course Mth 152 ?€?E???g??????assignment #025025. `query 25
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20:26:16 **** query 9.7.6 intersecting lines m, n parallel to k **** In which geometry or geometries is this possible and why?
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RESPONSE --> This is in Riemannian geometry. Riemannian pointed out the difference between a line that continues indefinitely and a line having infinite length. For example a circle on the surface of a sphere continues indefinitely but does not have infinite length. So if two lines intersect on a circle or a sphere it is occuring in Riemannian geometry.
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20:26:21 ** Two parallel lines intersect on a sphere (think of lines of longitude). So this occurs in a Riemannian geometry. **
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RESPONSE --> o.k.
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20:29:12 **** query 9.7.18 ruler r.b. CD wrench nail **** To which of the objects is the coin topologically equivalent and why?
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RESPONSE --> The coin is topologically equivalent to A (the ruler) and to E( the nail) because neither of these have holes in them.
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20:29:16 The coin is topologically equivalent to the ruler and the wrench nail because none of these have holes.
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RESPONSE --> o.k.
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20:32:16 **** query 9.7.27 genus of 3-hole-punched sheet of paper **** What is the genus of the sheet and why?
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RESPONSE --> The genus would be three because the paper has three (or more) holes.
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20:32:21 The genus of this sheet of paper is 3 becasue it contains 3 holes.
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RESPONSE --> o.k.
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20:34:50 **** query 9.7.42 3,3,3,3,4,4,2,2 **** Can the network be traversed or not and why?
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RESPONSE --> A network with more than two odd vertices cannot traversed. 3,3,3,3,4,4,2,2 has for odd vertices so it cannot be traversed.
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20:34:56 ** This network contains 4 odd vertices. A network with 0 or 2 odd vertices can be traversed; a network with 4 odd vertices cannot. **
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RESPONSE --> o.k.
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20:42:15 **** If you start on a vertex of order 3 can you traverse the network and end up on that vertex? Explain why your answer must be true.
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RESPONSE --> It would not be possible to start on a vertex of order 3 and end up on that vertex. When you start out you would be going along one of three edges. to traverse the second of the edges you would end up back at the vertex. If you leave again you would be going on the third edge and it would not be possible to get back.
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20:42:18 ** You can? start on a vertex of order 3 and end up on the same one. You leave the vertex along the first of the three edges. When you traverse the second of these edges you are returning to the vertex, and when you leave again you have to travel along the third and you can't get back. You can end up on a different vertex of degree 3 if there is one (and if there is one you must end on it), but you can't end up on the degree-3 vertex you started from. **
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RESPONSE --> o.k
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20:46:05 **** If you start on a vertex of order 4 can you traverse the network and end up not on that vertex? Explain why your answer must be true.
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RESPONSE --> If you have a vertex of order 4 you would have to >>>1 out on 1 <<<2 back on 2 >>>3 out on 3 <<<4 back on 4 you can not go again.
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20:46:08 ** If you start on a vertex of order 4 you cannot traverse the network without ending up on that vertex, since you leave the vertex on the first edge, return on the second and leave on the third. If you traverse the network you have to return to the vertex on the fourth edge, and you can? leave again. **
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RESPONSE --> o.k.
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20:48:09 **** If you start on a vertex of order 2 and traverse the network must you end up on that vertex? Explain why your answer must be true.
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RESPONSE --> You would end up on the vertex of order 2 because you would go out on 1 come back on 2 and that would be it.
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20:48:13 ** If you start on a vertex of order 2 and traverse the network you leave on the first edge, return on the 2 nd and you?e stuck there. **
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RESPONSE --> o.k.
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20:48:29 No, because once again this is an even vertex. One point must be the starting point and one the ending point.
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RESPONSE --> o.k.
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20:50:46 **** If you start off of a certain vertex of order 3 and traverse the network is it possible to end up somewhere besides this vertex?
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RESPONSE --> Yes, it would be possible to end up somewhere besides the vertex because you would go out on 1, come back on 2, and go back out on 3, since you could not return to the vertex you could end up somewhere else.
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20:50:53 ** If you start on a vertex of order 3 and traverse the network you leave on the first, return on the second and leave on the third edge. You can? travel any of these edges again so you can never return. Therefore you must end up elsewhere. **
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RESPONSE --> o.k.
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20:51:07 **** Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> NO SURPRISES
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