November 2007 - mathfun@mailman.xmission.com

[math-fun] Re: math-fun Digest, Vol 57, Issue 17
by Steve Witham 28 Nov '07

28 Nov '07

>From: "Fred lunnon" <fred.lunnon(a)gmail.com> > > > http://math.ucr.edu/home/baez/square.root.html > >I've read this webpage three times, and I still don't understand it. I didn't make much sense of it, but maybe he's leading up to the idea that while you can flip the number line with two rotations, trying to build a complex conjugate out of rotations is a blind alley? --Steve

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[math-fun] Mirror Mirror
by David Gale 28 Nov '07

28 Nov '07

If you Google "mirror, left, right" you get over 9,500,000 hits. Some of the discussions among the dozen or so I looked at are very good and most of them make the same basic point which I describe as follows. An experimental finding is called an ARTIFACT if it turns out to have been produced not by nature but by the experimenters themselves and their apparatus. Scientists are of course very much aware and wary of this possibility. Well, the famous mirror phenomenon we have been discussing is an example, par excellence, of an artifact. People say things like, a mirror reverses right and left but not up and down, or words to that effect. Not true. It is we, the observers who do the reversing or up-side-down-ing. Simplest example: You are reading a book and decide to hold the page you are looking at up to a mirror. Will the letters in the image be "backward" or "upside down"? Obviously it depends in whether YOU the experimenter turned the book about a vertical or horizontal axis. The case of, say, a room reflected in a mirror is a bit more subtle. It's certainly true that what you see in the mirror is different in an essential way from what you see when you turn your back on the mirror and look at the room directly. Namely, everything in the mirror is "backward". For example the hands of the clock on the back wall are at two o'clock but in the mirror they seem to be at ten o'clock, etc. Nevertheless, I assert that it's you, not the mirror that's doing the reversing. First you looked at the room directly, (or you imagine you have done so) and then you TURN and look at the image in the mirror. But how did you turn to face the mirror? Probably about a vertical axis, although (assuming you're sufficiently agile) you could have done it by standing in your head, in which case everything would be upside down. Finally, even the mirror is irrelevant. On the store window you see "MERRY CHRISTMAS" in big letters. You go inside and look at the same window and, -what do you know,- the letters run backwards. I've never heard anyone claim that windows reverse right and left. It's you, the experimenter with your apparatus, in this case your body, that accounts for the observed phenomenon. Yes? Again, many of these and similar examples are given in the articles I referred to at the beginning. dg

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Re: [math-fun] Computer Go and Euler number
by Dan Asimov 27 Nov '07

27 Nov '07

Euler number (often called "Euler characteristic") for nicely behaved topological spaces is standard terminology. One large class of "nicely behaved" spaces is those that can be triangulated, and this include all those that can be expressed as a union of simple closed polygons (including interiors) meeting only along common edges and/or common vertices. The definition of Euler characteristic X(S) for such a planar set S is X(S) = V-E+F (where the X is written as a script xi) and is a topological invariant. (So it doesn't depend on how the space is decomposed into a nice union of The area of a Go board occupied by certain pieces may be thought of as the union of squares, one per piece, meeting (when they do) along common edges. (Such squares would be thought of as centered at any of the 361 dots, with side length equal to the distance between adjacent dots.) So to calculate the Euler characteristic of a set of pieces on a Go board, just consider each piece to be a square with 4 Vertices, 4 Edges, and 1 Face. Each Vertices and Edges must be counted only once, even if it belongs to more than one square.) To implement this, maintain one array for all (19+1)^2 = 400 Vertices, one for all 2*19*(19+1) = 760 Edges, and one for all 19^2 = 361 Faces, zeroing these all out at the start. Whenever a piece in the set S of interest is encountered, ensure that all 9 = 4+4+1 relevant array values are set to 1, not 0. Finally, calculate X(S) = V-E+F. There is a simple relationship between X(S) and the number H of holes in S: H = 1 - X. (As a check, note that a full board has H = 1-(400-760+361) = 0 holes.) Hope this helps. --Dan

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[math-fun] Computer Go and Euler number
by Steve Gray 27 Nov '07

27 Nov '07

Hello Ray and all: "Euler number" (an ambiguous designation, not recognized by most mathematicians) is easily calculated this way. Let Q1=the number of 2x2 neighborhoods with this configuration: X 0 0 1, and let Q3 be the number of these: X 1 1 0, where X is don't care. Then E=Q1-Q3. The Q's are counted with a raster sweep that moves across and down by one cell, so a given pixel is examined four times. Also E=B-H where B is the number of blobs (connected sets of 1's) and H is the number of holes (connected sets of 0's inside blobs). This formula for E does not treat the following patterns optimally, but it can be improved by making it slightly more complicated. 1 0 0 1 and 0 1 1 0. Holes on the edge can be counted by surrounding the image with a frame of 1's, but separate blobs touching the edge will become connected. It's been 36 years since I published this work (IEEE Trans. Computers, May 1971) so I may not recall everything perfectly. Incidentally, 2x2 neighborhoods are much better for tracing boundaries than 3x3, being faster and less ambiguous in crowded areas. I thought about the Go problem a little but never wrote any code. I'm not sure how much help the Euler number stuff would be. I got to be about low intermediate as a player. Steve Gray Ray Tayek wrote: > steve, hi, someone using your paper on the go list. > > dave, my last post to steve went unreplied to, do you have a more > recent address? > thanks > > >> Stuart, >> >> Here is the deal on euler numbers and implementations: >> >> It's difficult to find any articles on the web that you don't have to >> pay for. So it was difficult for me to implement this. I resorted >> to scanned texts of the images that I found after a lot of work - then >> you have to work your way through the papers - which are designed more >> for academic style than understandability but I eventually ended up with >> an implementation. >> >> The basic idea of euler numbers is that a 2 dimensional bit map can be >> analyzed to determine the number of objects and number of holes in the >> object. The euler number is the number of objects minus the number of >> holes (and can be negative.) If you can figure out how, there is a >> way to calculate this number very efficiently. After a lot of pain and >> agony I eventually figured out how. >> >> In my implementation an object is connected groups of stones. A stone >> is connected to another stone even on the diagonals, although you can >> use other definitions of connectivity. A hole is exactly what you >> think it is in GO, a connected set of empty points (and diagonals are >> not counted in this case.) >> >> In image processing, this is the most visually intuitive definition. >> The object is connected groups of printed pixels on a white piece of >> paper and the holes are the white background showing through. >> Visually we see diagonally connected pixels as part of the same >> object. And that probably works best in Go too when used as an >> element in an evaluation function. >> >> I believe the canonical text on the subject is a paper by Stephen B. >> Gray called "Local Properties of Binary Images in Two dimensions." I >> eventually found a copy of this paper on the web that you don't have to >> pay for - it looks like a scanned image of it. >> >> It was even harder for me to figure out how to implement this >> efficiently using scan lines. It's mentioned on the web in several >> places, but there is no freely available code and no clearly available >> explanation that I could find. Perhaps I just missed it. >> >> - Don >> >> >> >> >> Stuart A. Yeates wrote: >> > Could you give us a quick reference for exactly _which_ Euler numbers >> > you're using? Wikipedia has three separate ones and the MathWorld site >> > a similiar number. >> > >> > Maybe I'm just being stupid. >> > >> > cheers >> > stuart >> > >> > On 26/11/2007, Don Dailey <drdailey(a)cox.net> wrote: >> > >> >> After reading the paper on solving go on small boards, I am curious >> >> about the use of euler numbers as a simple evaluation element. >> >> >> >> I implemented a little euler number test program and it works >> correctly >> >> from a sample of about 50 positions of various types. I'm using the >> >> fast version where you scan 2 lines at a time with a lookup table. >> >> >> >> However, it calculates holes inside of groups and this does not >> detect >> >> eyes or "holes" on the edges of the board. It's not clear how >> to deal >> >> with this. >> >> >> >> I'm experimenting with a version that wraps a border around the whole >> >> board so that even the empty position looks like a 1 group with >> one big >> >> hole. This causes a lot of silly anomalies - for instance if you >> >> surround a big chunk of safe opponent stones it looks like a big >> >> hole. If you own half the board and the opponent owns the other >> >> half, his half contributes favorably to your euler number (it >> looks >> >> like a big hole of yours.) >> >> >> >> Of course I realize that this is just a quick and dirty >> calculation but >> >> I was curious about any tricks that others use to deal with it. >> >> >> >> - Don >> >> >> >> _______________________________________________ >> >> computer-go mailing list >> >> computer-go(a)computer-go.org >> >> http://www.computer-go.org/mailman/listinfo/computer-go/ >> >> >> >> >> > _______________________________________________ >> > computer-go mailing list >> > computer-go(a)computer-go.org >> > http://www.computer-go.org/mailman/listinfo/computer-go/ >> > >> > >> _______________________________________________ >> computer-go mailing list >> computer-go(a)computer-go.org >> http://www.computer-go.org/mailman/listinfo/computer-go/ > > --- > vice-chair http://ocjug.org/ > > > >

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[math-fun] Re: mirrors (was The Axiom of Choice for roots of z^2 + 1)
by Steve Witham 27 Nov '07

27 Nov '07

>On 11/25/07, Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > > > > A mirror neither exchanges left with right nor up with down. It >exchanges front with back. This is precisely the definition of >"reflection in a plane". >From: "Fred lunnon" <fred.lunnon(a)gmail.com> > >[Not seeing that is] > an excellent example of how a badly-formulated linguistic trope may >confuse its users so badly that they become incapable of disentangling >its distinct meanings, even after those are explicitly pointed out. WFL >From: Gareth McCaughan <gareth.mccaughan(a)pobox.com> > >[...] I'd have added that we think of what a mirror >does as exchanging left with right because the things >we look at in mirrors -- most notably ourselves, but >plenty else besides -- are much nearer to being left-right >symmetrical than to being top-bottom symmetrical. [...] Or front-back. >(There's a *reason* why people commonly think that mirrors >swap left and right, which is logically prior to that trope.) I would say we code for the smallest edit distance, where "smallest" means most likely. So: the image in the mirror is rotated around the vertical axis and flipped left to right. It seems to me that the rotation happens unconsciously (pre-linguistically), and once it has, untangling the rest consciously is tough. Mirrored pairs are called "right-handed" and "left-handed"-- is that the "badly-formulated linguistic trope" you mean, Fred? Or is it the whole treatment of "swap" as a verb, or...? >From: Bill Thurston <wpt4(a)cornell.edu> > > Ultimately, I mellowed. I realized that people were unconvinced by >each other's explanations because they weren't actually confused >about what a mirror does. There's not an actual question here without >words. Maybe we can't *access* the question without words(*), but I think I was actually confused when I first came across this one. The mirror obviously flips one axis and not the other, and just as obviously can't. There's a real question of how to make sense of it. Motor/perceptual problem: sequences of transformations and outcomes. Math problem: equivalent expressions. Language problem: in practical (as opposed to tact & felicity) situations, once you have one expression to identify a thing, that's usually good enough: you've identified the thing and the thing is what matters. I guess this shows that a practical problem can have the same structure as a felicity problem. (It's a matter of "perspicuity." Check out Craig Swanson's http://content.perspicuity.com/?q=taxonomy/term/487 .) --Steve * "...just ask the axis." --Jimi Hendrix

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[math-fun] Re: n-D median?
by Steve Witham 27 Nov '07

27 Nov '07

I wrote an applet in Processing to play with the 2D version of my n-D median: http://www.tiac.net/~sw/2007/11/2D_median Processing is a nice dialect of Java for writing mathematical or artsy applets like this. It lets you focus on the math and graphics rather than the cruft of Java APIs. --Steve

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[math-fun] Re: The Axiom of Choice for roots of z^2 + 1
by Steve Witham 26 Nov '07

26 Nov '07

>From: "Andy Latto" <andy.latto(a)gmail.com> > >So we describe (but also think of) >the effect as a >"reflection in the axis of symmetry", by which we mean "some stuff too >unimportant to mention, followed by a reflection in the axis of >symmetry". What he said. >From: "James Buddenhagen" <jbuddenh(a)gmail.com> > >And speaking of psychology, I aways turn my head 90 degrees when I look >in a mirror. That way it reverses up and down for my head and left and right >for my neck and I feel no angst :) Yeah, viz. (: --Steve

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Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Eugene Salamin 26 Nov '07

26 Nov '07

----- Original Message ---- From: Fred lunnon <fred.lunnon(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Saturday, November 24, 2007 12:30:23 PM Subject: Re: [math-fun] The Axiom of Choice for roots of z^2 + 1 Which reminds me --- quite irrelevantly --- of a claim I fear I may have made previously in these august pages: that a most effective means of causing a polite gathering to descend into an incoherent brawl is to remark that "a mirror exchanges left with right" and then to enquire innocently "so, why doesn't it also exchange up with down?" A mirror neither exchanges left with right nor up with down. It exchanges front with back. This is precisely the definition of "reflection in a plane". Gene ____________________________________________________________________________________ Be a better sports nut! Let your teams follow you with Yahoo Mobile. Try it now. http://mobile.yahoo.com/sports;_ylt=At9_qDKvtAbMuh1G1SQtBI7ntAcJ

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Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Henry Baker 26 Nov '07

26 Nov '07

I thought that physics was such that if Pooh chose the wrong paw, time would then go backward... (CPT - invariant only if charge, parity & time are all simultaneously interchanged.) -- Also, in quaternions, x^2+1=0 has an infinity of solutions, as Hamilton points out. I think that Hamilton favored left-hand coordinates; does anyone know if he himself was left-handed ? At 02:27 AM 11/24/2007, Fred lunnon wrote: >On 11/23/07, Dan Asimov <dasimov(a)earthlink.net> wrote: >> I've often wondered if it is somehow a mistake for >> mathematics to distinguish between i and -i. >> >> There is absolutely no mathematical way to distinguish >> between the two. Perhaps they should only be referred >> to as a pair, and never one at a time? >> >> Opinions? >> >> --Dan > >"Pooh looked at his paws. He knew that one of them was the right, and >he knew that when you had decided which one of them was the right, >then the other one was the left, but he could never remember how to >begin." > [A.A.Milne (1926); > http://www.everfree.ca/2007/01/pooh_on_your_shoe_kid.html] > >And, using elementary physics at least, there's also no way to tell the >difference. But the existence of symmetry doesn't (necessarily) imply that >it's a good idea to factor out the symmetry group! > >One situation where something analogous does routinely occur is classical >projective geometry; and it's a significant nuisance when this discipline is >applied to computer graphics, etc. [In practice it can be repaired by retaining >the sign when normalising homogeneous coordinates --- easily overlooked >in a coordinate system based on points, the only subspaces which happen >not to be orientable.] > >Some authors have attempted to repair the omission --- see e.g. > Jorge~Stolfi \sl Oriented Projective Geometry \rm Academic Press (1991); > www.dcc.unicamp.br/~stolfi >--- but the extra complexity doesn't engender any apparent mathematical >interest, merely engineering practicality. > >Fred Lunnon

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Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Marc LeBrun 26 Nov '07

26 Nov '07

Either -i is different from i or it is the same. If we choose to make it different we get the usual system. If instead we choose to make it the same i = -i then we can add i to both sides obtaining i + i = -i + i i + i = 0 2 * i = 0 making i a divisor of zero.

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