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

week2, fourth dimension, Wei-Han OuYang

Monday, January 19th, 2009

One time my friend and me were standing outside of a party and just chatting about what is going on with our lives, we were amazed by how many people showed up and happy at the same time that everyone is having a good time. Suddenly my friend turned to me and said, “ this (we were watching the people inside moving in and out) is forth dimension…” I stopped talking, amazed by the concept that he just threw at me, and realized how amazing our world is.
“More importantly, once the artistic impact of the new geometric is understood, the art and critical literature of the early modern era regain a unity and a level of meaning that has long been lost.” Everyday we live in world that anything could happen. What is the forth dimension in our daily life? One could say that a perspective that is watching the world move as it sit still; like a person who is sitting down, just watching the time pass by, people walk by. But more importantly and realistically, newspaper records everything that happens near us everyday. In another words, newspaper could be an aspect of the forth dimension. And as the geometric being understood, the art, our living styles, and the critical literature, the newspaper will come together and work towards a better world. More specifically, the main purpose of existence of newspaper is to tell people what goes on everyday in their lives. Good and peaceful news such as a conflict between countries at one place is over will make people happy. Bad and horrific news such as global warming causing animals to endanger will make people paranoid. If people come to be aware of the importance of this geometric relation between our lives and the newspaper, they will improve their moral and work together to make the world a better place to live by doing good deeds such as: recycling, use more public transportation, treat people nicely and equally. And as people understand this forth dimension relationship between our world and the newspaper, they will advance their thoughts and further more realize that by reading these articles on the newspaper, the readers in another sense are in fifth dimension. And of course, as always, the decision is always back to us, the readers. Are we really aware of what is going on in the world, do we get the message that our world is trying to send to us?

blobattack_fullJust as the article “The Fourth Dimension and Non-Euclidean Geometry in Modern Art: Conclusion” mentioned that mirrors and shadows could be the fourth dimension, when we look into the mirror, do we see ourselves as good beings or do we just not care. The forth dimension always give us the awareness to something, it is always in a form of reflection to raise awareness. And the fifth dimension is always the decision part. If humans were watched and painted by some kind of alien form, will our shadows be painted differently than how it should be? More specifically, if shadow represents fourth dimension and is representing what we think and how we act instead of how we look externally, will our shadows be painted as monsters or human-beings?


Week2_Mathematics, Art, and Perspective by Catherine Yang

Monday, January 19th, 2009

In this week’s lecture, I found the most interesting topic that Professor Vesna touched was the subject on the Fibonacci sequence and The Golden Ratio. Fibonacci sequence goes along the path of 1, 1, 2, 3, 5… etc. Fibonacci sequence was created by Leonard of Pisa. This sequence is used frequently in mathematics and Leonard of Pisa also discovered Golden Ratio, Golden Spiral, and Golden Rectangle. The Golden Ratio is a pattern that can be found in nature and even the human body. Nature is itself creates works of art through plants and flowers. For example: pine cones grow around the stalk in a spiral until it reaches the top. This amazing phenomenon can illustrate to us how art and math collaborates.

I remember learning this sequence and high school and not until this lecture did I realize that it could be incorporated into art. In school, I would only use the sequence in order to solve an equation to get an answer. However, I discovered it is just more than mathematics because it can be applied in nature, architectural buildings, art, and the physical shape of the human body. The human body consists of many Golden ratios for example the ratio between the length and width of face, ratio of the length of mouth to the width of nose, ratio of the distance between fingertip and the elbow to the distance between the wrist and the elbow, and etc.

edenproject-745473 360px-eden_project_geodesic_domes_panorama1

In modern day, The Eden Project consists of an education center called The Core. The Core is designed by using the Fibonacci sequence and plant spirals. It is a place where plants from all over the world can be seen. The designer Peter Randall-Paige designed it after the spirals in the sunflowers and pine cones seeds. This combination shows that art also cannot exist without math. By using math within art, one can get precise drawings or sturdy buildings. However, I believe that math can exist without art. In mathematics, one does not have to draw precise shapes and use it to get an answer. Many times in math one must roughly sketch a shape that is an estimate of what it would look like. As a result, sometimes when one tries to get an answer from the shape that is shown, it is misleading. For example, on math test taken in school or the SAT, it may say figure may not be drawn to scale. In addition, most of the times in past history, we could see that many people tried to apply mathematics into perspective art. For example: Duccio, an artist, tried to find a mathematical way to determine depth. Albrecht Durer used single vanishing point and tried to connect it to mathematics. Lastly, Brunelleschi was credited for mathematic formulation of linear perspective. These artists tried to incorporate mathematics into arts, which further shows that art is more dependent on mathematics. In conclusion, art is not only affected by science, but also affected by mathematics throughout the history of time.

Catherine Yang

week2_Fractals by kirk naylor

Monday, January 19th, 2009

This week, the topic I found most of interest was that of fractals. Although I am a music major, I’ve always been deeply interested in math, and fractals are one aspect of math that absolutely enrapture me. Fractals are mathematically created infinite patterns, and many of them are actually really aesthetically pleasing as well. one of the most famous fractals is the Mandelbrot set, which is an excellent example of self-iteration. The basic pattern:can be found at all levels on the fractal.

kirk naylor

Week 2- Art Technology and God–Allie Gates

Sunday, January 18th, 2009

During our discussion last week, one of the other students touched briefly upon another dimension of our discussion in art and science–the aspect of spirituality.  Because religion and beliefs are so undeniably personal–yet far reaching in their implications–I’ve been thinking about the human relationship to that which we create.  And more importantly, how we manifest our ideas of “God” in that which we create ourselves.  

Walking around campus, its hard not to be taken aback by the beauty of Royce Hall.  The person who designed Royce modeled it almost exactly after a famous cathedral in Madrid.  The original architect was a priest who believed that nothing that man could create would ever be perfect; nothing could ever approach that which is “godly” in its perfection.  So he designed the building to be inarguably beautiful, but also with discrete imperfections– some noticable, like asymmetrical windows on the two towers, some not, like the fact that one tower is slightly taller than the other.  This was his combination of art, science, and god: the beauty and complexity of the building, his art; the construction of the building, his science; the symphony of perfection and imperfection, his relationship to god; the combination of the three, his mode of worship within his role of the humble human.

However, modern art and science seems to have taken a different path.  Now it seems that art and technology merry together with the goal of approaching–or rather careening toward, screaming toward startling perfection, rather than avoiding it. The astonishingly sleek lines, shiny surfaces, and flawless looking construction lend a modern quality to products and buildings of the 21st century and seem to be a nod to the human ability to innovate technology and take more and more control of their circumstances, things that were once things unchangeable to anyone but god. It seems to me that as we manifest our own perfection via science and art our ideas of God have seen a paradigmic shift.  Whereas the vast majority of European people across many religions in the early part of the last millenium widely believed that God was a more active, vengeful God.  Although there are exceptions, it seems that more people now think of God as a loving and passive entity, rather than one who often harshly and decidedly participates directly in human interactions. Because people have more control in their undertakings, it seems logical that the thought would follow that God either has less control, or more interestingly, is the type of dude who would choose not to participate.  

On a slightly different, but not totally unrelated note, I’ve been thinking about the role of technology in the communication of beliefs and ideas, especially those contained within art.  The most glaring example, the internet, provides an avenue for instant retrieval and pretty much any topic (blah blah blah, you know that.) Although this has spanned the reinforcement of a million unfair stereotypes and the propagation of some misinformation, on a very fundamental level it has also let people become familiar with ideas that are not accessible from the people they surround themselves with immediately.  This is a related musing because I have been thinking about the amount of inter-religion hate and persecution that has occurred during most of history, and its beautiful that people can and do become exposed to a lot of material on the internet that alleviates the strangeness of the ‘other’ and promote a truer sense of tolerance through simple familiarity.

In short, as our relationships with one another and our Gods continues to change, so will technology art and science be mutually changed.  

Allie Gates

Week 2: Math, The Bridge Between Art and Science by Ryan Andre Magsino

Sunday, January 18th, 2009

Week 2: Math, The Bridge Between Art and Science by Ryan Andre Magsino

Math, seperate from the sciences?

Math, seperate from the sciences?

Mathematics (or simply Math) is “the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically.” Although the definition sums it up quite eloquently, it does not specifically convey its relation to the sciences or arts. As we have explored throughout the week, there are several mathematical applications to both fields. As early as the 15th century, math has been utilized in order to explain perception and space. Artists have built on this application to determine vanishing points and formulate other artistic techniques. Science, on the other hand, is all about quantity, structure and changes. From the amount of blood pumping through the human body to the assembly of elements in a molecule, mathematics has played a somewhat integral part in determining those values. What then is math, a science or an art? American mathematician Benjamin Pierce refers it to “the science that draws necessary conclusions.” Yet another mathematician, Godfrey Harold Hardy, is “interested in mathematics only as a creative art.” Though it may seem one-sided at times, mathematical applications such fractals tie in both fields making it somewhat of a bridge between the fields.

Before looking into its applications, it is essential to determine what a fractal actually is and how it works. According to famed French mathematician Benoît B. Mandelbrot who coined the term, a fractal is “a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole,” however they can simply be defined as an image that could be infinitely found within itself. Fractals are often characterizes as having fine structures at arbitrarily small scales as well as being self-similar. A prime example of simple fractal would be the Koch Snowflake created by the Swedish mathematician Helge von Koch. It is created by first beginning with a single equilateral triangle then dividing all the sides into thirds thereafter. Last is replacing the center with a proportionate equilateral triangle. This process then continues onward for an infinite number of times.

Iterations of a Koch Snowflake

Iterations of a Koch Snowflake

Though we may be unaware, fractals have played a much larger and longer role in society than we have expected. Although it was only recently coined in the past century, American cyberneticist and ethno-mathematician Ron Eglash explores the implications fractals have left in African culture and society. ( Video - Ron Eglash: African fractals, in buildings and braids) Surprisingly, some African societies are structured in fractal iterations with multiple recursions. Looking now at modern technologies, fractals have also played a revolutionary role. It is due to fractal imaging we have such technologies as computer and video game imaging, especially when it comes to 3-dimensional modeling, and even fractal compression for image formats.

The first three iterations of the Square-flake

The first three iterations of the Square-flake

As for myself, I like to consider myself a mathematician (though nowhere as close as even amateur). Surprisingly, I along a few classmates in the past have developed a fractal we branded the “Square-flake.” Similar to the Koch Snowflake, the Square-flake is a fractal created by first beginning with a single square. In future iterations, the sides are cut into thirds with a square a third of the length on all sides is added to the given side. This process then continues onward for an infinite number of times. Taking it to a step beyond, we decided to integrate our fractal sequence and design. Our result was a fractal lamp. Taking the fractal to its third iteration, we were able to apply the concept into a work of art per say. (PDF: Square-flake Informational Pamphlet)

A photo of the fractal lamp unlit.

A photo of my fractal lamp unlit.

A photo of the fractal lamp lit up.

A photo of my fractal lamp lit up.

Related Links

week2_MPT&S; The evolution of the fourth dimension by Devin Quinlan

Sunday, January 18th, 2009

When I was a child, art was nothing more than finger paints, watercolors and crayons. Instead of incorporating ideas behind my work, I was focusing on keeping all of my colors inside of the lines so that I’d have something worthy of the refrigerator door. Nevertheless, art was still an outlet for my creativity, and allowed me to document the world as I saw it (and as my motor skills allowed). I find it interesting, however, that as a child, I always drew in two dimensions, and even into high school, the stick figure prevailed as my most commonly used representation of the human form. In fact, if it weren’t for a drawing class that I took and the instruction of my mother (an art teacher, go figure), I would still have very limited exposure to the idea of drawing and expressing anything in three dimensions. Essentially, I would not have even realized or discovered how to express anything in three dimensions if it hadn’t been taught to me.

While many of us take it for granted, expression in the third dimension is a relatively new phenomenon. For thousands of years, the only three-dimensional representations were sculptures, which were actually three dimensional themselves. It was not until the Renaissance that a two-dimensional medium was used to represent a higher dimension – something truly revolutionary in the evolution of art.

This had a profound impact on other areas as well, and in fact an interesting parallel to this development is the idea of three-dimensional mathematics. The development of math, much like art, started off in simple dimensions. When we are children, up until about the 6th grade all we learn is arithmetic, which is essentially one dimensional. As we progress into middle school and high school, we learn how things are represented on two dimensional planes. The idea of algebra, much like two-dimensional art, had been around for a very long time. The shift into three dimensions during the Renaissance, however, laid the foundation for the development of three-dimensional mathematics. Much like art’s movement from 2-D to 3-D, this parallel in math allowed the real world to be represented and understood much more accurately. Just as a man didn’t look completely human in two dimensions, the path of a ball thrown from one person to another in the real world can’t be completely explained through a 2-D graph. In order for us to have more perfectly understood and documented our world, three-dimensional expression was essential.

This brings us to an interesting question: If we have already discovered a means of expressing our 3-D world in two dimensions, and if both art and science are based on the idea of constantly pushing the boundaries of knowledge, then where is there for them to go? The answer, as you might have guessed, is the fourth dimension. But what exactly is the fourth dimension? The answer this time is not as clear. Some believe the fourth dimension to be represented by time, others by strange figures such as the tesseract (image shown below), while others believe that it is impossible to comprehend. Each interpretation is different, though all make sense and it is up to the creator to determine which one is “better”. As a result, art has become much more conceptualized, and is now focused not only on the final product, but on the process of creating the art itself. In a sense, this shift of focus represents the shift into the fourth dimension, as a process is a change of the work over a period of time. Other representations of the fourth dimension are interactive exhibits, where the art is constantly changing as a response to people’s movement, and in works such as the falling leaf by John Carpenter, where a space occupied by an object over time is represented rather than the object itself.

Thus, although it would seem that we will never be able to truly demonstrate the fourth dimension in art, we are still able to express its idea by pushing the boundaries of what can be defined as “art”.

A tesseract is a 4-D representation of a cube

A tesseract is a 4-D representation of a cube

- Devin Quinlan

Wk 2_Mathematics and Art by Alana Chin

Sunday, January 18th, 2009

This week in lecture, we focused on the relationship between mathematics and art. At first, I thought that numbers and arithmetic couldn’t be furthest from lines and shapes. However, upon further inspection, I saw that math and art frequently come together, especially in nature. The concept of the “Golden Ratio” completely threw me for a loop. I was conscious of the value of pi and its unique relationship between different properties of a circle but I had never heard of the similar value of phi. This golden ratio is a number around 1.618 that appears time and time again in nature. This ratio can be seen in many places, from the number of arrangements of branches along the stems of plants and veins of leaves to the branching of animal nerves and proportions of chemical compounds. It is also observed in famous historical across history. The Egyptian pyramids and the Parthenon in Greece were built using the golden ratio. I was very surprised to hear how often the golden ratio is observed but was especially surprised when the professor showed us a clip about the golden ratio in human faces. It claimed that people are most attracted to those whose facial features retain the golden ratio. Specific places for these ratios include the distance between your eyes or the distance in between your nose and your upper lip. When I heard this, I was incredibly surprised. I had heard before that people were attracted to symmetry, which made sense in my mind because symmetry might reflect healthy genes without defects. However, I have no idea why people would naturally be most attracted to this ratio. I could only label this as a phenomenon and go with it while trying not to think about it too hard. In the end I only generalized that for some reason, the golden ratio is very aesthetically pleasing. Because of this, many artists have thus utilized the golden ratio in their own artistic creations. Obvious examples would include Leonardo da Vinci’s “Mona Lisa” or any other representation of an attractive face. A less obvious example was the sound clip the professor showed us during lecture. One musician composed music based on the golden ratio where the frequency and length of note was determined by values from the golden ratio.

After hearing more about these instances of math in art, I wanted to find more examples of people utilizing math in art. I researched a little bit more and found the Mathematica Gallery on Mathematica is a computer program that incorporates algebra, trigonometry, calculus, and other math at the same time to describe and manipulate expressions. Because this system is so complex, it is said to be similar to natural human thought. And if these expressions were recorded or mapped on flat planes, you could create beautiful designs and images. The design could essentially be based on simple equations, but when put together, you can develop an intricate picture that is both beautiful and complex. I really appreciate the combination of art and math in Mathematica because it really shows that you the two can be related and can very well work together. It also shows that there is a connection between art and math, even if you hadn’t noticed it before.

-Alana Chin

Week 2- The Combination of Math and Art by Brandon Aust

Sunday, January 18th, 2009

This week we began the discussion with the number zero. Zero has always interested me since it is such a major component in problem solving and in the theories of mathematics. It is most likely the most used and written number. It is the only real number that is neither negative nor positive. Zero is also very interesting in the fact that it is not really a number at all. Zero is more of a description of an empty set. I found it interesting that Professor Vesna mentioned that zero was not used in the western world until the 1600’s. The Greeks debated the theology behind the zero as they were uncertain how something could classify nothing—one of the most interesting aspects of zero.  It wasn’t truly used as a number in the algebraic sense until the 9th century in India.

We continued the lecture discussing the combination of mathematics and art. I enjoyed the discussion on the golden ratio. I thought it was interesting how many students, including myself, discussed the golden ratio in the previous blog. It just goes to show how apparent math and science is in nature and art. I was not aware that the idea of the golden ratio was used on the Mona Lisa. The idea of the golden ratio and the differences between the facial features of opposite sexes was very interesting as well. It was also very intriguing to see that celebrities generally fit the golden ratio in terms of their facial features which explains why many people find them so attractive. We also discussed the ideas of perspective in art. Perspective helps to create a certain amount of depth in art to make a work of art appear more realistic. An artist uses various methods, and in a way must calculate, in order to discover the correct placement of an object to create the exact perspective that they desire.
Another large area of art which deals with math is Geometric art. Such art began in ancient Greece with various types of vase painting. The artists used lines which wrapped around the circumference of the vase. In between these lines, the vase was covered with different geometric shapes which helped the artist create a motif. In the 20th century, Cubism became very famous.  The movement was lead by Pablo Picasso, and the art centered around the basis of spheres, cones, and cylinders. The varying geometric forms helped create the images that were inspired by African and Native American cultures.
cubism A while back I read that mathematicians had discovered how to theoretically turn a sphere inside out. The theory requires a material that is able to pass through itself, but can one cannot crease or pinch it. The overall result is very hard to imagine, yet I found a video which demonstrates the problem: Sphere Turning Inside Out. The answer to the problem appears to be very difficult to imagine. It must have taken a very intellectual, yet creative mind to come up with the result. Just watching the answer itself is very beautiful and artistic. This idea has spread throughout the mathematical community and has caused various other geometrical problems. One such shape is the Klein bottle. It is a 3-dimensional shape in which you can move from the outside to the inside without moving across the edge of the shape: Klein Bottle. I feel that both of these strongly apply to the connection between art and mathematics. For; even though, these problems are greatly math based, the answer/creation that is formulated is beautiful in an artistic way.

-Brandon Aust

Week 2- On Math and Art by Kimberlie Shiao

Sunday, January 18th, 2009

As much as an integration of art and science intrigues me, art and math seems to discomfort me. Despite how things like the golden ratio, fractals, and Fibonacci numbers appear in nature, for me looking at spirals and other manifestations of math (such as the repetition of the efficient hexagons in beehives) make me feel physically ill. Part of the reason I suspect for this is the mind-boggling infinite nature of spirals, fractals and repetition. But that’s how math helps art evolve and explore new areas, i.e. space- an almost (relatively) infinite frontier I believe has and will continue to deeply influence the arts in various ways.

The meeting of art and math that interests me more than spirals and fractals is the idea of the fourth dimension. (And those dimensions beyond: Rob Bryanton nicely uses layman’s terms to help explain up to the tenth dimension in this youtube video, which includes numerous annotations directing the viewer to places for more research.) Bryanton explains the fourth dimension as time, unlike the spatial fourth dimension Henderson mentions in his text. While I’m not sure whether I believe the fourth dimension is spatial (does that make time the fifth dimension then?) I do like its manifestation in art, as discussed by Henderson. The artists he names are a good example of how not only art and math, but also philosophy, can contribute and develop ideas traditionally thought to only belong in one of those fields.

But the benefits of synergy are not merely confined to expanding hypotheses and trends of art, math, science and philosophy. Art also has the significant effect of being one of the most accessible methods to link more complex ideas of math and science to the general populace. Trying to better understand the controversy of the nature of the fourth dimension, I took a glance at Wikipedia, but failed to understand the proposed spatial fourth dimension (and there was something about time-reversal, though I understood even less about that). Videos like Bryanton’s (though debatable in factual content and not exemplary its art) use visuals to help the non-academics to understand the forays into new frontiers like upper dimensions, and then to contemplate on how it might effect themselves personally.

Last week I felt art and science were presented with a more one-sided relationship (with art taking more from science than science from art), but this week has led me to realize that art- manmade or natural- is often the application of math and science. (Is not math and science derived from observation of the first art that inspired humans: nature and the world itself?) No longer do I see this relationship as one way; we can look to art and nature for new ideas and new understandings about ourselves and the universe(s).

-Kimberlie Shiao

Week 2 Mathematics and Arts by Mindy Truong

Sunday, January 18th, 2009

During this week’s lecture, the topic that grabbed my attention the most was the golden ratio. It was fascinating to be introduced to it because it seemed to be related to a lot of paintings or buildings that are around. When the professor discussed the golden ratio in class, I was still a bit confused as to what it is. Of course, I knew the number was approximately 1.618 but I was confused as to where it was derived from. To make things easier for myself, I searched it up on Wikipedia; and I caught myself reading on all the links that related to the golden ratio. Apparently, works that relate to the golden ratio can date back to 5th century BC. The Parthenon, a temple of the Greek goddess Athena, which was built during this time showed portions that approximate the golden ratio.
Several other works that are more common are also related to the golden ratio. Works such as De Divina Proportione by Luca Pacioli published in 1509 researched the math behind the golden ratio. The following link contains works that relate to golden ratio:

The discussion about the golden ratio also brought something up to me. I watched a show on ABC called True Beauty. There was a part in the show where the cast was taken to a doctor who was a “real board certified plastic surgeon and beauty expert.” His job was to “take the face and body and compare it against ideal shapes, proportions, and angles.” The show didn’t exactly say that the doctor used the golden ratio in his daily job to critique people’s appearance and give them a rating out of 100. However, after seeing the youtube bit where the guy compared celebrities’ faces to the golden ratio, I figured that the doctor in the ABC show most likely used the golden ratio to critique the faces of the cast members.