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!

In response to Vorleak's post

Awesome post Vorleak. I'm really glad you posted about this because I got inspired and asked musician friends of mine on their opinion. My boyfriend who composes classical music told me about the "Shepard Tone," named after Robert Shepard.  In order to explain it technically, we would all need to understand composition terminology, but essentially, it is an auditory illusion that gives the impression as though a sound or upward/downward tone will reach its end. When you listen to a shepard tone, you either hear a falling or a rising in its tone and expect it to end but it doesn't. IT is explained as the following:

"The Shepard Tone fundamentally is based on sine waves. You start with a Sine wave say on note A4 which sits at 440 Hz and you have it glissando down to A3 at 220 Hz over a period of time. During the same time you have another glissando starting on A5 at 880 Hz and dropping down to 440 Hz.
If you were to repeat this cycle the glissando starting on 440 Hz would pick up where the glissando starting on 880 Hz left off. This creates the continued sensation of a falling pitch. However, if you repeat the cycle then you will quickly jump back to 880 Hz and will noticably hear it. So what do we do? In order to achieve a smooth and seemingly endless cycle we need to fade in the upper most glissando and fade out the lower most."
http://audio.tutsplus.com/tutorials/sound-design/sound-design-falling-forever-the-shepard-tone/
Here is a video of it if you want to listen. I think it's definitely a good representation of infinity through sound. Hear the illusion?