I'm now posting the rules for a sub-variant of "Chess on an Infinite Plane:

"Chess on an Infinite Plane with Huygens Option"

Introduction: "Chess on an Infinite Plane with Huygens Option" is a chess variant played on a chessboard of infinite size. This version uses Fergus Duniho's innovative and abstract piece images. The game includes two huygens for each color. Huygens can jump 5, 7, 11 and other prime numbers of squares in orthogonal directions (i.e. 5, 7, 11, 13, 17, 19, 23, 29, 31...). Since it jumps in prime numbers, other jumpers such as the knight and hawk, need to make an inefficient maneuver when pursuing a huygens, causing wasted moves. The huygens do not add a lot of strength to the army, but may have some effect on play. They are initially located to protect one of the pawns in each of the rearward formations (jäger units). Thus, it is safer for the hawks to leave their positions and join other battles. (The huygens is named after Christiaan Huygens, a Dutch mathematician who studied the rings of Saturn and invented the pendulum clock). As with the base version of this game, other new pieces are included not just for variety, but to help make enough material available in the endgame so that one player can force a win. Since there are no borders, the king cannot be trapped in a corner. The chancellors (R+N) provide immediate ability to create long distance attacks. There are also additional pawns, including those in the jäger units, which can create a threat of additional promoted pieces. A hawk is centered in each jäger unit, initially protecting the rearmost pawns, but can leave its formation and enter into spectacular play. Each player also has two guards, which can be used as king defenders or for other strategic play. Setup: Pieces: Black and White each have the following pieces: 1 king, 1 queen, 2 chancellors, 2 rooks, 2 bishops, 2 knights, 2 guards, 2 hawks, 2 huygens, and 22 pawns. All pieces move as in classical chess, with the "extra" four piece types moving as follows: Chancellor - Moves and captures as rook + knight.
Hawk - Leaps exactly 2 or 3 squares in any orthogonal or diagonal direction. The leaping move means it can jump over other pieces.
Guard - Moves and captures the same as a king but is not affected by check.
Huygens - Jumps 5, 7, 11 and other prime numbers of squares in orthogonal directions (5, 7, 11, 13, 17, 19, 23, 29...).
Pawns play the same and promote at the same rank as in classical chess. (Using the board coordinates shown in the diagram, white pawns promote at rank 14, and black pawns promote at rank 7). Pawns can promote to chancellor, hawk, or guard in addition to queen, rook, bishop, or knight. Pawns may capture en passant with the same rules as in classical chess. There is no castling. There is no fifty-move rule. Draws can only occur from stalemate, threefold repetition, agreement, or a proven case of insufficient material to force checkmate. All other rules are the same as in classical (FIDE) chess. The Chessboard: 1. Board for OTB Play: A playing area should be setup with at least 22 ranks and 20 files. Ensure provisions are available to expand the board if play requires. If this becomes inconvenient due to far-away pieces, a display board is used to indicate the location of remote pieces. If there is interesting play in small but remote areas, other playing areas can be labeled and used separately from the main board. 2. Diagram for Online Play: A chess diagram is used to indicate the position of pieces either after each move by white, or each move by black. The diagram should include 22 ranks and 20 files. If any pieces are moved outside of this area, the diagram is expanded or notes are shared to indicate the location of far-away pieces.  If there is interesting play in small but remote areas, other diagrams can be used to show piece positions separately from the main diagram. Ranks are numbered just as in classical chess; ranks 15 and higher are deeper into black's side; ranks 6 and less are deeper into white's side. Files visible in the initial diagram are labeled a - t. From white's view, they continue to the right u - z, and then aa,ab,ac, and so on. The file immediately to the left of "a" is "phi" ("φ"). Files further to the left are -a,-b,-c, and so on to -z, and then -aa, -ab, -ac, and so on. For convenience, the "φ" file can be written "0". (The zero file = "φ" = "0") Parenthesis are used around each square identification. Some examples of move notations are as follows (view from white): 1) A rook moving to the left from e8 to a8: R(e8)-(a8) or R(a8) (With a classical board, this would be "Ra8") 2) A rook moving to the left one square from the a8 square: R(a8)-(φ8) or R(φ8) or R(08) 3) A rook moving to the left three squares from the a8 square: R(a8)-(-b8) or R(-b8) 4) A rook moving vertically three squares from e8 farther back into white: R(e8)-(e5) or R(e5) (The game can also be played using numerical coordinates for both files and ranks, as is usually played at chess.com) Notes: Thanks to Fergus Duniho for making Diagram Designer available, and also the abstract piece collection. Also thanks to Kevin Pacey for helping with Diagram Designer. by Vickalan. I will also be happy to consider other variations of the rules, or to play with other pieces. Please contact me if interested in playing a game.