Archive for the ‘Week2_Mathematics, Perspective, Time & Space’ Category

Week 2_Musical Math by Angelica Merida

Saturday, March 7th, 2009

Math seems to be one of those subject that finds itself in pretty much every aspect of art. From geometric art, to the circle of fifths, to even help determing how much support a truss needs so that it can sustain several actors/dancers running back and forth onstage.

By far the most interesting to me, is its influence in music. Music theory has everything to do with math. The first to discover the relation between math and music was Pythagoras of Samos. To Pythagoras ratios were everything and in music, ratios play a big role. In Greek music at the time, each octave only had 5 notes. Pythagoras noted that each note in an octave was simply a fraction of a string. For example, if you had a string that played the note A, if you only allowed ¾ of the string to vibrate upon being plucked you would get approximately the note D, if 4/5 of the string vibrated then you’d get C, and so on.

Later on, we got 12 note scales because followers of Pythagoras started applying this notion to other notes.

Math and physics can also answer other questions in music. For example, why if a flute and a violin play the same note, do they sound different? Answer: harmonics. Music is made of sound. Sound is made up of repeating sound waves. In physics, harmonics are waves at proportional frequencies, and inversely, at proportional amplitudes. If we were to play an “A” we not only hear the 440hz tone, but also the 880hz, 1320hz, 1760hz, and so on until the frequencies get too low or high for us to hear them.

Many classical composers are said to have incorporated math in their music. For example, Beethoven’s Fifth Symphony is based on the “golden mean”. That is, the ratio between the sum of two quatities and the larger quantity is equal to the ration between the large quantity and the smaller quantity. This is actually a constant (1.618033988….) usually denoted by the Greek letter phi, Φ. Another piece that is said to have been influenced by math is Bartok’s “Music for Strings, Percussion, and Celesta” in which Bartok structured his music using the Fibonacci sequence in the first movement.

Here is a link to Bartok’s piece  Music for Strings, Percussion and Celesta , and Beethoven’s Fifth Symphony.

Furthermore there is also the relationship between learning math and music commonly referred to as the Mozart effect. It is said that if a child is exposed early on to early classical music, then it will lead to better performance on tests including spatial visualization and abstract reasoning, plus they tend to excel in subjects like math and science. Now where this is true or not, I am not sure, but it is definitely something to look into.

Friday, February 27th, 2009

Also, Please post your extra credit assignments to the category entitled “Extra Credit Blogs”

Week 2: The Golden Ratio’s Place in Art, by Sarah Lechner

Tuesday, February 3rd, 2009

Being a physical science major, it is extremely fascinating to explore the connections between mathematics, science, and art. I know that a lot of modern art has begun to use advanced technology, but I had no idea that even Leonardo DaVinci integrated mathematics into his art. One mathematical term in familiar is very fascinating to me: psi. Psi, also known as the golden ratio, is a number encountered when taking ratios of distances in simple geometric figure. The number turns out to be 1.618034…. At first glance, this would appear to be irrelevant in art, but it can actually be found in architectural masterpieces created in Ancient Greece. The Parthenon is famous for its use of the golden ratio in its construction; even the Greeks knew that the golden ratio was visually pleasing. Another ancient example of the use of the Golden ratio is the Great Pyramid of Giza.

This pyramid was built circa 2560 BC and the length of the base divided by the height is very near the Golden Ratio (1.5717…). Euclid also speaks of the Golden Ratio; Plato dabbled in the area as well. Plato philosophized that if a line were divided into two equal segments so that each would be related to the whole in the same way, then a special proportional relationship would result. Plato, as we know now, was quite correct.
So how, exactly, is the golden ratio derived? It turns out that there are multiple ways to do so. One is the Fibonacci sequence.
Here is a ‘Fibonacci series’.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..
If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers.
1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666…

8/5 = 1.6

13/8 = 1.625

21/13 = 1.61538…

34/21 = 1.61904…
The ratio seems to be settling down to a particular value, which we call the golden ratio(Phi=1.618..). (http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html)
The Fibonacci Sequence is even more intimately related to the Golden Ratio: shapes exist such as the Golden Rectangle and the Golden Spiral which combine these two concepts in different ways. The Golden Rectangle is a shape formed using proportions that are two sequential numbers in the Fibonacci Sequence. It has been theorized that this shape is one of the most appealing shapes to the eye. It seems like no large coincidence then that it is precisely this shape that Leonardo DaVinci used in his very famous Mona Lisa.
The Golden Ratio is also found in more modern artwork. The Sacrament of the Last Supper, by Salvadore Dali, is actually painted on a canvas matching the proportions of the Golden Rectangle.

Clearly, the Golden Ratio has stood the test of time. Its usage has spanned centuries upon centuries, and it has been utilized by dozens of artists and architects. Personally, I think this is a part of its appeal. The Golden Ratio and the Fibonacci sequence are found so often, in both math, art, and nature, but both concepts still remain largely a mystery. Why are shapes created by these mathematical concepts so visually pleasing? Did the ancient architects realize the aesthetics of these ratios and implement them on purpose, or was it just a coincidence? Personally, I think there is still a lot to learn about the use of the Golden ratio in history, but the ratio does serve one very important purpose: it shows another way that math and art are intimately connected.

http://www.youtube.com/watch?v=fmaVqkR0ZXg

P.S.–sorry this is late, I didn’t join the class until 3rd week so I had some catching up to do.

Week 4_ Extra Credit: Week 2 N/S Campus Mixer by Nicolina Greco

Sunday, February 1st, 2009

Two weeks ago I attended the North Campus/ South Campus mixer at UCLA, held in the main lobby of the California Nanosystems Institute. It was an exciting experience for me especially because I got to see artist Nina Weisman’s showcase that I helped assemble the previous week. It was amazing to see the finished product. Her art piece consisted of 10 different sized white columns that were placed in a path. Each column had sensors within them that made strange noises when people approached them, and the closer someone approached each column, the more distinct the sound became. The experience that the artist evoked through her work was that the human body was being scanned for nano particles. Each different sized column scanned different parts of the body, which made different sounds based on the area of the body it was concentrated on. Bursts of air shot out of every column to distinguish what part of your body was being scanned. The sounds that protruded from each column were scattered, random noises of beeps, robotic voices in different languages, and deep tones that actually gave a frightened effect. As I walked through the showcase I felt intrigued by the experience it gave me, and I thought it was a really neat exhibit.

At the mixer I got the opportunity to chat with many other students that were both art and science majors. I recall one conversation that I had with a student that was a theater design major, he wants to design sets and get into the industry of engineering props for theatrical performances. What was interesting to me was that he was previously a pre-med, chemical engineering major, but he decided that his other interest, theater, was his true passion and the field he wanted to pursue. I was fascinated about how much he know about both art and science, and it showed me how although the separations on campus are distinct, many people are equally balanced in some way or another and are passionate about both art and science. I also got the opportunity to talk to many grad students here, including one of my former math TA’s, and it was interesting to learn about the kinds of research they were doing. 

The North Campus/South Campus mixer was a great way to meet students of all majors, and taught me to appreciate both sides of campus equally. It was an experience that I was happy to have participated in. 

-Nicolina Greco

(I didn’t get the chance to write this blog until now instead of two weeks ago)

Week 2_The Fourth Dimension and a Change in Perspective by Beverly Okereke

Tuesday, January 20th, 2009

Buckminster Fuller and his famous Geodesic Dome

Buckminster Fuller and his famous Geodesic Dome

What does a simple playground, Walt Disney, and the vast world of science and mathematics have in common?

This week Professor Vesna talked about the Fourth dimension, mentioning several artists and exceptional thinkers who have contributed to a realm that most would consider quite unconventional.  This realm lying outside the confiedes of normal reality, makes the regular 3-dimensional world look primitive.

One particular man that I personally saw as interesting  that the professor mentioned in lecture was Buckminster Fuller. Fuller, born towards the end of the 19th century, wanted nothing more than to answer one of life’s most complex of questions about survival and longevity. How long would we as a human race survive on this planet? Devoting his life to this question, he wrote many books, and created many inventions, including his most famous, the geodesic dome, which is still used in a variety of instances today.  Although Fuller himself was not exactly the first to use this geodesic design, he was the first to have it patented.

One instance in which Fuller’s dome is still used today answers the aforementioned question of how a playground is related to science. Compare these two pictures, for instance:

3D Representation of a Geodesic Dome

3D Representation of a Geodesic Dome

A Dome Climber Jungle Gym
 Look similar? Well they should.  The design of the two look exactly the same. The design for this simple addition to a child’s backyard paradise matches the Fuller’s patented geodesic design.  It is also similar to Walt Disney World Resort’s EPCOT and variety of other structures. Can you think of any more?    

Walt Disney World Resorts EPCOT

Walt Disney World Resort's EPCOT

Linda Dalrymple Henderson’s The Fourth Dimension and Non-Euclidean Geometry in Modern Art: Conclusion, the reading for this week, dealt with the idea of a “Fourth Dimension” is associated with the idea of modernism,surrealism, and Einstein’s theory of relativity. In this time, the late 19th to early 20th centuries, was very important in relation to the art world. It ushered in a new era for the world of art, playing a significant role in the nurturing and “development of modern art and theory”. By inspiring artists such as the cubists, modernists, futurists, and surrealists, among others.  The idea did not only relate to art, however. It also dealt with science as well, linking ideas such as Einstein’s Theory of Relativity (Space-time Relativity Theory) to the movement in the earlier half of the 20th century.    

Space-Time Continuum

Space-Time Continuum

Einstein Theory of Relativity

Einstein' Theory of Relativity

 

Einstein’s Theory had much to do with the perception of the fourth dimension as time.  As the pioneer of his time in Germany, Einstein’s ideas and discoveries changed the way the whole world viewed the universe.

 

 

 

 

 

 

The idea of a fourth dimension flawlessly mixes the idea of art with science, technology, and mathematics.  Drawing off of Einstein’s famous theory comes another idea that existed around the time of the “Fourth Dimension”: “Non-Euclidian Geometry”. Einstein’s theory describes space as generally flat (i.e., Euclidean), but elliptically curved (i.e., non-Euclidean) in areas near where matter is present.  Remember those geodesic domes from before? Well, they can be described as a form of hyperbolic non-Euclidian geometry, which, again, is a significant way to describe the mathematical implications of time and space in relativity.

My favorite example of hyperbolic sculpture is the “Hypar” in the California Science Center. Created by Chuck Hoberman in 1996, this amazing 5-story sculpture expands and contracts continuously during the day, changing the viewers perspective of the sculpture every time it make another move. It is a  5,000-pound aluminum structure, and is known as the Hypar for its hyperbolic paraboloid shape, expanding from a diameter of 15 feet to 50 feet, with the motion of its 2,500 aluminum links controlled by a computer at the sculpture’s base:

 

Chuck Hobermans Hypar in the California Science Center

Chuck Hoberman's "Hypar" in the California Science Center

 

Week 2 Changing Dimensions By Brendan Ryan

Monday, January 19th, 2009

Sometimes I like to think of ants as two dimensional creatures, and so I put then on pieces of paper and fold the paper into weird three dimensional shapes and watch what they do while they crawl over corners.  If you put an ant on a sphere it will walk around the whole thing many times and really never figure out its going in a circle, that’s because its brain is too stupid to understand how dimensions work.  Then I think about how I am just an ant stuck on a 4 dimensional piece of paper going in some crazy shape that puts me back where I started. In the movie The Cube 2: Hypercube there are many people trapped in a theoretical four dimensional cube and almost all of them died inside (sorry if I gave anything away!) because no one could figure out what the heck was going on. That’s because moving around in a four dimensional cube works like this:

navigating 4th dimension

Walking around in a hypercube like that would be really difficult if you could only perceive it in three dimensions. If you marked all of the cube rooms in a hypercube house and walked around for a while so you could figure out what rooms lead where you would feel like an ant walking around on one of my crazy origami contraptions.

People often use time as a fourth dimension which makes sense for most practical applications. To meet someone you have to know where they are going to be and when they are going to be there in order for you to meet. Because people cannot conceptualize more that three dimensions we only see one slice of time at once, just the same as how we can only see three dimensional cross sections of four dimensional objects when we attempt to study them. From the perspective of higher dimensional aliens things may appear not only in different places but perhaps in all stages of its existence through time. Using this loose definition you could treat many different qualities as dimensions, at least for limited purposes. It is possible to effectively simulate higher dimensions by using any independent and infinite quality such as temperature or color. Imagine if to be in the same place as your friends you all had to be wearing red and if one of you got a fever he or she would go somewhere else where everything else was hotter too. Thinking of higher dimensions like this inevitably reminds me of the Simpsons episode where Homer is transported to three dimensional space. When Homer is asked to describe what it’s like living in an extra dimensional space he replies “Has anyone ever seen the movie Tron?” I think that’s probably what it would be like.

Mathematics and Art by Oscar Chacon

Monday, January 19th, 2009

The only connection between humanities and science that I could see connecting strongly are art and math.  The art produced from the Renaissance and more recently Mandelbort’s fractals are clear examples of this.  It is clear that the relation between mathematics and art is inevitable.  There is something in attempting to capture the exactness of something in a painting that requires precise measurements produced by geometric calculations.  Thus, there was the creation of linear perspective by Brunelleschi.  I have taken a couple of art classes and have had some trouble grasping the concept of linear perspective. The painting, “The Last Supper” by Leonardo Da Vinci is an excellent example that produces very exact measurements and more importantly a beautiful piece of work highlighting an important religious event.

The influence of the suggestion of a fourth dimension has had an enduring effect on art, thus elevating the intimacy of the relations between math and art.  It is captivating to think about capturing or expressing motion or time.   The manner in which artist have captured the concept of the fourth dimension is far more easy in understanding in comparison to its counterpart scientific explanation. 

The fourth dimension is another aspect of sciences of which is greatly represent by science fiction. The fourth dimension as being time can be seen in the classic movie Back to the Future where the wacky character, Doc Brown invents a time machine that allows him to travel and forward through time.  This movie of course holds inspiration from H.G. Wells’ The Time Machine, and many other versions modified from this version.  The idea of the fourth dimension as time is easier to understand rather than something spatial, which I think is open for philosophical interpretation.  Time is passing is easy to be conceived since we always experience it constantly.

The spatial conception of the fourth dimension is from what I understand harder to grasp.  This is perfect opportunity for artist to express this concept, and so they did.  This is greatly depicted in Star Wars and Star Trek where light speed is possible, and there is centered attention on the participation of humans in a spatial environment.  The movie series “The Matrix “ provides an interpretation of this other dimension not by time but of consciousness.  The Surrealist artists really capture this spacey quality and idea of a separate consciousness.  Salvador Dali’s “The Persistence of Memory” captures this concept very explicit while at the same time integrating time into his work.

            This work not only resonates the concept of merging spatial and time interpretations of the fourth dimension but also in a sort of joining of the sciences and humanities.  The spatial qualities of the fourth dimension represent the humanities since they are best represented by artist.  The time perspective of the fourth dimension is well the sciences since it is easily explained by physics, and such.

Oscar Chacon

Week 2_Extra Credit_North/South Mixer_by Adam Parker

Monday, January 19th, 2009

As I was walking to the California NanoSystems Institute building for North/South campus mixer, I had no idea what to expect. I had never been to a university organized art exhibit before, let alone a mixer. As I arrived at this part of campus which I had never ventured to before, it became obvious where the mixer was being held. I entered the building and scanned my surroundings. There was an entry table with various handouts and informational pamphlets, several tall tables scattered about the room, a bar in the back containing sodas and alcoholic beverages, a buffet of different types of food lined against the wall, a main art installation, and a few informational display boards.

By the time I arrived, most of the food had already been eaten and most of the professors had left. But I did manage to grab one of the last remaining Sprites. Although I admit I was not the most social at the event, I did manage to speak to several other Art, Science & Technology students about their views on the class and the mixer. I spent most of my time wandering around and through the main art exhibit. The display was made up of several different sized white blocks incorporated with motion sensing cameras. When someone came within a certain distance of the block (roughly 1-2 feet), a speaker in the block would produce some sort of ambient sound. Although I found the exhibit to be engaging, I wish I had arrived earlier to receive a proper explanation of the display itself. All in all, the North/South mixer was a great introduction to what academic campus events have to offer and I look forward to attending the next one.

Week 2_The Fourth Dimension and Beyond by Adam Parker

Monday, January 19th, 2009

As an engineer and a “math guy”, I always believed that the possibility of a fourth dimension was mostly a science issue, but after reading about non-Euclidean geometry in modern art, my views have drastically changed. I now know that all dimensions and ideas of space affect artists just as much as scientists. I found it very eye-opening that the fourth dimension wasn’t just a drawing or spacial tool but was “primarily a symbol of liberation for artists.” In retrospect, this completely makes sense. Before the concept of the fourth dimension emerged, artists were bound to representing reality as everyone saw it in its true form. Once fourth dimension ideas began to spread, artists were given the opportunity to portray the world in any way they saw fit. This led to a very unique era within the world of art. An era which I find fascinating.

The sequence of pictures above shows Jean-Francois Colonna’s representation of the fourth dimension. He begins with a hypercube and alters it until it depicts what he believes to be the fourth dimension. I found it interesting that the cube began as black and white, turned into an array of colors, and then ended with something entirely different - a mixture of gray scale and minimal vibrant colors. The alteration of the hypercube is also very peculiar. It changes from straight formulaic lines, to curvy psychedelic lines, and finally into something that I would like to call, “worms and amoebas”. This evolution of cubes shows the creativity that can be expressed through the idea of alternate dimensions.

Above is a drawing and sculpture from artist Tony Robbin. What I like most about Robbin’s drawing is the fact that it is made up of several different layers. On one hand, there is a complexity of white angled lines which seem to represent a vast blueprint, but one the other hand, the colored shapes give the drawing a sort of depth. By looking at the drawing from different perspectives you can either see a flat collage of colors and shapes or a mixture of overlapping cubes and other three dimensional shapes. Robbin’s sculpture seems to take the artistic ideas from his drawing and converts them into a three dimensional object. The sculpture is still created using white lines to bound the colorful shapes, but this time his creation is truly 3D.

One last example that I found visually appealing is Salvador Dali’s Crucifixion (Corpus Hypercubus). I liked the fact that he took a world renown symbol and put his own spin on it. Instead of using the usual cross that Jesus is nailed to, Dali decided to use his own form of a hypercube. As seen in the painting, Dali also decided to repaint the crucifixion without nails. Instead it seems that Jesus is invisibly bound to the futuristic crucifix by the small hovering cubes in front of him. In addition, hypercubic crucifix transitions into the rest of the painting by producing a shadow on the black and white checkered floor which repeats across the ground.

Week 2: Knights, Poets, and Prophets of Space

Monday, January 19th, 2009

Okay, so due to technical difficulties, this is late, but c’est la vie.

         After doing the reading for this week, I found myself thinking about what it means to involve science, and art, and what that has to do with the fourth dimension at all. I am a humanities major, and in all honestly, I have a very unclear vision of what the fourth dimension means, after all, I- as a creature in a what I consider a 3-D world- can really have not that much understanding of anything beyond that. When I see illustrations of the fourth dimension, all I can think about it how it makes sense, but then its on a two dimensional surface, and not actuality. I remember watching this video in my ninth grade science class which was demonstrating dimensions. The paper people living on the two dimensional surface had absolutely no clue about anything beyond that, I mean, after all they couldn’t even really see each other on their paper plane, so how are we different from them? 

      I think that this is where the artist steps in. Science can explain to me what the fourth dimension is, how I don’t live in it, and how its possible for me to fail a test on it even though it doesn’t really apply to my daily life. However, if I look at an art piece which is trying to show me the fourth dimension, usually I’m baffled and very curious. Maybe its because I have a large imagination, and the art makes me see the world around me in a whole new sense. Not like the movie the Sixth Sense but something more socially acceptable. Scientists discover these new worlds, but it takes art for me to get interested in them. Its like when I read the Discovery magazine, but I have to look at the pictures for the article to make sense to me. Maybe that’s because I’m more of a visual learner, but also because I find that the artist can enhance the new theory, and really illustrate it in a medium which gets you to start thinking about the subject in an applicable sense, like the fourth dimension. 

       So if were living is a very science driven world, as I see it, where is the line drawn between aspects of art, and those of science? The researcher on the fourth dimension must be creative: how weird is it to think that there is another plane beyond what we can sense? An artist is creative as well:  How could M.C. Escher come up with the incredible staircases that lead to nowhere and yet everywhere else too? The point to me is that the human mind is the most incredible thing in the world, and it belongs to scientists and artists both.