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.
Nice post Yumiko! BTW, I know now how hard it is to post pictures on the blog, so I appreciate your work! So as soon as I came home last Monday I made a Mobius strip to show my husband and the next morning I also made one for my oldest daughter. They loved it!! So thanks for making me look "cool" Professor Hamman, even if it was short-lived. As I was doing research for my post, I recognized some of Escher's works. Before last week I really didn't know WHO he was. I have become fascinated by this type of art and how it is related to math, infinity and paradoxes. It's so methodical, confusing and sort of calming.
ReplyDeleteThanks for the post and good pictures. I'm just gonna say the "Concentric Rinds" looks bounded to me, even though it depicts that it's going smaller non-stop inside. It's a cool visual example of the connection between infinity and finite:
ReplyDelete1/2 + 1/4 + 1/8 +...= 1
It's actually infinite = finite.
Great job on your summary, I hope I can add images to my post when I’m doing my summary. I went to check on his art work online at the official website and there were amazing gallery of art that was unimaginable. I was looking through the " Back in Holland, 1941-1954" art work where I saw one which look similar to the mobius Strips but it was spiral. You can actually see the beginning of it but you can't tell how it ended. It looks like its finite but on the other hand it appears to be infinite. The other fascinating one was the tetrahedral Planetoid which was also printed from two blocks. They are mansions moving around the earth or sitting on it. I feel like it’s infinite because you can go through the building without stopping. The behavior looks like the clock. If you get the chance please tell me what you all think.
ReplyDeletehttp://www.mcescher.com/
November 18, 2011
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ReplyDeleteYaa, I looked the other pictures, and they were all interesting, but I like the relativity one the most. It's like the tetrahedral planetoid, but gives more emphasis on the different sides because it's zommed in. I've actually seen it somewhere else before, but can't remember where.
ReplyDeleteYumiko, nice summary. I was trying to argue earlier why I think the fish is a better representation of infinity than the lizards, but had trouble putting it into words. What I think is that the fish represent a larger infinity. If you try counting the lizards, you can group them into sets all of the same size as you count inwards. i.e. you can group the outer ring of brown lizards with the touching red and white lizards, and do the same thing with the next set of brown, red and white lizards, and the pattern just repeats itself with the same number of lizards in each set. However, if you try grouping the fish in a similar way, you find that each next set of fish goes up by a factor of 2 or 4 depending on how you group them. And as we learned earlier, a power set is the same size as 2 raised to the original set. So I think it's reasonable to say that the lizards are comparable to N, and the fish to R.
Tell me if this made any sense at all to anyone.
hahaha i was right! the picture with the the big fishes in the center is a better representation on the lizards! Sorry, i can get pretty competitive and I love to know I was right :) especially in this class when everything I think infinity is always feels wrong. Julian, your explantation makes sense. I love the way Escher explains how the lizards can be infinitely small yet gathered at one point, but by having the bigger fish in the center, you get a greater infinity from all sides. Both can be infinitely cut in half, but the big fish in the center removes boundaries. the infinitely many halves can be reduced from multiple directions instead of reducing to one center like the lizards. You (Julian) just helped put it in perspectives through what we've been discussing in class. Thank you. Although power sets and one-to-one correspondences puzzle me, I can understand that the fishes represents a bigger infinity than the infinite lizards.
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