Never Ending Topic (Final summary)

Basically, we all started this class wondering what infinity was all about and everyone thought of it as something beyond our way of life. We compared it to nature by using the movements of the wind, sea waves, grass, human population, and water molecules. We also talked about some form of attributes of infinity which were boundless, irreducible, abstract, incomprehensible and endless, just to mention a few.
After further discussion, Professor Hamman started introducing us to simple way of relating infinity which he taught us One-to-One Correspondence by using two infinity set. Hilbert Hotel also showed us how an infinity hotel can accept each and everyone in the bus who needed a place to stay and the hotel never gets full. Cantor’s array of rational numbers and power set gave us more insight on infinity and we were able to create different subset from the original infinity set which were real numbers and natural numbers.
Zeno’s paradox which was the master of all infinity made everyone thought beyond their scope and Russell Paradox made it easier for us to get the details of the paradox and resolve it on our own. M. C. Escher and other great artist expressed their art in a way that we could express infinity through art work and we were able to identify them ourselves.
Finally, our final project work, the video and our poster summarized all we did in class but unfortunately one of our mirrors got broken with all our effort that evening but we were all happy to end the class with something we can continue to learn more about it. To believe in infinity or not to believe in infinity is the question?
On a personal note, I wish everyone all the best in the exams. We had a good team and everyone contributed in their own special ways. Professor Hamman, thank you for your time. Happy Hanukkah, Merry Christmas to you all and see you next semester if you are not transferring yet!!

Last class extension

We learned to look at infinity from different angles: mathematics, philosophy, artistic. We also discuss how sound and music relate to it. While doing research for this extension, I was so excited to learn that for the Pythagoreans, numbers, music and harmony were considered among the first principles guiding the universe.

Relating to mathematics, the Greeks first understood harmonics that vibrating strings and columns of air produced overtones. And, Pythagoras, specifically, “described the arithmetic ratios of the harmonic intervals between notes, for examples: Octaves, two-to-one ... fifths, three-to-two ... and fourths, four-to- three” (Rockmore) Pythagoras also noted that if two commensurate strings were strummed to vibrate, then the tones that they produced would be pleasing in harmony. Thus, the Greeks believed that all the harmonious things in the world must be based on whole number ratios. And all measurements must be rational.

Diagonal of a square

The Pythagoreans’ belief now changed because they discovered that the diagonal of a square is never commensurable with its side. No matter how many units they divided up each side of the square into, there was always a small amount leftover when they tried to measure the diagonal with this basic unit. 

 

In another words, “in any measuring system that gave a whole number of units to the side, the diagonal would have a length that has to be expressed as an infinite decimal expansion” (Rockmore). So, square root of 2 is irrational and therefore cannot be expressed as a ratio of two integers.

Response to Catherine's Email Post

I think your art teacher was right. She just didn't or couldn't go any further with it. She left out the 'reflection' part of it. She should have gone one step further. For instance, the leaves on the trees are green. They are reflecting the green light that doesn't get absorbed so they look green to us.

I like this idea of different sounds cancelling each other out and can create silence. I never really thought about it that way. The saying "if a tree falls in the woods and no one is around to hear it.." makes me laugh. Of course it makes a sound! We humans don't decide what makes a sound just because of our presence! I don't have a definitive answer for you, but I can say that you have piqued my interest and I am going to look in to it further.

Oh and thanks for the music reference! Gotta love The Police!

Last thoughts on a never-ending topic

"When the doors of perception are cleansed, man will see things as they truly are, infinite" _ William Blake
As i wrote this quote down, i had a feeling that it was for us. So when we first came in this class, we had the same ideas of what infinity was: endless, boundless, uncountable, incomprehensible, etc (the attribute list). As we learned more about the views and the sizes of infinity, about how to use one-to-one correspondences and the power sets to discover it, about philosophical problems with infinity, our "doors of perception" seem more opened.

After we got our first papers back and were assigned to be responsible for the final paper, professor Hamman explained some of the reading Escher on Escher. Next, he questioned if anybody ever related infinity with music. Some said yes and some said no. Then an argument came up between the answer: Is silence a sound? Again, some said yes some said no. Somebody on the Yes side basically mentioned that we see color because our mind can detect the wave radiation from light. If there's no light, the color is black. So black is a color. Using the same argument, we may say that the lack of sound (silence) is still perceived as a sound. But people one the No side argued that silence is independent. It is not created by anybody/anything so it's not a sound.

Then, we discussed how different it is when there's sound under water. The really high pitch of the whale was an example. We also talked about how fun and weird it is when we can make bubbles if we try to talk/breathe under water. After that, somehow, we ended up with what Escher said: "Infinity cannot be portrayed by sound."

Professor introduced us to the works of Ivars Peterson, a mathematician and computer writer.

 Triune Twists, at Philadelphia's City Hall

Fragment of Infinity, Designed by Ivars Peterson using the Sculpture Generator

 

Old Recycle Symbol, designed by Ivars Peterson

Petersons uses the power of math and computer to do all the work. Amazing. So from now on, we may want to notice all things that depict infinity wherever we go (sculptures, mobius strips, mirror, etc). 

 

 

 

 

Infinity in music

We had been spending time in class, looking at infinity from Philosophical, math, and art point of view. However, we haven’t spent much time, discussing what infinity is like in music. Is there such a thing as infinity in music or even in sound over all? In order to know if infinity exists in sound or not, one has to understand what sound is. This raised a question in class last week as to what considered as sound. Is silence a sound? According to the web, sound is a vibration that travel through the air or another medium and can be heard when they reach a person's or animal's ear. Thus, can silence be heard? Of course it can!! Have you ever heard of the expression “read inside your head?” It’s completely silence, yet there’s sound that read what was written on the page. Have you ever dreamt of people speaking in your dream? That’s too, completely silence from the outside, yet sound exists in the inside.  

One of the most amazing things that humans discover is music for deaf people. Deaf people sense vibration in the part of the brain that other people use for hearing – which helps explain how deaf musicians can sense music, and how deaf people can enjoy concerts and other musical events. "These findings suggest that the experience deaf people have when ‘feeling’ music is similar to the experience other people have when hearing music. The perception of the musical vibrations by the deaf is likely every bit as real as the equivalent sounds, since they are ultimately processed in the same part of the brain," says Dr. Dean Shibata, assistant professor of radiology at the University of Washington. To us, it might just be sound and sign languages, but to them, it is music. 

Just because it is silence, doesn’t mean you can’t hear the sound. Sound is everywhere around you; it’s outside of you, inside of you; it’s infinite. And when there’s sound, there is music. Thus, music is infinite.

Response to Catherine's Post Questions

Thanks Catherine! Ok, so I'm going to take a stab at the questions from what I know so far. The things that make up a tessellation are the shape, order/flow and colors. This is why Escher feels the fish tessellation best depicts the infinite. As far as I know, the non-regular tessellations don't necessarily need to be a paradox (see sidewalk illustration below). I think I like them more when they do though! I'm a glutton for punishment I guess. As for what distinguishes a tessellation from any other pattern has to do with shape, order/flow, color and ,I think, most importantly are the angles. If the "math" is incorrect and the angles aren't accurate I the tessellation sort of falls apart. Ok so I did some research on the beach art and it's an optical illusion. HOWEVER, some of those optical illusions ARE tessellations and OH do I have a treat for your eyes:

Can you see the black dots between the squares?! Cool huh? This could be considered a tessellation (from what I've read) but it also an optical illusion. So the beach paintings that we have all seen could qualify as tessellations (probably the non-regular kind). I'm not certain on that though.

Oh, the FedEx arrow on the truck.....I learned about that about 8 years ago....so sneaky! Thanks for the questions!

Escher The Magnificent!

M.C. Escher was one of the most profound graphic artists. His fascination with: division of the plane, near vs. far and high vs. low is mindboggling and beautifully reflected in his artwork. I could ramble on and give a complete history on him and compare him to other graphic artist in this genre, but the artists fall short of what he created and I feel the artwork speaks for itself.
However, I would like to give you a little information on the types of tessellations and what they are composed of.

· Regular Tessellations – triangles, squares or hexagons
· Semi-Regular Tessellations – regular polygons
· Non-Regular Tessellations – have no particular shape (infinite # of this kind)

Escher (Regular Tessellation)

Computer Aided Triangulation (Regular Tessellation)

Side Walk Pattern (Non-regular Tessellation)

Natural Beehive/without bees (Semi-Regular Tessellation)

Escher, Drawing Hands 1948 (Non-Regular Tessellation)

It’s interesting to see how infinity plays a role in each of these renderings. Some were created by man (whether by hand or by technology) and others occur in nature. Where does the tessellation/paradox begin or end? Does it move inward or outward? The last photo in particular is my favorite. Which hand started first? A visual paradox!