Chess on the Dot

Chess on the Dot is a chess variant invented by Joshua Chao in 2010, which uses a total sphere (not a sphere-like cylinder) as chessboard with standard rules of chess. The digital version of "Chess on the Dot" won the Red Dot design award (Germany) in 2010.
All rules of chess can be applied to Dot without change because the surface of Dot is essentially a 2-dimensional membrane (without borders), and all squares on it can be displayed as a 2-dimensional, flat and rectangular chart with correct relative positions.
Dot
"Dot" is the name of the spherical chessboard. It has 64 squares (32 white and 32 black squares) on its surface, the same as a standard chessboard. The surface of Dot is border-less, thus every route on it is an infinite loop.
Squares on the surface of Dot are technically not squares anymore because they have unequall sides, but the term remains in this article for convenience.
Transmogrification
Dot is transmogrified from the traditional, flat chessboard by the following three steps:
1) Connect east and west borders.
2) Cover north and south ends.
3) Squeeze and bulk it up to a sphere.

This process is similar to transmogrifying a Mercator projection world map into a terrestrial globe.
Diagram
To exhibit all parts of Dot on a diagram, the first step is to divide it into two hemispheres. Then use "Azimuthal equidistant projection" to project them as two circles and make them connected by single point because their borders (the equator of Dot) are actually full-connected. In this diagram, the upper circle is the north hemisphere, and the lower circle is the south hemisphere.
Dimension
Although Dot is a 3-dimensional object, which has height, width and depth, however the surface of Dot is just a 2-dimensional membrane (without borders). It needs only two sets of value to coordinate every single square on it.
Poles
The two ends labeled "N" and "S" are the two poles on Dot. They are not regular game squares, neither are they white or black. They exist to complete the surface of Dot, and keep all squares four-sided. Since the two poles are not game squares, pieces must not stay on the poles. And as the poles are meant to complete the surface of Dot, pieces may pass the poles when they move.
Polar circle
The 8 squares alongside a pole are jointly called a "polar circle". This term will be used later in discussion of the "castling" and "pawn promotion" rules.
Origin of name
The name "Dot" originates from a photograph of planet Earth taken in 1990 by the Voyager 1 space-probe from a record distance of about 3.7 billion miles from Earth, which is called "Pale Blue Dot".
Rules
All rules of "Chess on the Dot" are exactly as the standard rules of chess.
Starting position
When the chessboard transmogrifies into the Dot, all pieces stay on the very same squares as they are. The starting position is shown as the following diagram.
Move and capture
The diagrams below show the application of the standard rules of chess to the Dot. On the diagrams, little black dots at the center of squares mark the piece's destinations, lines between piece and dots mark all routes from piece to its destinations.
King
The king can move only one square along any rank (latitude), file (longitude) or diagonal. It moves the same way when capturing an opponent's piece.
Queen
The queen can move any number of squares along any rank (latitude), file (longitude) or diagonal, but may not leap over other pieces. It moves the same way when capturing an opponent's piece.
Bishop
The bishop can move any number of squares along any diagonal, but may not leap over other pieces. It moves the same way when capturing an opponent's piece.
Knight
The knight can move to any of the closest squares that are not on the same rank (latitude), file (longitude), or diagonal, thus the move forms an "L"-shape: two squares along rank (latitude) and one square along file (longitude), or two squares along rank (latitude) and one square along file (longitude). The knight is the only piece that can leap over other pieces. It moves the same way when capturing an opponent's piece.
Rook
The rook can move any number of squares along any rank (latitude) or file (longitude), but may not leap over other pieces. It moves the same way when capturing an opponent's piece.
Pawn
The pawn may move forward (approach to opponent's pole) to the unoccupied square immediately in front of it on the same file (longitude), or on its first move it may advance two squares along the same file (longitude) provided both squares are unoccupied. The pawn may not move backwards (away from opponent's pole).
The pawn may capture an opponent's piece on a square diagonally in front (approach to opponent's pole) of it on an adjacent file (longitude), by moving to that square. The pawn may not move to these squares if they are vacant.
Special rules
Castling
This consists of moving the king two squares along the "polar circle" toward a rook and then placing the rook on the last square the king has just crossed.
En passant
A special pawn capture which can occur immediately after a player makes a double-step move from its starting position, and an opponent's pawn could have captured it had the pawn moved only one square forward (approach to opponent's pole).
Pawn promotion
The transformation of a pawn that reaches its opponent's "polar circle" into the player's choice of a queen, knight, rook, or bishop of the same color.
Gameplay
"Chess on the Dot" requires players to be more observant since the Dot is a border-less sphere and there are more chances for both players to initiate attacks. While it takes longer to think, the game can possibly end in a shorter time.
Geometry
What make the Dot a total sphere (not a sphere-like cylinder) is its true status of spherical geometry. If an piece on the surface of Dot moves along longitude, latitude or diagonal and keeps its route straight, it will move along a loop and eventually back to its starting point. There are eight longitudinal loops, four latitudinal loops, and eight diagonal loops on the surface of Dot.
Without contradiction, all squares on surface of Dot can be displayed as a 2-dimensional, flat and rectangular chart with correct relative positions. It takes seven steps to make this chart:
1) Replicate infinite quantity of chessboard and make them all West-East connected stream of chessboards.
2) Duplicate the stream.
3) Perform horizontal reflection on a stream to make it only North-South reversed (but not East-West reversed).
4) Shift a stream by 4 squares to align rank "a" with "e", "b" with "f", "c" with "g", and "d" with "h".
5) Connect south borders.
6) Duplicate them all, and connect north borders.
7) Replicate infinite quantity of them all, and connect all north borders to eventually create a 2DFARCSOD which is an infinite continuum.
See this article to get more information.
Misinterpretations
The most common misinterpretation about "Chess on the Dot" is: "The game will over in very few turns because there is no borders on the Dot and powerful pieces (i.e., queen, bishop and rook) will easily put the opponent's king in checkmate." This is not the truth; Although the surface of Dot is border-less, valuable pieces are completely surrounded and protected by their pawns, just like they are on the standard chessboard.
Another common misinterpretation is: "When a piece pass the East border it will arrive the West border. Likewise, when a piece pass the North pole it will arrive the South pole." The truth is: When a piece pass the North pole it will move away from North pole and approach to South pole, when it pass the South pole it will move away from South pole and approach to North pole. The situation is identical on planet Earth.
Conclusion
The only difference between chess and "Chess on the Dot" is the number of borders; chessboard has four borders, and "Dot" has zero borders. All rules, pieces and everything else are exactly the same. That makes "Chess on the Dot" a unique chess variant.
Implementation
Physical
A physical implementation of "Chess on the Dot" primarily made of Fraxinus americana (name of wood), with axis, rings and base for rotatability and magnets to provide "gravity" for pieces to stick on its surface. Designed and crafted by Joshua Chao.
Digital
The digital implementation of "Chess on the Dot" on iOS device. Designed and rendered by Joshua Chao.

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