SaxonViolence Posted July 8, 2013 Report Posted July 8, 2013 (edited) Take a Chess or Checker Board—I mean, they're the same. One can warp a set of Squares into a Cylinder Quite easily. But what if I want to warp a regular n x n grid into a "Sphere"... HMMMmmnn..... Let me rephrase. I don't care if the resulting figure is any sort of regular polygon... But I want to be able to trace each Row, Column and Diagonal Back to its starting point. I want No Ambiguities; No New Connections {Except where the former edges join} and No Lost Connections. Geometrically—I don't really care if all my Corners are 90o. I don't care if alternating squares are either Positive or Negative Curvature. I don't care how distorted my Squares get—just so long as it is easy to trace my Diagonals. I don't mind minor gaps—so long as the gap couldn't be mistaken for a square in it's own right. I've played around with the idea of a Spherical Chess Variant in my mind, for many years—Probably on a bigger grid than the Standard 8 x 8—Probably with some extra pieces to fill up the greater space... But the Sphere is only accessible on a Computer Simulation and one can only examine one half of the Sphere at one time—though you can rotate the Virtual Sphere around till you get dizzy, if it's your turn. Obviously the pieces would probably be arranged surrounding the King in some fashion. I hear that Symmetrical games often lack dynamic tension—so the Two sides would be a Square or two closer going "West" {just for instance} than "East"... It would also be handy to have the Computer tell you—especially for beginners—if one was attempting an Impossible Move—Moving into Check—Being Given Check—or had Just Been Mated. {It is concievable that two beginners might play on without realizing that Mate occurred.} But anyway, unless I know how to roll various boards into "Spheres" I can't rerally get past Step One. Saxon Violence Edited July 8, 2013 by SaxonViolence Quote
CraigD Posted July 9, 2013 Report Posted July 9, 2013 If you don’t care about building a physical, playable board, what you're describing, SV, is easy – just use a regular, flat board with a “wrap around” movement rule. Googling “wraparound chess” finds what appears to me to be a common variant called “cylinder” (example) where the wraparound rule applies to the east and west edge only, which folk seem to say is comparable in game length and quality to regular chess. A less common variant called “donut” (or “torus”) ( example) allows wraparound of both pairs of opposite board edges, but I read that, even with special restrictions on slider piece movement, games are shorter and less profound than regular chess. Both cylinder and torus chess boards could be physically built – you’d just need something to keep pieces from falling off of them, such as pegs, magnets, or Velcro. They could be rendered using a computer program without too much trouble – though I’m personally so rusty at this kind of programming, I imagine it’d take me weeks to get up to speed enough to write such a thing. I don’t think these “true” boards, or computer renderings of them, offer much practical advantage over a flat board with wraparound movement rules, though they’d look pretty and cool. If you truly want a board that topologically equivalent not to a cylinder or a torus, but to a sphere, tiled with faces of roughly similar size and shape, it’d have to be a regular or semiregular polyhedron. There are only 5 regular convex polyhedral (AKA Platonic solids), the one with the greatest number of faces having a mere 20 triangles, too few, I’d guess, to play a very chess-like game on. An infinite number of semiregular polyhedra are prism and antiprisms, which can have any number of faces, but for chess board purposes, are practically equivalent to cylinders with big “cap” faces. That leaves the 13 Archimedean solids. The 3 largest have 62 through 92 faces, comparable to the 64 on a regular chess board – I think they’d be interesting candidates for a chess variant game board. Practically, all these polyhedral can be drawn as flat boards with helper arrows showing how disjoint edges connect, so playing games on them shouldn’t be too hard. Promising! :thumbs_up Googling “Archimedean solid chess” finds nothing like this (chess.com has a page on them, but it’s just a encyclopedia page, with no mention of using them as game boards), so it’s not a heavily trod field. Quote
SaxonViolence Posted July 9, 2013 Author Report Posted July 9, 2013 Thanks. The beauty of having a Virtual Sphere viewable only on a monitor, the control of which passes between the players on the move... Is that Memory and Visualization skills would be at a premium. I wanted to use the idea of Spherical Chess—i.e. "Sphess" as an element in one of my SF Stories... A game that is very popular with Genetically Enhanced Geniuses (who aren't that uncommon) but all but impossible for a Non-Enhanced Human to play at all well—or even purposefully. Be all that as it may. I'm a little puzzled by the reference to Archimedean Solids. Ever hear of Dürer's Pentagons? http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/pentagon.htm If I remember correctly, Squares, Triangles and Hexagons are the only regular polygons that will tile properly—though there is at least one mixture that will work—square, hexagon and pentagon mixed (That's also the pattern on Soccer Balls...) But if you will be satisfied with a rough approximation... If I had generous supplies of modeling clay and various sizes of square tiles; And if I still had the energy and obsessive powers of concentration from my youth; I think I could fiddle-faddle until I got a pretty close approximation to what I mean. {Though having to work anything out by Brute Force Empiricism is rather Vulgar...} When I was done, you might say to me: "Dude, there are small triangular gaps between your tiles." And/Or, "You used two (or more) slightly different sized tiles to get them to fit." And/Or, "Your White Squares all have a slight Negative Curvature (they're slightly Concave); While your Black Squares are the Opposite." And/Or, "Some (all) of your "Squares" aren't Square—being slightly rectangular, trapazoidal—or whatever." That's why I said that I thought that it was a Topology Problem. Given an N x N Square array, printed on one of those Imaginary "Infinitely Flexible" Topological Sheets, I have every confidence you could warp one into shape. Since I've never studied Higher Geometry or Topology though, I thought that I might be going at it the hard way. {It is also possible that a good Geometer could tell me that any sort of Clay and Tile Approximation was categorically impossible.) O yeah, getting them to stick to a toroidal or other 3-D Surface is easy, make your board metal and your pieces with a Strong Magnet in the base... Saxon Violence Quote
CraigD Posted July 10, 2013 Report Posted July 10, 2013 The beauty of having a Virtual Sphere viewable only on a monitor, the control of which passes between the players on the move...:Exclamati I gotta object in principle to the concept of “viewable only on a monitor” for chess-like game boards. Then a computer program is being used to render an image of a physically possible object, it kinda by definition isn’t viewable only on a monitor. Chess played on unusual boards topologically equivalent to a sphere could also be viewed on a 3-D physical model, or flattened onto a 2-D board with helper lines to show the less obvious edge connections. Chess played in more than 3 dimensions – I’ve never seen such a variant, but it’s geometrically easy to describe – might be considered closer to “must have a computer to see”, though they can be decomposed into multiple 3-D or 2-D boards, either with a computer, or drawn on paper or physical 3-D models. Here, computer programs are valuable, as they can be written to avoid showing contradictory/impossible states that might mistakenly occur when hand-drawing multiple boards. Any game with discrete board spaces and reasonably small number of pieces can be represented just as lists of the locations of each piece, which can be done in simple text, in a text editor or written with pen/pencil and paper. I wanted to use the idea of Spherical Chess—i.e. "Sphess" as an element in one of my SF Stories... A game that is very popular with Genetically Enhanced Geniuses (who aren't that uncommon) but all but impossible for a Non-Enhanced Human to play at all well—or even purposefully.Cool idea. :thumbs_up Reminds me of Piers Anothony’s 1970 novel Macroscope, though that only involved naturally occurring geniuses playing a pencil and paper game called “sprouts”. I'm a little puzzled by the reference to Archimedean Solids.My though was using one of the larger of these – say the 62-face rhombicosidodecahedron -- the 62-face truncated icosidodecahedron -- or the 92-face snub dodecahedron -. The truncated icosidodecahedron would be my pick, because it has no odd-number-of-edge faces, so can support straight and diagonal movement (eg: rook and bishop) more obviously. My guess is that even very smart neurotypical people would blunder confusedly trying to play chess on one of these boards, never being able to achieve a checkmate. I can imagine a natural or genetically engineered neuroatypical person with savant talents being able to promptly trounce anyone not similarly talented on one. Ever hear of Dürer's Pentagons? http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/pentagon.htmSierpinski pentagons, I studies way back in school. This is the first I’ve heard of Dürer's "Tile Patterns Formed by Pentagons”. It’s a neat bit of art-presaging-math history. If I remember correctly, Squares, Triangles and Hexagons are the only regular polygons that will tile properly—though there is at least one mixture that will work—square, hexagon and pentagon mixed (That's also the pattern on Soccer Balls...)The 5 Platonic and 13 Archimedean Solids I cited above are all of the possible convex polyhedra that can be made of regular polygons, excluding prism and antiprisms, of which there are an infinite number. A standard soccer ball is the 32-sided truncated icosahedron, one of the Archimedean solids. If you relax the requirement that the polygons be regular – have all angles equal – there are an infinite number of ways to tile a sphere. Just pick a collection of vertexes on it, then draw a collection of line connecting the vertexes. :) I’ve not played with this idea much, but it sounds pretty fun, so I expect I will. Everyone’s invited to join in and share – the more the merrier! Quote
SaxonViolence Posted July 10, 2013 Author Report Posted July 10, 2013 Points well taken. If the rules say that you play on a monitor and are not allowed to use either pencil drawn diagrams or flattened boards to aid your visualization and planning, then the fact that they're possible doesn't change the basic idea. I also thought of each player having his own Virtual Sphere... Like while I'm waiting for you to make your move, I can rotate my own Virtual Sphere however I chose, to study the situation. I decided that while there may be two monitors, there is only one Virtual View and while your opponent has the move you are a passive spectator to his manipulations. Goes along with stressing memory and visualization. They may have discovered some of the Equations that define "Good Games" But it's my understanding that one creates a game and he can't tell if it will be very fun to play, totally boring or impossibly complex... And there are games with built-in ambiguities that occasion much arguing every time they're played. {Though there are folks who seem to have a great intuitive talent for designing games.) But I understand that the only way to really tell if a game is good is to persuade multiple people to try it—and perhaps give feedback. O yeah, while I made an aside into Polyhedra with my mention of the Soccer Ball, I was briefly digressing into 2-D with my talk of Squares, Triangles and Hexagons making regular tilings. I know that there is one way to mix Squares, Hexagons and Pentagons to make a Tiling. Quilters often use it. Off the top of my head I don't know if there is any other way to mix polygons (not polyhedra) to get 100% coverage with a regular repeated tiling... Nor do I know why it should matter. Saxon Violence Quote
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