Friday, September 05, 2008

What Is The Difference Between Multiplication And Addition In The Context Of Mathematical Poetry?



What is the difference between multiplication and addition in the context of Mathematical Poetry?

I posted the image above a few days ago hopefully to spawn some thought on the matter. The piece is titled: “THE QUESTION OF DE CHIRICO” and it poses the question: “Is the image on the right side of the piece equal to the ideas of Baltimore times De Chirico or is the image equal to the ideas of Baltimore plus De Chirico? My apologies for not including in the original post the fact that the image is of a tower in Baltimore Maryland USA

I did get a couple of responses that proved to me that it is an interesting question and the answer is not as esoteric as one might imagine. I will display and discuss the responses at the bottom of this blog entry.

I gave a lecture on Polyaesthetics and Mathematical Poetry last year at the Salk Institute and within the boundaries of my presentation I had a section that addressed this very issue. From that lecture I am going to borrow a few images to help illuminate this most interesting idea.

Let us think about the equation 3 + 4 = 7 and let us look at a pie chart to help illuminate our quest. When we add 3 and 4 together we can distinctly see the separate pieces within the pie as well as seeing the entire seven pieces. (Shown below)



The Bottom line is that it is easy to remove the 3 slices or the 4 slices from the mix of 7



Now let us think about the equation 3 x 4 = 12

When it comes to multiplication our task gets a little trickier tracking where the numbers 3 and 4 end up (visually). The difficulty is due to them get integrated into each other to produce the number 12. It is though they form an augmentation from which each other play a part in constructing. If we look at a pie chart again we can see that the 12 pieces can be viewed as 4 groups of 3 or we can view it as 3 groups of 4. Both numbers influence the whole in their own way.



Above we have 4 groups of 3 to yield the product of 12
Below we have 3 groups of 4 to yield the product of 12



So what we see is that the multiplier and the multiplicand both augment each other to produce the product.

So how does all of this relate to mathematical poetry? How can we multiply concepts or even images?

Let’s look at the next image titled "Americana Mathematics" and analyze its components.



We see an the popular American icon depicting a NASCAR racing machine added to an 8 ball from the game of pool to yield a strange vehicle that is part race car and part pool table. Here in this example as in our pie chart we can see the two concepts added in such a way that it would be easy to pull them apart and break them out of the whole. The two concepts can be clearly separated in addition however; in multiplication it is again trickier. Let’s look at 8 x 8 = 64 Here again we can refer back to our pie charts showing how the multiplier and multiplicand each augment the other idea to create a whole that possesses much more amplitude than the originating two concepts. Here our product is not a race car but a rocket ship that is obviously involved in some sort of pool game.

Now that we have the tools to understand the mechanics of this artwork we can then spend our time experiencing the interacting metaphors involved to come to our understanding of the signified.

I now want to post two responses to the original question of De Chirico from the blog entry on August 7, 2008.

The first being from the Math Poet TT.O.

The text in Blue is from TT.O. and the text in white is mine

My attempt at a solution to the difference in addition and multiplication in mathematical poetry is as follows:-----

As the difference in nomenclature suggests, the above problem of A+B=C and A*B=C may be a issue of semantics, and in the case of "mathematical poetry" the said equations NOT equal.

Consider One: A + B = C may mean let A abut B i.e. let image A physically touch image B, a kind of concatenation, a bringing together. Which would then go on to suggest that A + B = C1, and B + A = C2 since A + B ≠ B + A, and as their relative positions read from left-to-right would imply, the bringing together would result in an AB versus BA result. Notice that the collapse into a visual representation would suggest a kind of visual multiplication.

I want to add for any mathematicians that are reading this -- when he says A + B ≠ B + A we all realize that this is definitely not true in pure mathematics however, it is debatable within the context of mathematical poetry due to syntax having some bearing on the results. From my perspective the influence of syntax is minimal when performing addition, although, I am willing to listen to all arguments.
I will say that syntax is more important with multiplication. One can see the importance within the recent post I made called a+b+c does not equal c+b+a in this post our attention is brought to a problem with the order in which one experiences a phenomena. The author titled his observation a+b+c does not equal c+b+a however I believe that he should have realized what he was performing was multiplication not addition.

Consider Two: A * B = C may depend on how it is read i.e. a issue of semantics (again) i.e. the number to be multiplied is called the "multiplicand", while the number of multiples is called the "multiplier". Perhaps this is better seen in the following equation A ( B + C ) = D. Here, the multiplier is A while the multiplicand is (B + C). The semantics of the equation would then suggest that ( B + C ) A ≠ A (B + C) in mathematical poetry, since it would depend on which was the multiplicand and which the multiplier, and in what order they were being taken to be (or read) i.e. what was to be infused by what, or what was to be increased by what i.e. a kind of what is being "acted on" (passive) and what active.

Here TT.O. has provided a good argument to warrant attention being paid to the syntax of the equation within the context of mathematical poetry. However, there could be an argument that within the realm of pure math syntax makes no difference and therefore the poet needs to create his/her metaphor to reflect this mathematical truth. In other words make the product reflect an equal amount of the conceptual essence of the multiplier and multiplicand.

From where I stand, in the equation A + B = C, A cannot infuse into B (or visa versa), but can only stand-by it. Multiplication, in the equation A * B = C, on the other hand (to carry on the metaphor) "impregnates" B but not visa versa.

I don't understand your poem properly, because I don't understand the basic essence of De Chirico's work (i.e. a specific painting???) or who or what Baltimore is i.e. a City? An Artist? An attitude?

However, I would suggest that Baltimore × De Chirico is different from De Chirico × Baltimore and different to Baltimore + De Chirico, and De Chirico + Baltimore, and that we should be mindful of it in our equation making.
TT.O.

I want to thank TT.O. for commenting on “The Question of De Chirico” and I must ask forgiveness for not explaining that the image is one of my photographs of a tower that resides on the inner harbor landscape in downtown Baltimore, Maryland USA. I modified the image to be in the style of the twentieth century painter Giorgio De Chirico. (See Google)

Here is a few excerpts from a response from Todd Smith:

Here's my take on it: The painting on the right seems to fit the style of the painter Giorgio de Chirico, so I assume that it is his work. If this is the case, I would vote for the equation: de Chirico (Baltimore) i.e., multiplication.

Multiplication implies a combination (almost a mixing of two elements) and it generates something more than the sum of the two entities being combined.

I would suggest that a snap shot of de Chirico with Baltimore in the background to be represented by the equation de Chirico + Baltimore.

But a work of art produced by de Chirico in which Baltimore is featured would mean multiplication to me. The painting is as much de Chirico as it is Baltimore. The two are inextricably intertwined.

Multiplication seems to be a more complex combination than addition to me. Two spools of thread might be added together when placed in a shopping bag, but they would be multiplied together if they were woven into a shirt.


Here is an image (above) which illustrates Todd's idea of a mathematical weave between two axes. The image is titled "Distance" and it uses the distace equation:
Distance = velocity multiplied by time.

Also, addition seems to be one-dimensional, while multiplication seems to create two dimensions. Addition happens along the number line, while multiplication can be graphed along the x and y axis.

They say you can't add apples and oranges. In addition you have to find a common denominator before you can add. This implies the number line again. As soon as two things are on the same dimension they can be added. For example, de Chirico and Baltimore are both physical things and so they can both be photographed together and said to be "added together" in the picture.

But with multiplication there is less restriction. You don't need a common denominator to multiply two things. The combination creates something new that is not merely more quantity of a common denominator. In pure mathematics 3 x 4 creates a rectangle of area 12. Before there were only lines (one dimension), after multiplication there is area (two dimensions). New space is created. In the example of de Chirico, Baltimore x de Chirico created a new vision of Baltimore colored by de Chirico's own inspiration. No one had seen Baltimore in quite the same way. It is as if a new dimension was opened when these two were combined.

Well, I didn't plan to write this much, but it's fun to think about.

Thanks,
Todd

I also want to thank Todd Smith for his wonderful comments as well. I think the point that we all would like to assert is that this idea of adding and multiplying images (or concepts) is easy to understand. I would love to see more from everyone out there.

Thanks.
Kaz

20 comments:

Todd Smith said...

The pie charts explain it very clearly. great idea!

Kaz Maslanka said...

I find it very encouraging that we all saw it the same way … I would bet that cognitive scientists have already mapped this out from a linguistic point of view however, it is now up to artists and poets to make use of it.

Thanks again for your insight,
Kaz

anandi said...

hi kaz, really a nice post. It gave me an insight into the very basics of mathematics but with a different perception. I would accept that it is pretty difficult for me to understand everything in the context but yes, it is fascinating. I never knew about de Chirico before this.
But I do have something to say here. We always reach a value after multiplying/adding variables(of any kind). Even in the case of images, we reach some value. But do we have any logical way of specifying that value or should we think in terms of logic/abstractness/axioms/tautology? I think that we consider all the scenarios? is it?

Kaz Maslanka said...

These are great questions Anandi however; I don’t know that there are good answers for them. As far as value in the arts is concerned; I think it is meaningless because these days value is determined by art galleries instead of art critics. It is very possible that many artistic geniuses die in obscurity while other lesser artists thrive due to having talent at playing the games necessary to get their work seen and have their work in demand.

The fact that we have a mathematical structure within mathematical poetry makes this work no more able to discern or express value than any other art form. What is important here is not the mathematical answer but the mathematical aesthetic.

anandi said...

Thanks for your answer. By value in art, I meant value found by performing mathematical operations on images/artworks (similar to the pure mathematical fundas). When I think in Algorithmic terms, it becomes much more complex and more intriguing. Sometime back I read about Steganography on Wiki. Here is the link (http://en.wikipedia.org/wiki/Steganography)
Just have a look at the two images.
The transition is really beautiful.
In the similar manner, I asked that can we specify rules for perfoming mathematical operations on images? I know it is not proper but yes, can't we still define some rules?

Anonymous said...

Dear kaz
In the NASCAR example, the initial addition equation only works one way i.e. if you put the 8-ball image first (assuming left to right reading) it takes on a macabre and gothic association as its starting point (metaphor), and presumable the result would be a massive toy(?) car pile-up on top of the billiard table. In both versions of the addition equation the pool table and car image are placed next-to-each other (in this case on top of one another) as opposed to an "infusing" into each other.
The multiplication equation, on the other hand "is" reliant on the addition equation that came before it for its full impact i.e. having established a "premise" (the initial association between billiards and car) it goes on to extend itself into the rocket metaphor (by multiplication) i.e. a rocket is better? bigger? than a car, more balls in pockets etc?? and as such metamorphizes into a rocket, completing an act of infusion. If the 8-ball came first in the multiplication table (or was flipped round into first position as its next and subsequent equation) we would have (again) some sort of metaphor of carnage (an infusion into disintegration?).
In this NASCAR example, I don't think we are dealing with a single equation, and as such cannot take the multiplication equation on its own i.e. have it extract for examination from its context (as a couplet?).
I think you are right that the predictivity of metaphor-making is nigh impossible, but I don't think the NASCAR example necessarily contradicts what I said earlier. That's as best as I can do for now tho if its of any use.
TT.O.

James said...

The pie charts are a very nice visual representation, but I don’t agree with the NASCAR analogy in the image. If I were to say I have, in my garage, “a car AND a pool table”, you would imagine them sitting side-by-side; not combined. The fact that they are combined in any way shows that it’s more than simply addition. A car that is also a pool table is more than the sum of its parts. Not to mention, rockets have nothing to do with NASCAR; it’s over extrapolating. The produce the rocket image, I'd use an equation similar to NASCAR multiplied by pool then cubed.

On another note, is the following mathematical poetry?

http://www.phdcomics.com/comics/archive/phd112107s.gif

Anonymous said...

James,
What it seems to me is that Kaz is working with ‘ideas’ as opposed to ’objects’. If he drew a picture of a car and a pool table and said (one car + one pool table) = (a half of a car and half of a pool table) then you may have an argument. I don’t see him putting coefficients on his ideas. They aren’t objects. Also a car plus a pool table is not the same thing as a car and a pool table.
GB

Kaz Maslanka said...

James,
Your link is extremely appreciated! … yes I would call the result of your link a mathematical poem for two reasons one because it possesses a quality that I call “reflexive didactic” … which means that it makes a statement that asks a question. Furthermore the reason that it possesses a “reflexive didactic is that it is presented in the form of what I call a “mathematical paradigm poem” (see my taxonomy in the sidebar of this blog). If you notice it is extremely close to Newton’s law for the force of gravity. F=G(m0+m1)/r^2 The only thing it is missing is the gravitational constant ‘G’… I need to create a blog entry for this … it is wonderful example of a mathematical paradigm poem.
Thank You!
K

Kaz Maslanka said...

GB,
Thank you for your very perceptive comment!
K

pourmoslemy@yahoo.com said...

hi
i really enjoy
i am alireza pourmoslemi MS in mathematics and a poet
i want to find a mathematician and poet to continue my study
cand you help me to find some one (poet and mathematician) that accept me as a Phd student ?
Reagard

Kaz Maslanka said...

Hi Mr. Pourmoslemi,
I know of no school that possesses a curriculum on mathematical poetry. However, my best advice would be for you to attend the next Bridges conference organized by Reza Sarhangi. The next conference will be in Banff Alberta Canada next summer. There are many professors who attend and are interested in the connection between mathematics and the arts.

Good Luck,
Kaz

anandi said...

Hi Kaz,
I want some help from you. I have been working on a mathematical artwork which is based on high dimensions. I have taken some hint from the 'E8' figure which is supposed to be the pictorial representation of 248 dimensions. But I am unable to find enough of help on visualizing higher dimensions. Also, I had a talk with Carlo during Bridges. He asked me to look up some pape presented some years back in Bridges. Can you help me out with this? In my artwork, I have 'Coins' as the artistic element. It is named Infinitum - COINS. I hope to present it in the next Bridges.

Kaz Maslanka said...

Hi Anandi,
There is an extremely strong group of polytope explorers within the “Bridges School” in which all of the main players hold the late Donald Coxeter in reverence. Coxeter wrote a seminal book titled regular polytopes and it seems that a lot of folks reference it. I think the current King of polytopes on earth is probably George Hart and he has a good PowerPoint on polytopes at this link.

http://www.kazmaslanka.com/Hart-Polytopes-Forms.ppsx

The King of polytopes not on this planet is Jonathan Bowers (Google him) There are many other who have contributed greatly on helping people discover polytopes. Mark Pelletier is definitely one who was a friend of Coxeter as well as Mangus Wenninger. I don’t want to forget Paul Hildebrandt and the Zometool boys in Denver Colorado for they provide us with great tools for building three dimensional projections of hyper-dimensional objects.
I remember being at a great party at Mark Pelletier’s house in Boulder Colorado a couple of years ago where George Hart, Mark Pelletier, Carlo Sequin, Chris Kling, myself and others were brainstorming on ideas surrounding polytopes. I put in my two cents worth by asking someone to design a hyper-dimensional cad system. I am still waiting hehe. Anyway the link I provided will help you out with some ideas how to visualize hyper-dimensional objects. (If there is such a thing as visualizing them)
Cheers,
Kaz

anandi said...

Hi Kaz, Thanks a lot for the information. I guess I missed out discussing the same with George. I'll try to get information on the people and their works as per your suggestion. Thanks once again.
:-)

D. Coys said...

Kaz

You were right - this is a good introductory post. Forgive me if this has already been discussed, but it struck me that the above equations act (grammatically) in the infinitive. So, for example, while de Chirico * Baltimore <> Baltimore * de Chirico in mathematical poetry because the multiplier precedes the multiplicand, the order could be reversed but with the same product if view it in the past tense. Thus, de Chirico * Baltimore could be expressed as Baltimore de Chiricoed.

Kaz Maslanka said...

Hi D.Coys,
This is a fascinating observation that you have made. I love the way everyone brings something to the table to teach us (me in particular). So would the present be a gerund? Baltimore de Chiricoing? Hmmm … To Be Or Not To Be DeChiricoed. Ha!

Thanks so much!!
Kaz

Inner Piece said...

A small example of my work.

Jealous Moon

Towering Quintessence
Indulging Interior Essence
Decadent Self Hypnotic Presence
of my soul

3×4=12

Syllables per line: 6,9,9,3 “3″
Letters per line: 20,24,28,8 “4″
Words: 12


inner_peace2006@yahoo.com

Anonymous said...

It is extremely interesting for me to read that blog. Thanks for it. I like such themes and anything connected to this matter. BTW, try to add some images :).

Kaz Maslanka said...

Thanks Anonymous,
I should have more time for my blog after my March show in NYC.
Thanks for stopping by!
Kaz

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