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

Week 2_Fourth_dimension by Nikola Kondov

Sunday, February 8th, 2009

During week 2 we have discussed how mathematics  connects with art. Although something “boring” like math could not be easilly associated with art, both of them are inseperable. Whether we talk about a painting, a sculpture or music, mathematics is everywhere. The Golden ratio seems to appear all the times. In the same time, the concept of “the fourth dimension” became important among artists during the 20th century. The types of art that come closest to that concept are, according to me, the three-dimentional street art, that creates the illusion of space where it does not exist. Artists like Kurt Wenner challenge people’s perspective about art.06-hell

Another type of art that approaches the idea of the “Fourth dimension” is the “flashlight art”, in which street performers use a flashlight in order to create images in the air. What is unique about this art is, that the artist must be quick in making his art, simply because the image disappears. Of course, high-speed cameras may save and reproduce this work of art.

This type of art can be pushed to a whole new level, as we can see here:

Paris by Light

or here:

light graffiti

These works of art show that today’s art still strives to overcome the three-dimensional boundary it is restricted to.

week 2_less dimension while more dimension by josh bohbot

Monday, January 26th, 2009

for 1000s of years artits have been trying to acheive as realisticly as possible 3 dimension in their art. artists have developped techniques such as the vanishing point, to more greatly improve their illusion. today, with computer graphics we can acheive ultra reaistic 3 dimensional effects, a 3d face can now  look  almost as real as anyones in real life. but our advanced 3d media can also go in the opposite direction. especially in video games and animation, new rendering capabilities  enable artists to take 3d to fake 2 dimesionality. my favorite example is from the game The Legend of Zelda: Wind Waker, wich was one of the very first games to use this technique

2d looking image, of link on his  boat, completely made with 3d graphics.
this rendering technique is called cel shading. in animation this technique can even go further, by creating outlines around object to give it a cartoon style. -this link is a short animation entirely created with cel shading.

Week 2_ Math is Artistic by Christine Vu

Sunday, January 25th, 2009

     Math has snuck itself into the vast world of music and art. Nowadays, artists and musicians are gaining popularity because they have found a way to cater to two very different cultures. Artists and musicians are forced to take on the role as mathematicians in order to offer a new medium for their audience.
     More than ever, math has become a key asset to starting trends. Many young adults are being intrigued by musicians, artists, and scientists who have strayed away from conformity. Musicians, like Tatsuya Yoshida, are now incorporating mathematical formulas into their drumming schemes, something that may very well be appreciated by a math group as well as a rock band. New genres of music, like math rock, are finding its way to the top, offering a new aspect to music. Soon enough, the role of music will shift from being just an entertainment tool to solving a math problem.
     As human beings, it is of our nature to be attracted to anything that challenges our knowledge. This has been evident even in the 1400’s when artist Brunelleschi discovered the idea of the vanishing point. This new concept attracted attention because individual perception of reality was challenged. Even for scientist Albert Einstein, he was puzzled by the integration of space and time. As a result, he discovered the special theory of relativity, integrating his artistic ability to construct a spacetime continuum. This concept was the idea that space was in 3D but time played the role of a 4th dimension. According to “The Fourth Dimesnion and Non-Euclidean Geometry in Modern Art: Conclusion” by Linda Dalrymple Henderson, this belief encouraged artists to depart from visual reality and to reject the one-point perspective system. This led to the formation of abstract art. Math has changed the way we view art, and it is time we acknowledge its importance.
     Math is a necessity in moving us forward. Long before I can remember, our nation’s growth in intellectual design, technological advances, and scientific input is credible to math. Without advances in mathematical strategies and formulas, our knowledge of computer design, engineering, physics, and architecture would reach a constant standpoint. Our nation’s man made artistic creations-
Statue of Liberty Statue of Liberty Golden Gate Bridge Golden Gate BridgeHoover Dam Hoover DamMount Rushmore Mount Rushmore- are all attributable to the beauty of math. With much evidence, math has proved itself to be artistic.

Week 2-Math/Perspective/Time/Space By Mary Tam

Tuesday, January 20th, 2009

Professor Vesna talked a lot about the number zero that I had no idea about before. Greek astronomer began to use the number zero. The eastern countries used the number zero before the western countries did. During the middle ages, zero meant the devil himself. Many people did not accept zero because everything would change in the learnings. Another thing that was very interesting is that I have had experiences with math, perspective, time, and space. In high school, I had to do a math project for my math analysis class. I did research on Galileo Galilei. Galileo Galilei was a great astronomer, mathematician, and physicist. He had many inventions and new theories. One of the experiments that he came up with was that he dropped two cannon balls of different weight at the Leaning Tower of Pisa. It was thought before his experiment by Aristotle that the mass of the particle is dependent on the speed of the particle. He was skeptical of that theory so he did a demonstration on it. Clearly, his demonstration said otherwise. As a result of his experiment, he came up with the Principle of Inertia, which is  a body moving on a level surface will continue in the same direction at constant speed unless disturbed. He also came up with a kinematic law that distance is porportional to time squared. His ideas were adopted from Newton’s  Law of Motion. The mass was in fact independent from the speed. It is interesting how visuals can demonstrate and show a lot of new perspectives.

I also found Mandelbrot very fascinating. There’s an infinite of images that goes on and on. When Professor Vesna showed some fractuals in class, I really found them enjoyable. I realized there is an interpretation of these images like Rorschach inkblot printing that I learned in psychology. These images are what you interpret them to be. But how you interpret them determines the way you think. When I saw one of the fractuals YouTube video, I saw happiness and a calming feeling. However, some people might not understand or find the same meaning. Overall, I want to say that art has no boundaries. You can do whatever you want as long as you are creative with it. Although science has some boundaries, one can always come up with new theories and perspectives, like Galileo. We cannot be ignorant and accept everything we hear or see.  With science and art incorporated together, impossibilities can happen.

Here is one of the Fractual video that I found interesting:


Week 2-The Dimensions and Time Travel- Idy Tam

Tuesday, January 20th, 2009

We are familiar with 3-dimensional space because it is our perspective of the world we live in now. But there is an extra dimension that we should dwell on, the fourth- dimension, which is usually interpreted as time or as space (Euclidean space).

In the lecture, Professor Vesna talked about the evolution of art from a 2-dimensional perspective to Escher, who illustrated how humans began seeing the world in other perspectives, as his art pieces included dimensionality. Escher incorporated mathematical relationships among shapes, figures, and space. He is one of the first artists to sketch planes with what people called “irregular” perspective. Escher not only made contributions to the art field but also to the mathematics field. This demonstrates the bridge of the two cultures.

In another scope of this topic, I would like to talk about the fourth dimension relating to time traveling. Time travel is a concept of moving between different moments in time and different points in space. I am interested this particular area because at some point in a person’s life she may wish to travel back in time to alter history or travel to the future to explore how life would be like.
But many physicists argue that traveling back in time is impossible due to the “grandfather paradox.” This theory questions, what if one travels back in time kills his own biological grandfather before conceiving his father? Although many may wish that traveling back in time may some time be made possible but the chances are very unlikely.



Wormholes, which is shortcut between space and time, is permitted by Einstein’s field equations of relativity, suggests that it is impossible to travel through a wormhole. But the wormhole can allow timetravel by accelerating the objects speed to a high velocity, relative to the other then sometime later bringing it back together. Scientists believe that it may not be possible to convert a wormhole into a time machine in this manner: some analyses attempted to incorporate quantum effects into general relativity, indicate that a feedback loop of virtual particles would circulate through the wormhole with ever-increasing intensity. This destroys it before any information could be passed through it, in keeping with the chronology protection conjecture.

Instead of inventing a time machine that travels to the future or back in time, why don’t we just think of the events in our lives as a fourth dimensional perspective. From the day we were born, till the day we die, our memories act as the fourth dimension. We can constantly revisit our past mentally, even though not physically. But the closest “time machine” that there is, our memories, is actually what we take for granted. There is no point in altering the past.  What has been done, is done.

-Idy Tam

Week 2 – “Mathematics, Perspective, Time, and Space” by Derek Spitters

Tuesday, January 20th, 2009

It is clear that mathematics is intricately intertwined with art, nature, and beauty. For example, the concept of perspective allowed painters to portray a more realistic and believable representation of the real world. On the other hand, abstract art often ignores these principles. This does not invalidate the importance of mathematics in art, but simply illustrates that a mathematical approach to art is one possible avenue towards creating a masterpiece.

A more compelling argument for the importance of math in art is its apparent prevalence in nature. Fractals are an example of the presence of math in nature. A fractal is a geometric shape that is made up of increasingly smaller parts, each of which approximates the shape of the figure as a whole. These types of shapes are common in nature. It was once thought that many of the beautiful shapes found in nature were simply random occurrences. It is now known that many of these phenomena can be represented by iterative equations. In fact, great advancements in creating computer models of nature have been made thanks to the use of fractals. An illustration of this is that equations that model fractals can be used to render more accurate landscapes.

Examples of fractals found in nature:
A cross between broccoli and cauliflower

An additional instance of mathematics in nature is the golden ratio. This proportion, which is approximately 1 : 1.618, is widespread in both geometry and nature. For some reason, this ratio has an intrinsic aesthetic appeal, and is therefore commonly used by both architects and artists. However, more importantly, this ratio can be found in many places in nature. The branching of tree limbs and veins in animals follows this ratio. Even human bone structure is in accordance with this proportion.

Mathematics also appears in music. Even a cursory study of harmonics illustrates how math governs music. Conversely, it is still surprising that sequences of numbers, such as the Fibonacci Sequence, can be use to create an attractive musical composition. One would expect a series of random sounds to result from such an approach. The interconnectedness of these two seemingly unrelated fields demonstrates the overlap between the “two cultures.” Under a rigorous analysis, it is impossible to completely separate one culture from the other. There will always be some overlap.

The fourth dimension is an interesting concept both in art and science, and is an idea that has evolved over time. Originally, artists of the Cubist movement attempted to portray objects in four dimensions by depicting them simultaneously from many different angles. These artists were attempting a spatial representation of the fourth dimension. Einstein’s theory of relativity postulated that the fourth dimension was time. Some of the artists in the Surrealist movement attempted to adopt this new theory into their portrayals of the fourth dimension. One way of doing this was through movement since movement shows the passing of time. Other artists felt that the two concepts of the fourth dimension (space and time) were not mutually exclusive. As the scientific definition of the fourth dimension began to change, artists depicting four-dimensional objects began to change their methods as well. This is a perfect example of the integration of the arts and sciences.


Using fractals to model nature:

The Golden Ratio:

–Derek Spitters

influencing the fourth dimension_Nicole Carnarius

Tuesday, January 20th, 2009


I see all the connections, I don’t know what they mean.

Huddled inside a pool of light, I chose wisely.

I paste together the fragments of Humanity.

Those perfect glances opening portals in time,

Shrieks of laughter cutting away remorse,

Endless hours where words flow like water,

Confirmation lifting us up gently from the constriction of doubt.

Don’t look too long at these scattered scraps,

They’ll draw you in with Devil’s promises,

But you’ve come all this way, care for a glimpse?

I’ve sculpted time into a string of pearls, I’ve sown the ends together

with the thread of continuity.

I know where this one goes, and that one goes, but still it is unfinished.

I feel pregnant with the burden of creativity.

I’ve carried this weight up and down the crowded streets, the lonely dirt roads.

The wisdom of life-times has taught me patience and humility.

But the seed was there all along.

Slipping from moment to moment, it grew out of longing.

Maturing as I matured. Seeking shelter where I sought redemption.

A lump of coal caught in the current of my spiraling soul,

Until pressure can release it, scalding hot, as a diamond.

I have dreamed, oh I have dreamed.

I am the diamond.

Rays of light bounce off my back in infinite angles.

Their voyage is eternal, each moment frozen in time.

So stay awhile, oh blinding light, chat with me for I am willing to listen.

You know very well I am as old as you. 

Our wisdom dances with a harmony reminiscent of the very notes that sang this plane into existence.

I see all the connections and don’t know what they mean.

I sleep father. I have slept.

I am the wasp.

Conceived behind such a wispy impediment

Though I see pot marks of freckled light from the pressure,

I can glimpse from the top like through the eye of a storm

A horde of geese flying with such violent freedom as to inspire me

To abandon my greed hover,

To let my soul draw breath, to stretch its limbs within this cramped vial,

To fall silently until the restraints of my stature, growing tight with a fractioning resistence,

sling me back into a busy, scared existence.

I have dreamed. I don’t remember what I have dreamed.

Hush my child, sleep here awhile.

It won’t be long until you too depart

To search for silk roads untraveled.

Fear Nothing

Accept Everything

Week 2_Maths, Art, and Biological Data Visualization_Wenjing Wu

Monday, January 19th, 2009

Most people tend to take it for granted that the form of art is wild and free with pure imagine. However on Tuesday Prof. Vesna talked about the artful beauty expressed by mathematics, a discipline which is usually linked to a bunch of formulas and rules. The part of perspective grasped my attention immediately, since that was also the topic of my drawing lessons this week. Our assignment was to draw a picture with four different perspectives. From Duccio and Giotto’s first attempt of introducing perspective into art, to Duerer and Vemeer’s  masterpieces that followed strictly to the mathematical rule, to the modern art styles like Cubism, which totally heads the opposite way of perspective, and then the way of teaching perspective in today’s drawing classes, artists are employing maths as useful tools and sometimes frames to break in order to creat “hacking” effects. I found a website discussing similar issues on maths and arts in National University of Singapore(poke me).

Do you see the strange object on the floor? Close your left eye, put your face close to the computer screen near the right side of the picture. You will then see a skull!

Do you see the strange object on the floor? Close your left eye, put your face close to the computer screen near the right side of the picture. You will then see a skull!

Artists use geometrics to fool our eyes, too. The famous painting above would be a perfect illustration. Besides that, there are also some artists using  Anamorphosis to produce astonishing 3D illusions, like Julian Beever.

On Thursday, our TAs talked about good jobs done by several students and also introduced briefly what they were working on. What I found the most intriguing is the part of John and Gautam’s–Biological Data Visualization. The works are simply COOL! As a senior student major in biotechnology, I know how overwhelming the biological data could become if it’s not well-organized–for example, our genome is consists of 80,000~10,000 genes, which roughly equal to 3000,000,000 base pairs. Moreover, since there’are needs to study biological process on different scales, organization and visualization turn out to be extremely urgent for biologists.  Some of the works appear to be very interesting, such as a project from the University of Tokyo(figure 2) and the research of Institute for Systems Biology (ISB).

Figure 1. High-Precision Three-Demensional Bio-Structure Modeling

This project enables the observer to track the distributions of different ingredients of foods through Internal Structure Microscope. Here, scientists is utilizing artful presentations to reveal its secret. As for the artists who are interested in latest scientific or philosophical ideas, as Henderson concludes in “Geometry in Modern Art”, are motivated by a desire to complete their subjective experience by inventing new aesthetic and conceptual capabilities. I suppose this might be a better answer for Gauvain.

Week 2 Art and Math by Isaac Arjonilla

Monday, January 19th, 2009


In this week’s lecture we saw how two naturally contradicting topics, actually have much in common.  Both Art and Math have always been two ideals that seem to have structures that are very opposite, while art is seen as leisure and creative, math is structured and organized. What I didn’t know is that mathematics have been used for many years to create some of what today’s standards are considered masterpieces. Filippo Brunelleschi was the father of engineering during the Italian renaissance was able to use linear perspective in art. Along with linear perspective, the vanishing point also came from his work, which was later used in many of the paintings and structures that are seen today around the world. During high school in my precalculus class, our teacher had her room decorated with fractals which are which are a geometric shape that are infinitely complex which are created using mathematics such as calculus.


These shapes began to be known in the seventeenth century when mathematician Gottfried Leibniz though of the idea of fractals. All of this was back in the 17th century proving the long span that math and art have had and how it has evolved over the centuries. More recent artists have alluded to mathematics in their art. Pablo Picasso is known for his abstract paintings which are mostly made up by shapes broken up.  Again, In my math class we used math to make up a picture, the main objective was to use algebra and calculus equations, and plot them on a grid, and when the equations were put in, the lines would display a picture, I was inspired by Pac-Man, I used sine graphs to form the mouth of the ghosts, and the formulas to graph circles to make Pac-Man’s body. As I began to do some research on paintings and the techniques that were described in lecture,  I saw how Brunelleschi’s techniques would continue to be used and further developed by other artists. Leonardo da Vinci’s famous painting: “The Last Supper” has clear signs of the vanishing point technique, and symmetry as it shows balance throughout. Other artists who were shown to be connected with math is Escher was also proven to have reached mathematical perfection by his lifelong friend Coxeter, as shown to us by Professor Vesna during lecture.



I enjoyed seeing how mathematics and art have been in existence for more many centuries. Seeing how it has changed over the years and how it has now helped conduct research in the fields of medicine is very inspiring. There was a video showing how the brain operates and it uses graphic animation, the reason that I say its inspiring is because I always thought that art wouldn’t really be able to help the world in the way that medicine does, but looking as to how technology and art have been recently growing together, it is easy to see that art will always play a big impact on the world. 


-Isaac Arjonilla

Week2_Mathematics,Perspective,Time&Space by Dennis Yeh

Monday, January 19th, 2009

Throughout history, humans have attempted to explain the universe with mathematics.
Einstein managed to relate two facets of life: Energy and Mass with the equation E=mc^2. This demonstrates a law that is obeyed by everything in the universe, with just a simple mathematical equation. In fact, his equation was not proved until last year, by scientists that “envisioned space and time as part of a four-dimensional crystal lattice, with discrete points spaced along columns and rows.” I am unable to find a sketch or model of their work, but I’m sure that if it were to be drawn or modeled with a computer, it would be fascinating to see how energy and mass are equivalent.

What is the universe? When Professor Vesma discussed perspective and dimension, it reminded me of a video that demonstrates what all of the theoretical dimensions are, and where our universe lies in that scale.
-Theoretical Dimensions:
Professor Vesma talked about how all early paintings were simply 2-D sketches until artists experimented with the mathematics of perspective and lines that illustrate a third dimension when your medium (canvas, computer screen, paper, etc.) has only two dimensions. These techniques, such as the idea of a vanishing point, or transforming from higher dimensions into lower ones were covered extensively by my linear algebra class last quarter. Using vectors and matrices that represent linear spaces, it is possible to use math to calculate and model linear transformations to the linear spaces, and convert 3-D images into 2-D images easily and mathematically accurately using a computer.

Another interesting example of how math is found in art is with photography. A photograph is simply a 2-D image of how a certain 3-Dimensional setting appeared at a specific point in time. Math is involved with many aspects of photography:
-Single Lens Reflex cameras use math to calculate how the light will travel through the lens and inside the camera (as well as the positioning and angle of the mirrors inside the camera)
-Aperture size and shutter speed is used to calculate how much light will enter the camera as well as how long the light is exposed to the film.
-Photographers usually try to obey the rule of thirds: that is, an image is divided equally into 9 squares, and important compositional elements should be placed along the lines.

A demonstration of the Rule of Thirds.

When I thought about how artists these days almost always use techniques to give their paintings three dimensions, I realized that these days our technology is able to replicate 4-Dimensional images on a 2 or 3 Dimensional screen. Movies and video recording are now a cheap technology that everyone can afford and use.  Like photography, video recording is also related heavily with math.  Cinematographers deal with several variables when they compose their shots, which they have to get perfectly right every time.

Some of the variables cinematographers deal with:
frames/second: how many images are shot every second.
film gauge: thickness of the film emulsion.  Thicker emulsion = better quality = more expensive.
film speed: how sensitive film is to light.  Lower speed = less sensitive to light; higher speed = more sensitive to light.

filters: Manipulate the light entering the camera to create emotions or moods (ie. colored filters, polarizing filters, diffusion filters, etc.)
focal length: Changes the angle of the shot and the field of view.
aperture: Used to control the exposure of the image.  Affects image quality and depth of field.
depth of field: How much the background, midground, and foreground are in focus.
aspect ratio: the ratio of width to length of the shot.  Standard ratio = 4:3 (4 units wide, 3 units high).  Widescreen ratio = 1.85:1 (1.85 units wide, 1 unit high).  In the 1990s, with the advent of high-definition video, television engineers created the 1.78:1 (16:9) ratio as a mathematical compromise between the theatrical standard of 1.85:1 and television’s 1.33:1, as it was not physically possible to safely create a television tube with a width of 1.85:1. Until that point, nothing had ever been originated in 1.78:1. Today, this is a standard for high-definition video and for widescreen television.
movement: The video camera has a fourth dimension: time.  Therefore, cinematographers have to consider if and how the camera will move while it is recording.
mood: The “art” part of lighting.  Creating an effective mood in a video composition is greatly affected by how the set is lit.  The quality of lighting affects how effectively the mood is conveyed
video camera technology: Gives the cinematographer more options when considering how to light the set.  The quality of technology affects how effective the lighting is.