is an example of mathematical poetry that shows there are no boundaries for this aesthetic form. It also has elements of mathematical vispo. (Namely the square root of evil equating to the root of all evil)

It is also curious to note a bit of a disconnect in how the author of this poem mathematically defines women as purely time and money. But verbally states that they require time and money.

I feel there can be philosophical discourse revolving around this poem but I wouldn’t touch it with a ten foot pole.

Defining Bloglusions of Grandeur

Delusions are such a wonderful thing.

Verbogeometry -- installment VIII --- Distance Formula and Verbogeometry

2.7. Distance Formula and Verbogeometry. As we have seen, to calculate the distance between two points, we need to describe our points by its coordinates using the nomenclature of the coordinate pair. Let me reiterate, describing a point in verbogeometry is no different from numerical coordinates except we use words. Lets look again at the example in figure 15 where we used the midpoint formula to find the exact point between the points: P1(love,praise) and P2(hate,punishment) but instead of putting them in the midpoint formula lets put them in the distance formula. (See figure 32)

Figure 32

Here we have an expression for the distance between the points P1(love,praise) and P2(hate,punishment) in two dimensions. But we can also use verbogeometry in any number of dimensions including hyper-dimensions. But before we look at hyper dimensional verbogeometry lets look at another example which we will express in the third dimension. The following example uses a three dimensional Cartesian coordinates system with 3 simple antonym word-axes. (See figure 33) The first axis is noble / ignoble the second axis is just / unjust and the third axis is loyal / disloyal.


Figure 33

Now lets look at the expression for the distance between the points P1(noble,just,loyal) and P2(ignoble,unjust,disloyal) see figure 34


Figure 34

Notice the green line in figure 33 is the visual representation for the mathematical expression above. However, it would be much easier to visualize if we were able to rotate the axis. Figure 33 is an isometric view, which I chose to use because it is best for viewing the axis but unfortunately at the expense of viewing the spatial orientation of the green line.Now let us look at verbogeometry in a hyper-dimension. Let us look at the distance formula used in seven dimensions:Figure 35 shows the mathematical poem 1+1+1+1+1+1+1+1 =1 This is a metaphorical piece that creates a metaphoric path from the concept of confusion, to where seven deities meet. The piece uses the analytic geometry distance formula in a seven dimensional space where each dimension is a gradation from confusion to a point where a deity exists.

Figure 35.

Here is a detail:Figure 35



Lets look at the coordinate pairs for these two points P1(confusion, confusion, confusion, confusion, confusion, confusion, confusion) and P2(Allah,Buddha,Jesus,Spider woman,Vishnu,Yahweh,Zeus)In conclusion what I have shown is scratching the surface of the possibilities of verbogeometry. Verbogeometry can be taken in vast directions that I have not covered or will be able to cover. I hope, in the future, more people join in to explore the possibilities of verbogeometry.

Kaz Maslanka
San Diego, California
February 3, 2006

Verbogeometry -- Installment VII --- Pythagorean Theorem and The Distance Formula

2.6. The Distance Formula and Pythagorean Theorem: The distance formula uses the Pythagorean Theorem to calculate distances on the Cartesian coordinate system. The Pythagorean Theorem says that the hypotenuse of a right triangle is equal to the square of it sides. (see figure 26)


Figure. 26

solve for c and we get the following see figure 27


Figure. 27


Let us plot two points on a two-dimensional axis system P1(-9,10) and P2(4,3) and If we draw a lines between the points and lines parallel to the axes we can obtain a right triangle. (see figure 28)


Figure. 28

To solve the length of the hypotenuse we first find the difference between the x values and the y values to create the sides of the triangle then we plug the values into the Pythagorean Theorem.
x value is 9 - 4 = 5 and the y value is 10 - 3 = 7
Now we plug it to the equation and we get the expression in figure 29


Figure 29

The distance or length of the hypotenuse would be the square root of 74 or approximately 8.602
The distance formula is thus:
Given two points (x1,y1) and (x2,y2) (see figure 30)


Figure 30

Now what we have been looking at up till now has been in two-dimensions but we can also express a distance in many dimensions. Let us look at two points in three dimensions.
P1(x1,y1,z1) P2(x2,y2,z2)
If we were to put these two points into the distance formula it would look like what is shown in figure 31


Figure 31

It is evident that we can use the distance formula in uncountable dimensions all we have to is to add another dimensional term to the formula for every dimension we wish to express.

Verbogeometry -- Installment VI --- Prismatic Structures

2.5. Prismatic Structures in Verbogeometry: Notice that 'barren and infertile' is a complex antonym and 'fertile and infertile' is a simple antonym as previously defined. This is interesting because we can see that there exists in verbogeometry a geometric construction where a line expressed as a simple antonym is normal (90 degrees) to a plane containing all of the complex antonyms related to the line which is expressing the simple antonym. To illustrate this idea lets look again at the relationship between the simple antonyms 'fertile' and 'infertile' and the synonyms 'barren', 'fruitless', 'unproductive', 'sterile', 'impotent' which reside on the plane that is normal (90 degrees) to the line created by the simple antonyms. Furthermore you can draw lines from all of the synonyms back to the complex antonym 'fertile'. (See figure 19)


Figure 19.

This idea also lends itself to prismatic structures where we have a group of parallel simple antonyms whose endpoints construct polygonal faces on two parallel synonym-planes. (See figure 19) Example: Let us define one synonym plane containing the words 'pleased', 'content', 'affected', 'satisfied', 'enchanted' and 'sympathetic'. The other plane contains the following simple antonyms for the previous group of synonyms: 'displeased', 'discontent', 'disaffected', 'dissatisfied', 'disenchanted' and 'unsympathetic'. Due to the synonyms of one plane have corresponding simple antonyms which create lines 90 degrees from the synonym-plane then the simple antonyms are synonyms of each other and reside on their own individual synonym plane and because the lines are 90 degrees to each other the planes must be parallel. The former verbiage is a lot easier to understand visually (See figure 20)


Figure 20.

On a side note: Everything that we have been talking about up to this point has been viewed in Euclidian space. Not much time has been spent exploring verbogeometry outside of Euclidian space. However, I see that there can be a lot more thought evolving models of verbogeometry in other spaces. For example, Instead of viewing antonyms and all the varying meanings spread out across a number-line, we may think of the antonyms as magnetic dipoles. Furthermore, all the varying meaning between them is analogous to a magnetic flux. Yet again, another view may be looking at the antonyms residing at the endpoints of a major axis of a three dimensional ellipsoid and the varying meanings reside on the surface of the ellipsoid. (See figure 21)


Figure 21.

Figure 22 shows an ellipsoidal flux-like paradigm for the previous prismatic structure. (See figure 20)


Figure 22.

Figure23 shows an unlabeled side view


Figure 23.

It may be easier to see the ellipse structure if it was wireframe (See figure 24)


Figure 24.

Figure 25 shows 5 parallel word axes in a wireframe structure


Figure 25.