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!

A Further Look at Series

When Professor Hamman first brought up the topic of series, and pointed out how not all series need add up to one, he reminded me of something I had heard a few years ago. When I was in Calculus II, and Prof. Wheatley was explaining series, he said that he could take a series and have it add up to whatever he wanted it to. However, he could only do this with certain series. Prof. Wheatley's argument was thus: Because we are adding up all of the terms of the series, we should be able to add them up in any order we want to and have the correct value.

So let's look at the series {1/2, 1/3, 1/4, 1/5, ...} again. As Prof. Hamman showed, this series should add up to infinity. So let's change the order around some; let's move the 1/4 in front of the one third. We now have {1/2, 1/4, 1/3, 1/5, 1/6, 1/7, ...}. As you can see, the sum should still be the same. However, let's move the 1/8 in front of the 1/3 also. And let's do the same thing with all of the powers of two we can find. The series now looks like {1/2, 1/4, 1/8, 1/16, 1/32, 1/64, ...}. Intuition would say that this should add up to the same number, but Prof. Hamman showed that this series adds up to one

So what went wrong here? If we count the series in that order, we will only ever be counting powers of two. In doing so, we will be skipping a large part of the series, which will never get counted. This is like Hilbert's Hotel, where we had a bus with an infinite number of people onboard.  If we gave the people on the bus rooms in normal order, everyone would get their own room. However, if we were to try giving all of the odd-numbered people rooms first, not everyone would get a room, as we would have no end to the number of odd-numbered people on the bus.

So what does this mean? Just like before, if we take a set with a finite number of terms, no matter what order we count in, we will count them all with no exception.  However, if we take an infinite set, we can count in some order so that we can skip terms and never count them, even if we do not stop counting.  This also relates to the idea of one-to-one correspondence.  Just as we can take an infinite set and form a one-to-one correspondence with a subset of itself, we can do the same with the terms of a series.  The important thing to note here, though, is that doing so with a series changes the series, changing what it will add up to.  So, in the end, we cannot have a series add up to whatever we want it to without changing the series itself. Sorry if I got your hopes up.

Infinity...More Personalities Than Cybil

"The known is finite, the unknown infinite; intellectually we stand on an island in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land."--Thomas Huxley

After clarifying some fuzzy spots on our old friend Zeno, and discussed our mid term paper; we delved into the topic of series that shed new light on Zeno's paradoxes. It made me see them in a different way, a better way.

Tear a piece of paper in half. Then tear one half into quarters, then that quarter into eighths, etc. When will you have the entire piece of paper elsewhere? When will the piece of paper you are tearing disappear from your hands? The answer of course; lies within the study of the infinite.

Professor Hamman introduced us to the concept of series: basically it is a sum of a bunch of numbers. If we take the series {1/2+1/4+1/8+1/16+...} and never stop adding terms this series will come to equal one. If you never stop, you will never reach the number one. Another series goes like this: {1/2+1/3+1/4+1/5+...} If you continue to add these terms you do not reach one. If you never stop adding; the sum is -----infinity. You may ask,"How can one series add up to a finite number and another series add up to infinity?" I could say "because Prof. Hamman says its so" or "it just does." But I implore you to think of reasons why this is. Intuition leads us there but it is hard to put into words. Here lies the hidden beauty within the math.

We also discussed the concept of a repetend equaling a finite number, such as .9 rep. =1

If we assign "x" to .9rep. then 10x=9.9rep. If we set these two equations up as a subtraction problem such as 10x-x, and 9.99999999-.9999999; we get 9x=9. Therefore x=1. But we assigned x=.9rep. It is both! So it follows that x=1 AND .9rep.. Another way of saying this is an infinite decimal can equal a finite number. But not ALL infinite decimals equal a finite number. An example of this would be if we set x=.01001000100001... no equation would be possible to use and get a finite number.

I hope this helps everyone with their paper!

Don’t Go Near the Water Until You Learn How to Swim

Before we get into the “oooh” and the “aaahh,” or maybe even “huh? I don’t get it,” we took a little of time to talk about what is it mean by “countable?” A couple of our classmates argued that if infinity is boundless and has no limit, there is no way it’s countable. However, at the end we came to an agreement that countable is a special kind of infinity and that in order to be countable, it has to have one-to-one correspondence, and it can be put in order. For instance, natural numbers (N) are considered to be countable because we can say that 1 comes after 0, 2 comes after 1, 3 comes after 2, and so on, whereas P (N) or even the real numbers (r) are uncountable because they do not have one-to-one correspondence, nor can they be put in order. We don’t exactly know what comes after 1.278563; is it 1.278564 or is it 1.2785631 or is it 1.27856305?

After understanding that the idea of infinity has the first, second, third, and so on is the essence of it being countable, we went on to talk about paradox. What is paradox? It’s not just a contradiction, but it’s a statement that could be false and true at the same time, which does not suit very well with Aristotle’s “law of excluded middle.” There are different types of paradoxes, one of which is the self-referential type that can be shown in a famous Russell’s paradox, stated “R be the set of all sets that are not members of themselves.”  There is no right or wrong answer here, thus in order to “resolve” this problem, Russell decided that a huge set is not a set but a class.

Regarding the paradoxes, we went ahead and looked at 4 of the Zeno’s paradoxes that Aristotle tried to compile things to give an answer to each of them.  Each paradox does not work individually; hence they work as a group. The four paradoxes are:

1.      Dichotomy: in order to reach a certain location, you first have to get to half of that distant, and to get to that half of that distant, you have to get to half of the half of that distant, and so on. This is the infinite half-way points, which has no end, yet we have to pass all these infinite steps to get to that certain location.

2.      Achilles: By letting his tortoise starts out first, it doesn’t matter how fast Achilles runs, he will never catch his tortoise. This is because before he can overtake the tortoise, he first has to reach p1, but when he reaches p1, the tortoise is still ahead, reaching p2. Again, when Achilles reaches p2, the tortoise is still ahead, reaching p3, and so on.

When looking at these two situations, one might think that this is easy; if time and space has the smallest unit and is not infinitely divisible, of course, we will be able to move one unit of space at one unit of time to reach that certain location and catch that tortoise. However, let move on to the third and the fourth paradoxes:

3.      Arrows: clearly, we can definitely see the movement if the arrow moves from point A to point B. Assume that time is indivisible like we mentioned above, we will use the word “instant” as the smallest unit of time. If the word instant cannot be divisible, there wouldn’t be a start or an end, and thus the arrow never moves at an instant. The entire of time that the arrow moves from point A to point B will just consist of a bunch of “instant” moments. This means that the arrow has never moved, yet we see movement as the arrow goes from point A to point B.

 

4.Stadium:  The picture below assumes A is stationary, B is moving on unit to the right of A, and C is moving one unit to the left of A at the same time. If it takes one smallest unit of time to move the purple box of B to match with the red box of A, the time that it will take for the black box of C to match with the blue box of A is also one smallest unit of time. However if time can be atomic like one assumes in the first two paradoxes, then how long will it take to have a black box of C match with the purple box of B?