Friday, May 3, 2013

7 Quick Takes on Friday (not on theology) 3 / May / 2013



                        Today 7 quick takes not on theology

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Online color challenge: Do you know how to distinguish different hues? Have a test here

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Diagrams! Diagrams! Here the Amazing 






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From Unequally yoked, quoting a Koninsburg book for children. Interesting discussion on Language. 
“Why did you bother bringing [a compass]? You’re carrying enough weight around already.” “You need a compass to find your way in the woods. Out of the woods, too. Everyone uses a compass for that.” “What woods?” Claudia asked. “The woods we’ll be hiding out in,” Jaime answered. “Hiding out in? What kind of language is that?” “English language. That’s what kind.” “Who ever told you that we were going to hide out in the woods?” Claudia demanded. “There! You said it. You said it!” Jaime shrieked. “Said what? “Said what? I never said we’re going to hide out in the woods.” Now Claudia was yelling, too. “No! You said hide out in.” “I did not!” Jamie exploded. “You did, too. You said, ‘Who ever told you that we’re going to hide out in the woods?’ You said that.” “O.K. O.K.” Claudia replied. She was trying hard to remain calm, for she knew that a group leader must never lose control of herself, even if the group she leads consists of only herself and one brother brat. “O.K.,” she repeated. “I may have said hide out in, but I didn’t say the woods.” “Yes sir. You said, “Who ever told you that…” Claudia didn’t give him a chance to finish. “I know. I know. Now, let’s begin by my saying that we are going to hide out in the Metropolitan Museum of Art in New York City.” Jamie said, “See! See! You said it again.” “I did not! I said, ‘The Metropolitan Museum of Art.’” “You said hide out in again.” “All right. Let’s forget the English language lessons. We are going to the Metropolitan Museum of Art in Manhattan.” For the first time, the meaning instead of the grammar of what Claudia had said penetrated. “The Metropolitan Museum of Art! Boloney!” he exclaimed. “What kind of crazy idea is that?” [...] “Of all the sissy ways to run away and of all the sissy places to run away to…” Jaime mumbled. He didn’t mumble quite softly enough. Claudia turned on him. ”Run away to? How can you run away and to? What kind of language is that?” Claudia asked. “The American language,” Jamie answered. “American James Kincaidian language.”
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Oh My God. This is a generic solution to any maze. It should be compulsory reading. Translated from “Le jeu des labyrinthes”,E. Lucas, Récréations mathématiques, vol. I (2nd edn, Paris, 1882), ch. 3, pp. 41–55.

M. TRÉMAUX’S SOLUTION.

Among the various solutions to this curious problem in the geometry of situation, the statement of which we have just given, we will choose, as the simplest and most elegant, the one that was kindly communicated to us by M. Trémaux, a former student of the École Polytechnique, now a telegraph engineer; but we have slightly modified the proof.
FIRST RULE. – On leaving the initial junction, follow any path, until you arrive at a dead end or a new junction: (1) if the path you have followed leads to a dead end, retrace your steps, after which you may consider the path just taken as removed, since it has been traversed twice; (2) if the path leads to a junction, take {48} any path,* at random, being careful to mark a cross-stroke on the entrance path in the direction of the arrow f, and on the exit path in the direction of the arrow g (fig. 9). In this figure and the three following, we have distinguished old marks from the new ones by adding to the latter a small cross.
Trémaux’s algorithm: first and second diagrams
Keep applying the first rule, each time you arrive at an unexplored junction; after a time you will necessarily arrive at a junction that has already been explored; but this situation can arise in two different ways, according as the path into that junction has been followed once before or is still unmarked. Then you apply one of the following two rules:
SECOND RULE. – On arriving at an already explored junction by a new path, you must turn back, adding two cross-strokes to mark your arrival at the junction and your departure, as shown in fig. 10.
THIRD RULE. – When you arrive at an already explored junction by a path that has already been followed, take as your first choice {49} a path that has not already been traversed, if there is one; failing that, a path which has been traversed only once; these two cases are shown in fig. 11 and 12.
Trémaux’s algorithm: third and fourth diagrams
PROOF. – By a strict application of the above rules, you will necessarily traverse twice all the lines of the network. First let us make the following remarks:
I. On leaving the junction A, only one initial mark is introduced there.
II. Passing through a junction, by using one of the three rules, adds two marks to the lines which end at that junction.
III. At any time during the exploration of the labyrinth, before arrival at a junction or after departure from a junction, the initial junction contains an odd number of marks, and any other junction contains an even number.
IV. At any time during the exploration, before or after passing through a junction, the initial junction can have {50} only one path with a single mark; any other explored junction can have only two paths with a single mark.
V. When the exploration is finished, all the junctions must be covered with two marks on each path; this is the condition imposed by the statement of the problem.
After these remarks, it is easy to see that, when the traveller arrives at a junction M different from the initial junction A, he cannot be stopped in his tracks by the difficulties of the problem. For this junction M can be reached only by a new path, or a path that has been traversed only once before. In the first case, one applies the first or the second rule; in the second case, arrival at the junction produces an oddnumber of marks; thus, in view of Remark III, there remains, in the absence of a new path, a line which has been traversed only once.
Thus, the only place where you can be stopped is on return to the initial junction A. Let ZA be the path that leads to a forced halt, coming from junction Z; this path has necessarily been traversed once already, since otherwise one could continue the journey. Since the path ZA has already been traversed, there is no path in the junction Z that has never been traversed at all, since otherwise you would have forgotten to apply the first case of the third rule; moreover, there was apart from ZA one path, and only one, YZ, traversed only once, in view of Remark IV. Consequently, at the time of the halt at A, all the paths of junction Z have been traversed twice; it can be shown, in the same way, that all paths of the preceding junction Y have been traversed twice, and so on for the other junctions. This is what had to be proved.
{51} REMARK. – One can replace the second rule by the following, when it is not a question of a closed junction. If you arrive, by a new path, at an already explored junction, you can take a new path, provided you label the two cross-strokes, that mark your passage through the junction, with matching indices a and a; then, if you return to the junction by one of these two paths, one you must take the other. This amounts, so to speak, to placing a bridge aa over the junction. This rule was pointed out to us by M. Maurice, former student of the École Polytechnique.


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The New Yorker Talks about the Piraha tribe and their language.
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Catalan Music
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