Angličtina pro matematiky IV
COURSE MATERIALS AND HOMEWORK VIII.
Read the text, fill in the missing words, and decide whether the statements are true or false.
experiments construction middle observation form properties
The Mobius strip (from Wikipedia, the free encyclopedia)
The Mobius strip is a 1. …………… which has some very strange properties. It is named after Mobius (1790 – 1868) who first wrote about it in 1865 in abook called Uber die Bestimmung des Inhaltes eines Polyeders.
The 2…………… of the Mobius strip can most easily be discovered by 3…………….. To do this, take a long strip of paper. Now twist it once. Finally stick one end of the paper to the other.
Now you can try a few 4…………………….. with this Mobius strip. If you try to color only one surface of the strip, you will find that it is impossible. Drawing a continuous line along the 5…………….. of the strip produces a line on both sides of the paper. By cutting along this line, we would expect to divide the strip into halves, but in fact we 6……….a longer, thinner strip which is still one piece.
a) A Mobius strip has only one surface.
b) A Mobius strip has two edges.
c) A Mobius strip is a three-dimensional figure.
d) Observing a Mobius strip is the easiest way to discover its properties.
e) Sticking together the ends of a twisted rectangle produces a Mobius strip.
5. Read the text on Topology and answer the following Qs.
a) What examples of deformations are mentioned in the text?
b) What does the word topology mean, where did it come from?
c) What are the most important subfields in topology?
d) What is a topological space?
e) Where do compactness and connectedness belong within topology?
6. Read the text again and try to summarize each paragraph in one sentence.
a).....................................................
b)....................................................
c)..................................................
d)..................................................
Topology (from Wikipedia, the free encyclopedia)
Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as 1736, and toward the end of the 19th century a distinct discipline developed, called in Latin the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”), and later gaining the modern name of topology. In the middle of the 20th century, this was an important growth area within mathematics.
The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line.
Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division.
7. Now read the short text on one of the important concepts in Topology, homeomorphism, and decide whether the statements are true or false.
a) Topological isomorphism and bicountinuous function are synonyms.
b) Two spaces with a homeomorphism are identical.
c) A topologist can not distinguish between a cup and donut since he is not very good at cooking.
d) Henri Poincaré thought that the relations between object are more important than their shape.
e) Homeomorphism is a more general term than isomorphism.
Homeomorphism
From Wikipedia, the free encyclopedia
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings which preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.
Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations (isomorphisms for instance) between them.
7. Now have a look at the definition below and then decide whether the examples below are homeomorphisms or not.
Definition
A function f: X → Y between two topological spaces (X, TX) and (Y, TY) is called a homeomorphism if it has the following properties:
- f is a bijection (one-to-one and onto),
- f is continuous,
- the inverse function f −1 is continuous (f is an open mapping).
A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism
Examples
- The unit 2-disc D2 and the unit square in R2
- The open interval (−1, +1) and the real numbers R.
- The product space S1 × S1 and the two-dimensional torus
- Every uniform isomorphism and isometric isomorphism
- and ............................. for .
- Let A be a commutative ring with unity and let S be a multiplicative subset of A. Then Spec (AS) ........................ to .
LISTENING - Ron Eglash on African Fractals
Do you understand these words?
iteration mission scale altar mound recursion crinkle
Listen and answer the questions.
- What did Georg Cantor discover? What were the consequences for him?
- What did von Koch do?
- What did Benoit Mandelbrot realize?
- Why should we look at our hand?
- What did Ron get a scholarship for?
- In what situation did Ron use the phrase “I am a mathematician and I would like to stand on your roof.” ?
- What is special about this royal palace?
- What do the rings in a village in southern Zambia represent?
Listen for the second time and decide whether the statements are true or false.
1. Cantor realized that he had a set whose number of elements was equal to infinity.
2. When he was released from a hospital, he lost his faith in God.
3. We can use whatever seed shape we like to start with.
4. Mathematicians of the 19th century did not understand the concept of iteration and infinity.
5. Ron mentions lungs, kidney, ferns, and acacia trees to demonstrate fractals in nature.
6. The chief was very surprised when Ron wanted to see his village.
7. Termites do not create conscious fractals when building their mounds.
8. The tiny village inside the larger village is for very old people.
I want to start my story in Germany, in 1877, with a mathematician named Georg Cantor. And Cantor decided he was going to take a line and erase the middle third of the line, and take those two resulting lines and bring them back into the same process, a recursive process. So he starts out with one line, and then two, and then four, and then 16, and so on. And if he does this an infinite number of times, which you can do in mathematics, he ends up with an infinite number of lines, each of which has an infinite number of points in it. So he realized he had a set whose number of elements was larger than infinity. And this blew his mind. Literally. He checked into a sanitarium. (Laughter) And when he came out of the sanitarium, he was convinced that he had been put on earth to found transfinite set theory, because the largest set of infinity would be God Himself. He was a very religious man. He was a mathematician on a mission.
And other mathematicians did the same sort of thing. A Swedish mathematician, von Koch, decided that instead of subtracting lines, he would add them. And so he came up with this beautiful curve. And there's no particular reason why we have to start with this seed shape; we can use any seed shape we like. And I'll rearrange this and stick this somewhere -- down there, OK -- and now upon iteration, that seed shape sort of unfolds into a very different looking structure. So these all have the property of self-similarity: the part looks like the whole. It's the same pattern at many different scales.
Now, mathematicians thought this was very strange, because as you shrink a ruler down, you measure a longer and longer length. And since they went through the iterations an infinite number of times, as the ruler shrinks down to infinity, the length goes to infinity. This made no sense at all, so they consigned these curves to the back of the math books. They said these are pathological curves, and we don't have to discuss them. (Laughter) And that worked for a hundred years.
And then in 1977, Benoit Mandelbrot, a French mathematician, realized that if you do computer graphics and used these shapes he called fractals you get the shapes of nature. You get the human lungs, you get acacia trees, you get ferns, you get these beautiful natural forms. If you take your thumb and your index finger and look right where they meet -- go ahead and do that now -- -- and relax your hand, you'll see a crinkle, and then a wrinkle within the crinkle, and a crinkle within the wrinkle. Right? Your body is covered with fractals. The mathematicians who were saying these were pathologically useless shapes? They were breathing those words with fractal lungs. It's very ironic. And I'll show you a little natural recursion here. Again, we just take these lines and recursively replace them with the whole shape. So here's the second iteration, and the third, fourth and so on.
So nature has this self-similar structure. Nature uses self-organizing systems. Now in the 1980s, I happened to notice that if you look at an aerial photograph of an African village, you see fractals. And I thought, "This is fabulous! I wonder why?" And of course I had to go to Africa and ask folks why. So I got a Fulbright scholarship to just travel around Africa for a year asking people why they were building fractals, which is a great job if you can get it. (Laughter)
And so I finally got to this city, and I'd done a little fractal model for the city just to see how it would sort of unfold -- but when I got there, I got to the palace of the chief, and my French is not very good; I said something like, "I am a mathematician and I would like to stand on your roof." But he was really cool about it, and he took me up there, and we talked about fractals. And he said, "Oh yeah, yeah! We knew about a rectangle within a rectangle, we know all about that." And it turns out the royal insignia has a rectangle within a rectangle within a rectangle, and the path through that palace is actually this spiral here. And as you go through the path, you have to get more and more polite. So they're mapping the social scaling onto the geometric scaling; it's a conscious pattern. It is not unconscious like a termite mound fractal.
This is a village in southern Zambia. The Ba-Ila built this village about 400 meters in diameter. You have a huge ring. The rings that represent the family enclosures get larger and larger as you go towards the back, and then you have the chief's ring here towards the back and the chief's immediate family in that ring. So here's a little fractal model for it. Here's one house with the sacred altar, here's the house of houses, the family enclosure, with the humans here where the sacred altar would be, and then here's the village as a whole -- a ring of ring of rings with the chief's extended family here, the chief's immediate family here, and here there's a tiny village only this big. Now you might wonder, how can people fit in a tiny village only this big? That's because they're spirit people. It's the ancestors. And of course the spirit people have a little miniature village in their village, right? So it's just like Georg Cantor said, the recursion continues forever.