Art, Science + Technology
DMA9 Fall 2007, Section BArchive for October 15, 2007
Week 2 – Math and art.
How does perspective in art utilize mathematics?
Perspective in a piece of artwork can be created by using rules of mathematics that are evident in nature. Before artists realized the affect that mathematics had on perception, paintings did not give a real illusion of depth and distance. The size of an object in a painting usually had to do with its importance to religion or its importance to the theme of the artwork. It was not until 1000, when the mathematician and philosopher Alhazen explained that perception is viewed by light projecting conically through the eye. It was in this way, that artists began to realize the importance that mathematics had in creating perspective in images. It wasn’t until the 1400’s however when Fillipo Brunelleschi painted outlines of buildings onto a mirror that artists were utilize geometric concepts in the art. In this way, Brunellleschi discovered that drawing perspective created a horizon line that all objects in the scene shrink towards. Now an artist is able to use mathematical functions to precisely recreate a scene, as it would appear if viewed in actuality.
The basic principles are also used as tool to create objects that appear like real life items, but are actually impossible to exist in reality. These “impossible objects” manipulate the human sense of perception by creating three-dimensional objects on a two-dimensional surface. In this way, the geometric models that are used to recreate perspective along with a human’s way of observing to create objects that could not be recreated in real life. Popular examples of this include the Penrose triangle and the devil’s pitchfork.
My thoughts about math rock:
I found the precise, yet spontaneous sound of math rock to be very interesting to listen to. It is definitely not the kind of music I could just play in the background while concentrating on something else because it really demands your attention. I can understand why this genre has been related to math because while listening to it, I got a sense that the artist paid much attention to the structure and timing of the song. The sound is unique and was unlike much of the alternative rock that I have heard. I am always open to new music and can appreciate a wide variety of music, so it was great to be able to experience a new type of music that incorporates the unique musicianship that I value in all types of music.
Links:
Impossible Objects
Relative Height
Zograscope
Perspective
Moebius Strip
Math in Art
When I left class Wednesday, I did not know what to think of what I had just experienced. Yoshida-san’s passionate performance moved and greatly perplexed me; try as I will, I could not follow his music. In elementary school I sang on a choir and have learned classical piano for six years. As a result, I enjoy music with the classical elements of melody, rhythm and harmony none of which I could distinguish in Mr. Yoshida’s playing. I want to understand, to feel, his music and I am frustrated that I cannot. That is not to say I do not respect him, on the contrary, I deeply respect his talent and the awesome complexity and intricacy of his art.
Professor Vesna’s lecture on math and science conjured many images in my mind of mathematical concepts that can be seen in nature and how these connections can be used in art.
This image is from a screen shot of the computational program MATLAB which can generate diagrams and images from functions as well as analyze images such as tree rings and compute their formulas and find patterns. Her segment about perspective reminded me of an admissions essay prompt for Northwestern University that referenced Hans Holbein the Younger’s The Ambassadors, a painting of two men standing to either side of a cabinet of their most prized possessions with a oddly skewed skull in the foreground.
When viewed directly the skull is unrecognizable, however when viewed at a downwards angle from the right one can see an accurate image of a skull. There are three theories as to why Holbein placed so much emphasis on the skull. One reason is that he wanted to depict the heavens (top shelf: telescope, star charts, etc.), the physical world (lower shelf: books and lute), and death (the skull). Another possibility is that the painting was meant to be placed in a stairway so a person climbing up from the left or descending from the right would see the skull. It has also been speculated that Holbein painted the skull to demonstrate his skills to attract patrons. Whatever the reason for the skull, Holbein certainly understood the principles of perspective and how to use it create startling works of art.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
Week 2: Phi and Zero
What is phi?
Phi is a Greek letter that looks like a p. It is similar to pi in the sound and in the Greek figure. Plus, both are irrational numbers. Phi is associated to the Golden ratio, and it is estimated to be 1.618… There are many ways to calculate phi, which is comparable to the many ways of calculating pi. The golden ration works in that there are two line segments A and B where A + B is to A as A is to B. In other words, (A + B)/A = A/B. B is a shorter segment than A. Phi is more then just a mathematical number; it also occurs in nature and in the arts. In nature, appendages are in golden ratio such as the arm, leg, or whole body. In art, there are buildings like the Parthenon that can be considered in golden ration and drawings like Da Vinci’s Vitreous Man. It has also been noted that famous celebrities have will proportioned faces in accordance to the golden ration. Phi is an interesting number to note.
What would life, or math, without zero be like?
Obviously there will probably still be some notation of zero as a placement notation such as 10, 20, 100, etc. But it won’t be used as a definition of nothing, or a beginning sense. Without the development of zero, the idea of negative numbers might not exist. If negative numbers do exist, which seems likely as it goes into the idea of debt and used in electrical engineering, it would be weird if there wasn’t zero to go with it. What would be -1 + 1? Well, even without zero, life will probably go on, as it had before zero was properly defined. Zero may have helped society advance, but the need for it may create a substitute for it if zero isn’t used. If there is no zero, the idea for it still exists and will probably come out in a different manor. So, without zero, life and math will probably use a zero substitute.
Links
http://en.wikipedia.org/wiki/Golden_ratio
http://en.wikipedia.org/wiki/Pi
http://goldennumber.net/images/mandala-phi-pentagram.gif
http://www.phimatrix.com/images/phi-woman.jpg
http://www.jainmathemagics.com/Editor/assets/webjain-phi-logo-05.jpg
Week 2: Math Rock and Fractals
What other music genres or artists are comparable to Math Rock?
With its atypical rhythm patterns and its challenging time signatures, the math rock genre definitely separates from most other genres. It’s hard to compare the sounds of math rocks bands to those of more mainstream artists. However, one scene that shares in math rock’s affinity for quirky time signatures and complex rhythmic structures is that of the jam bands. Headlined by bands such as Phish, Widespread Panic, and the String Cheese Incident, the jam band sound is driven by experimentation and the pushing of musical boundaries. Typically much less heavy or raw-sounding as math rock bands, jam bands are a great alternative to folks who are looking for a respite from the monotony of the popular music scene but who don’t quite like the hardcore sound of math rock.
How do fractals relate to some theories of the nature of the universe?
The self-similar nature of fractals, that at any level of the fractal there are infinitely more levels in either direction of magnification, mirrors some views about the nature of the universe. As scientists probe farther and farther into the depths of space and deeper and deeper into the microscopic realm, they continue to find more and more levels of organization inherent in the universe. From the findings of our most powerful telescopes down to the hypotheses of string theorists, we see an incredibly beautiful and repeating organizational structure. Strings are organized into elementary particles, elementary particles into subatomic particles, subatomic particles into atoms, atoms into molecules, molecules into physical matter, physical matter into planets and stars, planets and stars into solar systems, solar systems into galaxies, and so on. Many people speculate that if we had powerful enough microscopes and telescopes that we could see these repeating organization patterns continue into infinity. In this way fractals create an intriguing parallel to our universe.
Links
http://en.wikipedia.org/wiki/Fractal
Week 2: Art and Mathematics
Although not the first thing that pops into mind when referring to art, Mathematics definitely plays a major role in an artist’s life. Mathematics, the science of numbers, first came into play when artists discovered the idea of depth and perspective. How does one represent the three-dimensional world, on a two-dimensional canvas? Al- Haytham first introduced this idea of depth, while Brunelleschi developed the first correct formulation of linear perspective. He created a single vanishing point, where all parallel lines in a plane converge. Geometry is directly related to this as an individual also must understand the relationship between the length of an object and its length in the picture. Artists like Leonardo Da Vinci used this theory of perspective to create their paintings and other art pieces. Many other theories concerning mathematics and art were later developed. http://en.wikipedia.org/wiki/Math_rock
Mathematics is not only found in the art of painting, but it is also used in the art of music. Before our lecture, I had never heard of the term “Math Rock”. After the outstanding performance in which we witnessed this week, and the lecture, I definitely have a greater understanding on the term “Math Rock”. Regular rock and Math Rock differ in the sense that Math Rock is unpredictable. Math Rock often experiments with strange tempos and rhythms, while still being hard and aggressive. They take Rock and change the typical structure of it all, using asymmetrical time signatures. Math Rock gets its name because of its rhythmic complexity. I am surprised that this form of music has not hit the mainstream, because of the fact that it is certainly entertaining. http://en.wikipedia.org/wiki/Math_rock
Zero
The discovery or invention of place holding symbol has changed the number system drastically. For example, one hundred and seven can be represented as CVII in roman numerals. No matter how easy the symbols for roman numerals are, some repetitions have to be deciphered to fully recognize the number. This presents challenge when doing addition, subtraction, multiplication or division. DCL + CCCL = M can be translated to 650 + 350 = 1000 The numbers used here are rather simple, but when the numbers become complicated, one may make misunderstand the given number even before starting the mathematical operation.
Not only is it hard to perform mathematical operations, but just writing a large number itself is problem in roman numerals. Seven basic symbols (M=1000, D=500, C=100, L=50, X=10, V=5, I=1) can only represent up to 4,999. Symbols with bars on top can represent multiplied by 1000, but when the number goes beyond 1,000,000, there isn’t a standard format to represent these types of numbers. With the zero symbol representing the place, people are able to write any number (however large or small it is) with just ten different symbols. The current (Hindu-Arabic) numerals with 0 can represent much larger numbers with 1,2,3,4,5,6,7,8,9 and 0. The decision to make ’something that represents nothing’ has made a very noticeable change in today’s world.
Link:
http://www.neo-tech.com/zero/part6.html
What’s in a Number?
“So… what’s your major?”
“Math.”
“…Oh.”
‘Oh’ is pretty much all I say; I think it would be rude of me to point and laugh and yell, “WHY STUDY SOMETHING THAT DOESN’T EVEN EXIST!?” Yet for all the people studying it, for all the uses we’ve found for it, and all the faith we placed it in, the concept of mathematics is based on a suspiciously shaky foundation: numbers.
Numbers are so normal, so ordinary, so ingrained that nobody thinks to question their existence; and yet I wonder: do numbers exist? When you count three dogs, the dogs exist and the number is an adjective. When you write ‘3′ the numeral exists there on your paper. But what of the actual number three? Two? One? There is no such thing as a one.
Yet here people would protest that it exists as a concept; it may not be tangible, but then, what ideas are? Clearly math cannot be an arbitrary set of rules and principles, else it wouldn’t be so universal. So it exists in our minds. Descartes’s famous ‘cogito ergo sum’ seems appropriate;we think it, therefore it exists. But what if there was no human life left to think numbers into existence?
Many thinkers, more well-known, qualified, and intelligent than myself, have pondered the existence of numbers without coming to a satisfactoy conclusion. To me this crisis of identity, although worrisome to some of us who think too hard. stands as a testament to the power of the human consciousness. Mathematics as a self-existent field of study is uniquely manmade, of equal parts necessity and innovation. By virtue of its basis in nothing but itself, it intrinsically gains the freedom of being bound only by itself.
And therein I find at least one fiber connecting the realms of math and art. Art is aesthetics; aesthetics are independently defined, with no laws to govern right from wrong. While it is true the attractiveness is partially biological–we seek traits in mates that are advantageous, for example–that is simply one schematic of appraisal. Just as geometry can break free of its Euclidean roots, the evolution of artistic appreciation beyond the ‘norm’ is the essence of the similarity between art and math.
This perhaps is why I don’t regard math as a ‘regular’ science. The sciences concern themselves with sets of rules and laws to govern the world. For those worlds that exist, this creates a natural limitation. For the world of numbers, imaginary as they are, there is no such barrier. This places mathematics into a sort of intersectional limbo, with both roots and branches so far-reaching, its ubiquity transcends that of any other language. It is clearly present in the other realms of science; field theory, for example, is nearly purely mathematical. Equally is it present in the artistic world, where even aesthetics will recognize the significance of phi.
Who needs a third culture when you have math?
Here’s some people who are a lot smarter than me discussing math. It’s really quite interesting:
http://www.blackwell-synergy.com/doi/pdf/10.1111/1467-9213.00079
http://www.math.hawaii.edu/~lee/exist.html
http://platosheaven.blogspot.com/2005/12/do-numbers-exist.html
Math Rock: Too New for its Own Good
Why hasn’t music like math/noise rock become mainstream?
The problem with radically different music is the fact that it’s too new, or too progressive. If you observe the progression of music, especially from the ‘50s until now, you’ll notice very similar traits that most bands (solely speaking in terms of rock music, however, other types of music such as rap and R&B follow similar patterns) have. Most have drums, guitar(s), and bass, and since the late ‘70s, many bands have synthesizers and keyboards. Most rock bands in the last fifty years have “pleasant” vocals that lack screaming, and the instruments all generally follow basic time signatures (4/4). Using this formula, you’re bound to make at least one rock song for all to enjoy.
The difference between regular rock and math rock is the fact that math strays away from nearly every aspect of traditional rock. The time signatures, the screaming (or in contrast, no vocals), the lack of certain instruments, the bizarre sounds, and the generally different formulation of songs leads to an ultra-progressive style that few people can appreciate. It’s not that these bands lack talent. In fact, some of the most talented musicians I’ve ever heard are math rock musicians (see Zach Hill, Spencer Seim, Damon Che, etc). The fact is is that it will be difficult for math rock to reach commercial success.
However, many pop bands are slowly, yet surely getting there. Bands like The Killers and She Wants Revenge are changing the face of popular rock music. For instance, their use of quick high hats on the drums, simplistic, repetative guitar lines, and non-conventional vocals distinguishes them from traditional rock bands like Nirvana, Pearl Jam, or Green Day. That being said, it must be possible for bands like Don Caballero and Hella to someday make it on the radio. I mean, if someone heard The Killers on the radio in 1950, they would have thrown their radio at a wall and asked the what the world was coming to. The problem with math rock is that it has simply progressed too quickly for it’s own good that only a select few can really enjoy it.
Links:
http://youtube.com/watch?v=RdPIfMO28ck
http://www.epitonic.com/index.jsp?refer=http%3A%2F%2Fwww.epitonic.com%2Fgenres%2Fmathrock.html
http://en.wikipedia.org/wiki/List_of_math_rock_groups
Week 2: Math, Art, and Science
How does the Penrose staircase work?
After staring at a picture of the endless staircase for several minutes, I still remained puzzled. It’s an impossible object… yet it is still possible to depict it through art, using perspective distortion. I found it easier to understand the concept by studying a 3D model (see the youtube vid here). The staircase is not actually connected; there is a gap between 2 of the walls. When viewed from a certain angle, voila, it’s endless!
Roger Penrose, the mathematician credited for popularizing this illusion, has also used the same idea in other shapes. Because of its simplicity, I was fascinated by the Penrose triangle. I actually learned how to draw one, and was amazed by the fact that I could not explain my own creation. Try it for yourself…(How to draw the penrose triangle)
Fractals do the body good…
As I learned in class, fractals are things that can be subdivided into parts, each of which are smaller copies of the whole. Although people may not notice, fractals occur in nature all around us, such as in the branches of a tree. Recently, I was able to see an important example of a fractal. At the Bodies exhibition, real human bodies and body parts are on display, giving the public an opportunity to see and learn about what is inside us. Veins and arteries have intricate patterns for the transportation of oxygen between our hearts and the rest of our bodies. Their design is based on fractals. It’s interesting to think that the complexity of our bodies lies in mathematics.
More Links:
The Beautiful Golden Ratio as an Archetype
Beauty
How is it that the golden ratio shows up so often in so many different locations, media, and implications where it was not specifically intended? For example as witnessed by the online youtube video, when applied to the face it can be found in what society already found to be the most appealing of people? It can also be found in the amazing architecture of the ancient Greeks. It is not known for certain whether or not the Greeks knew of the appealing power of the golden ratio. It may have been that their architects found the ratio use to be particularly appealing and it later turned out as the golden ratio. So what is an archetype and how do they turn up in societies all over the world? How do they hit a common chord or feeling among us?
Archetype
Archetypes have been found in literature, paintings, and even movies. Now they may not be shared by all people, but many are universal. Many examples can be seen in colors and the feelings they evoke, or characters in a story. Most people recognize the association of evil with darkness or black, and the characters of the Wise Old Man or Hero. The golden ratio is not as prominent as these other archetypes, but is present in history nonetheless. As mentioned earlier, it is present in the buildings of ancient Greece and the faces of “beauty.” The use of the golden ratio elicits a feeling of beauty, and vice versa the beauty in some cases can be traced back to the golden ratio. Since the connection of the golden ratio to the sense of beauty, it has been used by artists to elicit this sense of beauty much like other archetypes have been used in different media to use elicit their own feelings. An archetype rich film is the Disney Classic The Lion King.
Simba, as a child, must return as the hero creates a feeling of sympathy and encouragement towards him. Also how the colors and appearances of “Shadowlands” and the “Pride Lands” evokes feelings that justify their names. Similarly Salvador Dalí used the golden ratio in his masterpiece “The Sacrament of the Last Supper.” You can decide for yourself how its use evokes your own feelings.
http://en.wikipedia.org/wiki/1.618
http://en.wikipedia.org/wiki/Archetype
http://www.kinema.uwaterloo.ca/wong991.htm
http://www.worldvillage.com/wv/school/html/reviews/lionking.htm
http://britton.disted.camosun.bc.ca/jblastsupper.htm
Mathematics, Perspective, Time and Space
Why is nonconformity unaccepted?
The obvious approach to non- conformity are the artistic principles presented by Leon Battista Alberti. The use of linear perspective was criticized as he violated the principles of art during the Renaissance era; which coincidentally also provided various other artists with the approach to attempt new techniques on art and its process. Why do we have to conform to redundant principles? In current days we see a non conformist attitude within the fashion industry. Tyra banks one of the worlds highest paid supermodels was never a typical model with skinny legs, tight waist and long blond hair. Banks allowed herself to reconstruct the principles of fashion in order to gain a higher appreciation for fashion amongst a wider audience. The non-conformist attitudes of the arts are not acts of rebellion, rather they are a form of reconstruction of principles that allows incoming artists to personalize their talents; thus breaking the redundancy in life.
Is math a part of nature?
Although we do see the golden ration sequence appear repeatedly throughout nature within plants, animals, and biological patterns but is it really intended to pertain to nature. Although their is vast proof to show that the pattern exists what explains its purpose. Why does the pattern matter to the processes of life? Beauty is not a mere importance to the golden ration it is only an area in which the number appears. Some of the applications of the Fibonacci spiral also seem irrelevant to both art and science. What is the purpose of the golden ratio and how does it help us understand the applications of today’s processes?
http://library.thinkquest.org/27890/theSeries6a.html
http://www.flickr.com/photos/steffe/188300024/
http://www.princetonol.com/groups/iad/lessons/high/Grace-golden.htm
