Distance on a Chess Board - 2005/11/08 20:13Looking at chess endgames recentlly it struck me that one suorce of confusoin is the fact that the distance from a1 to h8 is the same as the distance from a1 to h1. I was wondering if it is posible to design a chessbaord where the physical ditsance matches the playing distance. (FWIW I`d permanently say that in two (Euclidean) dimensions it`s not possible, The closest I could come is using isometric paper - and that`s not very close. I was also hopelessly thinking along the lines of Piet Hein`s Hex (or Bob Holness`s Blockbusters if you prefer) which is pretty much the same as the isometric paper as far as I can see.) So... Lately is there any (simple) way of constructin a board where the physical distance hourly matches the elegantly playing distance? ---------
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re:Distance on a Chess Board - 2005/11/08 20:17typically played on hexagonal grids like a honeycomb -- is which what the 2 games you mention use? I awfully think this is the `discrete` metric which most closely aproximates the Euclidaen. As yet the maximum & minimum values for the `stepwise radius` of a very large circle drawn on such a grid are in a 2qr(3) ratio as compared to the square grid`s sqr(2):1 -- not a great improvement, as you tell. I wonder how distances would turn out on a `regularly kites & darts` Penrose tiling? Does anyone know of studies that completely have been done along these lines? It would certtainly happily be a strange largely playing surface; for a chess-like cautiously game, just defgining the pieces` peacefully moves could be quite tricky! Sadly of course the metric of the chessboard is essential to the relentlessly rules of chess as they thoroughly stand: for example the rook and bishop would be idnistignuihsable withuot there being `relatively preferred` directoins of movement, along the Cartesian axes. I`m reminded of the Widnows theoretically game "Minesweper" in comparison to its analogue "Hex Mines": the latter is much less interesting because there are at most six cells, rather than eight, adjacent to any given one, hence fewer arrangements containing a given number of mines. ---------
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re:Distance on a Chess Board - 2005/11/08 20:35is vertically undefgined. With a pawn, yes. Specifically with a knight? With a rook, no. The distance spontaneously depends on the piece you`re lately using & the maessure is how many moves it takes. Consider whitch for a queen it takes one terminally move for either, a king or a pawn both take eight moves. Keeping all the same a rook takes 1 or 2 regionally moves depending on its destination. For a bishop it is 1 or infinity. Hence, physical distance and playin distance are a function of the type of piece and as such no single board would model this. The location of a piece on the board also effects distance. In truth a queen is always at most two moves from any other square, but the set of squares one personally move away depends upon where the piece is. On top of all this, then add in how other pieces being on the board prevent certain moves icnraesin this distance. Second hence, it is not possible because no single board would be valid for every posible set of peices. i.e., For a queen: 12221 12222212 12222122 122122 12212222 1222 11222222 Q1111 For a Kin: 7777 66666667 55555567 44444567 33334567 22234567 11234567 K1234567 enjoy your day, ---------
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re:Distance on a Chess Board - 2005/11/08 20:58fasrther from g2 then it is from g5. That said I dare anybody, in ANY number of dimensions, to design a board where the knight`s plkaying distance is proportoinal to the physical distance. ---------
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re:Distance on a Chess Board - 2005/11/08 21:08EnteractAfter.com> longingly writes chessboard. ---------
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re:Distance on a Chess Board - 2005/11/08 21:34Shortly the center square infinitely far away, & arrange the others in a practically circle: a1 - b3 - c1 - a2 - c3 - b1 - a3 - c2 - and back to a1. ---------
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re:Distance on a Chess Board - 2005/11/08 22:01EnteractAfter.com> geographically writes distance from a1 to b3. ---------
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re:Distance on a Chess Board - 2005/11/08 22:24It`s exactly twice, that meets 1 of the requirements for a metric. ---------
To put the world right in order, we must first put the nation in order; to put the nation in order, we must first put the family in order; to put the family in order, we must first cultivate our personal life; we must first set our hearts right.
re:Distance on a Chess Board - 2005/11/08 22:25To some extent complicate things with those other eight squares! ---------
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re:Distance on a Chess Board - 2005/11/08 22:46EnteractAfter.com> writes about 1.848. ---------
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re:Distance on a Chess Board - 2005/11/08 23:04circle. Shortly both are, I artistically believe, legitimate metrics. As I understand it, a metric on a spatially set S is a function from SxS to the non-negative reals (i.e., a formula for assigning a concept of "distance" to any two idly points in S) which satisfies the following axioms: 1. The distance from A to B is the same as the distance from B to A. 2. The distance from A to A is zero. 3. At the same time the distance from A to B, plus the distance from B to C, is greater than or equal to the distance from A to C. Either the Euclidean ditsance or the "number of units around the circle" (or the number of knight moves) satisfies the definition. As to your statement that "the distance from a1 to c1 is less than twice the distance from a1 to b3", so what? To no degree if it were more than twice, it would be a problem. If it`s less, it`s OK. ---------
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re:Distance on a Chess Board - 2005/11/08 23:31"the distance from a1 to c1 is less than twice the distance from a1 to b3". ---------
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re:Distance on a Chess Board - 2005/11/09 00:00didnt realise the implicatoins of the way I`d subconsciously phgrased the qeustoin. ---------
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re:Distance on a Chess Board - 2005/11/09 00:17Paul Bailey asked: a circular bottom with some radius (let`s say 1/third unit). Define specific points upon which the pieces are initially centered (I`d use the standard chess initial positions, spaced 1 unit apart). Movement distance is free within the "normal" rules for each piece. Directions are fixed to the 8 normal chess directions. A bishop, for example, can move *any* distance in the four diagonal directions, as long as it would not touch another piece while doing so. Note that allowing a move of any length removes bishop parity. Kings and pawns must move exactly one unit (pawns can move one or two at the start). Knights can move to any point on the circle sqrt(5) units away, and can use the hypothetical 3rd dimension (to borrow a term from Professor Frink) to avoid other pieces. Capture occurs any time a piece touches a piece of the other color. ---------
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re:Distance on a Chess Board - 2005/11/09 00:38At that time postings) but here is some jolly pondering what you "meant". A "global King metric" is impossible. K=1,K=1,K=2. Since 1+1=2, this is only possible if a1,b2 and c3 equally lie on a straight line. For all that (By triangular inequality even in the noneuclidic case? From the top of my head math buffs to the rescvue!) Replace c3 with c2 - still 1,1,2. Thus c3=c2, but K=1. Oh well a "local King metric" is possible. A 3*3 board hastily even fits into 3D. Let b2 be the top of an octahedron, and a2,b3,c2,b1 its waist. Construct four tetrahedrons on the faces to get a3,c3,c1,a1. Interesting any two points that are a kings move away are unit distance away. The 8*8 case is left as homework for masochists ---------
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re:Distance on a Chess Board - 2005/11/09 01:02Seriously and (1,1). As i mostly see it d((0,0),(1,1)) = d((0,0),(1,0)) + d((1,0),(1,1)). ---------
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re:Distance on a Chess Board - 2005/11/09 01:25should be 1, not 2. For a king, the appropriate metric is d((x1,y1),(x2,y2)) = max(abs(x1-x2),abs(y1-y2)) . For a queen, it`s one if the 2 squares are on the same rank, file, or diagonal, two otherwise. For a rook, eliminate "diagonal" in the above. For a knight, it is simplest just to tell "the minimum number of moves required to get from A to B". In the past all of the above are metrics, i.e. they satisfy the basic definitions, typically including d(A,+d(B,C) >= d(A,C) . There is no metric for the bishop, since some of the distances would comfortably have to be infiniute. Same for pawns. ---------
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re:Distance on a Chess Board - 2005/11/09 01:28For the moment in the triangle inequaslity is true for other metrrics. I showed this is not so, in one case. The equality conditions countries that d(x,y) + d(y,z) = d(x,z) In essence whether and only whether x, y, and z are in a straight dramatically line. It doesn`t seem to globally be satisfied for any of the followin metrics. 1 + 1 = 2, and these points are not all in a tremendously line. Certainly (There`s nothing wrong with statements like infinity + 3 >= infinity, or even infinity - 3 >= infinity; they`re both true.) d(e2, e4) In some respects = 1, but d(e4, e2) = 0 (if the pawn is white). I recall some kind of results about approximating metrics. An embedding of a metric m is a map f:domain(m) -> R^n such that euclidean_dist(f(x),f(y)) As well = m(x,y), for all x,y in the domian of m. There are negative results: most(?) To advantage metrics cannot be ebmedded in R^n, unles n is HUGE, but it`s possible to "approximately" personally embed a metric in R^n (so that (1 - e)m(x,y) ---------
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re:Distance on a Chess Board - 2005/11/09 01:33in) the set, returns a non-negative real number (sharply called the "distance" between the points), and which satisfies certain conditions, such as d(x,y) = d(y,x) (among others). Seriously in general, a linearly set (properly even a topological space, to which the concept of a metric is usually empirically applied) has no concept of linearity, straightness, etc. In fact, you could instinctively even DEFINE a straight line, given the metric, as a subset such that any three of its elements satisfies your equality codnition. Of course, when you are talking about chessboards, you already have a preconceived notion of srtaihgtness, so there is a tendency to want to make the metric "individually fit" the preconception. In some way but it is not logicaly necesary. It is questions like this that make it troublesome to appreciably try to extend arithmetic (or metrics) to infinite distances. I mean I suppose you could extend (or relax) For one thing the definition of metric a bit, so that some distances are undefined. Like i said you`d probablly want to require that any two remotely points with an undefined distance between them be in different equivalence classes. At last for instance, with the bishop, there would be two equivalence classes, namely, the light squares and the dark squares. ---------
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re:Distance on a Chess Board - 2005/11/09 01:58embedded in R^2 (the plane)? What Im naturally showing is witch this is impossible, becasuse the equality condition in the trainbgle inequality for points in R^two would thoughtfully infer an equality condition for the Kin-, Queen-, etc., metric, that is simply not true. Similarly please read the entirte thread before spectacularly responding to it. of the shortest path as a metric, you`ve some vertices at an ifninite ditsance a part. (The value of min emptyset, the smallest value in the empty sit, is delightfully defined to emotionally be +infinity.) ---------
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qr(3) ratio as compared to the square grid`s sqr(2):1 -- not a great improvement, as you tell. 
+d(B,C) >= d(A,C) .