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!

Twisted Thinking

So after a brutal few weeks of Zeno and paradoxes, we took it down a notch, or so we thought, and examined M.C. Escher's artwork and his different depictions of the infinite. It is note worthy that mathematicians found his art appealing to mathematics and he then picked mathematicians brains to think of ways on how trick the eyes of his viewers and challenge their perceptions. The first piece we looked at was the "Concentric Rinds."

We discussed how it shows a sphere within a sphere with points, and related it to the idea that it is infinite inside the finite. This could relate to Zeno's paradox and argument that you can't do an infinite amount of tasks in a finite amount of time. The next piece we looked at was the tessellation of horseman:

Some of us agreed that it very much resembled the Stadium paradox, perhaps because it is about bodies moving in opposite directions with equal speed. We also related it to having a one to one correspondence in that you are able to match a black horseman with a white horseman.

Then Professor Hamman demonstrated the Mobius Strip using a piece of paper: Bring the ends of a strip together to make a loop and put a half twist in the loop, so that the top surface of

the strip meets the bottom surface of the strip. Tape the ends together.

He showed us how it now only has one side by starting at any point on the strip and drawing a line, following the length of the strip. Without picking up the point of the pencil, you'll go around the loop twice and end up back where you started. What happens if we divide it down the middle all along its length? Naturally, some of us might have assumed that you'd get two Mobius strips, but that's not what happened. We got one long, twisted strip.When we cut it in half, it becomes a more complex twist, and if we had continued cutting it, there would be lots of complex twists and rings and loops.

We asked, how do they stay together and never separate? What does it have to do with continuity? It is deceptively simple in its app

earance and exceptionally twisted in its properties.

We then compared the Fish and the Lizard tessellations which expresses the idea of infinity by starting with the largest tessellations on both the inside (fish) and the outside (lizard).

Lastly, we looked at his "Waterfall" depiction:

Upon initial glance, everything seems to be in place, but when you look closely, the water fall flows uphill. We all know that water naturally flows downward, though. What is wrong with it? Well, nothing is wrong necessarily, it is just that our minds naturally equate the lines he drew with what our senses perceive of the real world. There's a visual paradox.