The Book

of

3D-Chess

 

developed by
Björn Karlson, Ralph Puchta, Steffen Herrmann
 

submitted and presented at
 

Jugend forscht

Erlangen and Munich 1992
 
 

Table of contents

1. Introduction
2. Basic concept
2.1 Chessboard

2.2 Chess pieces
2.3 Moves
3. Definitions
3.1 Notation

3.2 Initial position
3.3 Moves
3.3.1 Fundamentals

3.3.2 Rook
3.3.3 Bishop
3.3.4 Knight
3.3.5 King
3.3.6 Queen
3.3.7 Pawn
3.4 Capturing and checking
3.4.1 Definitions

3.4.2 Pawn
3.4.3 Others
3.4.4 Capturing en passant
3.5 Castling
4. Construction
4.1 Components

4.2 The chessboard
4.3 The chess cubes
5. Outlook
6. Cube template


 

Copyright 1992, 1999 by Björn Karlson,
                     Ralph Puchta,
                     Steffen Herrmann
The chessboard was submitted for copyright protection at the German Patent Office (Geschmacksmusterschutz). For more information please contact one of the authors:


1. Introduction

In Spain of 1474 the game of chess, which is so widely known today, was given its final form. This game has its roots in old India, where a similar game was known as early as the 6th century. Since the middle of the 15th century no fundamental changes were made to the game of chess in its present form.

In a three-dimensional world, in which everything is constructed spatially, it takes to wonder that the venerable game of kings chess is still "merely" played in two dimensions - on a plane. The conquest of space seems sensible and necessary to us, not only from the aspect of an adaptation to our environment, but also from technical and strategic aspects of the game.
The introduction of space into the game brings dynamics and speed to the process. This may at first seem as a contradiction, but it becomes understandable with the realization that because of the immense increase in playing space a complete precalculation is virtually impossible, and the game is therefore carried by spontaneous decisions.
After the conclusion of the development of 3D-chess it became evident that our work has to be regarded as a game of its own, and not just as a further development of the ordinary, normal chess game. Contrary to all fears the game did not become more difficult through the addition of the 3rd dimension, but can be played just like normal chess after a short period of getting used to.
We wish all players many riveting matches, and a lot of fun with 3D-chess.
  
Nürnberg, March 1992                      The authors
 

2. Basic concept

2.1 Chessboard

Bild The chessboard for 3D-chess ideally consists of a cube with a side-length of eight measuring units. The spaces that can be occupied by the chess pieces are 8x8x8=512 cubes inside the chessboard with a side-length of one measuring unit. This layout is represented by the arrangement of eight regular boards as are used for normal chess. (They are aligned in such a way, that black and white squares alternate in all the three basic directions of space.)

In order to appreciate the concept of 3D-chess one has to realize that such a chessboard can not only be represented by the arrangement of eight regular boards stacked vertically (in z-direction), but just as well by eight regular boards stacked horizontally (either in x- or in y-direction).

With the concept in mind that the three basic directions of space should be regarded as equal, it becomes obvious that the pieces may move in their "normal" fashion not only on the eight physical boards, but equally on any of the virtual boards which lie at right angles to the physical boards. Indeed the chessboard could be randomly rotated by 90°-increments without affecting a game in progress, if it weren't for gravity.

The expression "normal" refers to the well-known rules of 2D-chess.

2.2 Chess pieces

The chess pieces have the form of a cube, with the symbol of the piece represented on all its six sides. These cubes admit an observation of the board from all sides, without giving a priority to the "up and down"-direction of space, as the shapes of "normal" pieces would.

The cube shape also emphasizes the concept presented in chapter 2.1, of the 3D-chessboard actually being a cube-shaped array of 512 small cubes. This board geometry can be approximated by trying to restrict the gaps between the physical boards, but of course this is limited by the size of the pieces and by the space required for a hand to make a move.

2.3 Moves

The moves allowed in 3D-chess are a logical extension of the moves in 2D-chess.

With the equality of the directions of space as a premise it is obvious that the pieces may move in any plane like they would move in the "normal" x-/y-plane. (One can simply imagine the board tilted by 90 degrees.)
A detailed discussion of the moves followes in chapter 3.
However, the enormous increase of the board from 64 squares to 512 cubes made amendments to the rules of some pieces' moves inevitable. The Pawn became much stronger by allowing partial "sideways" movement, and capture on the internal diagonals.
Also, regarding promotion it has become much more interesting and dangerous, as it can be promoted on the whole plane opposite its starting position and orthogonal to its primary direction of movement.

Example:
The pawns are located on the alpha-plane, on the line A2 ... H2 and move in the direction towards A8 ... H8, respectively. In this setup they may be promoted on the whole plane with a y-coordinate value of 8.
 

3. Definitions

3.1 Notation

The system of cartesian coordinates of the conventional chessboard-plane (with x-axis from A to H and y-axis from 1 to 8) has been extended by a z-axis; for the notation this axis is fitted with the greek letters from alpha to theta.

3.2 Initial position 

Any positioning whose orthogonal projection onto any one of the base planes shows the initial position of 2D-chess is allowed. This may be independent from the opponent's positioning and his chosen projection-plane.

3.3 Moves

3.3.1 Fundamentals

The movement of a piece is described by a vector of movement v which can be expressed as the difference between the positions of the piece before and after the move and is a linear combination of the basis vectors.

Or as a formula: Formel

Basic rules:

  • Not all three coefficients of the linear combination may be zero, for this would not constitute a move.
  • Moves which extend outside the boundaries of the board are illegal.
  • Any position may only be occupied by one piece at a time.
  • The vector of movement of any piece but a knight may not pass through a position occupied by another piece.

    • 3.3.2 Rook

    The rook may be moved in such a way that its vector of movement is linearly dependent on exactly one of the three basis vectors.

    3.3.3 Bishop

    The bishop may be moved in such a way that its vector of movement is linearly dependent on exactly two of the three basis vectors, with the coefficients of the linear combination being equal.

    3.3.4 Knight

    The knight may be moved in such a way that its vector of movement is linearly dependent on exactly two of the three basis vectors, with any two of the coefficients of the linear combination having the values 1 and 2.

    3.3.5 King

    The king may be moved in such a way that its vector of movement is linearly dependent on all three basis vectors, with the coefficients of the linear combination having values of -1, 0, or 1.

    3.3.6 Queen

    The queen may be moved in such a way that its vector of movement is linearly dependent on all three basis vectors, with any coefficient of the linear combination having the value of another coefficient or the value 0.

    3.3.7 Pawn

    The pawn commands a plane given by its "forward-vector" and its "altus-vector", the forward-vector being the basis vector which points from the officers' rank to the pawns' rank in the projection of the initial position. The altus-vector is the basis vector which is orthogonal to the chosen projection plane. Altus- and forward-vector are the same for all pawns of one side, but can be different to those of the opponent's pawns.

    A pawn may be moved in such a way that its vector of movement is linearly dependent on either its forward-vector or its altus-vector. In case of the forward-vector the coefficient of the linear combination must be positive, in case of the altus-vector it can be positive or negative. The value of the coefficient is usually 1, except when the piece makes its first move, where it can be 2.

    3.4 Capturing and checking

    3.4.1 Definitions

    Since the object of the game is identical to that of 2D-chess, the opponent's pieces can be captured by conquering their position, and check is given by threatening to capture the opponent's king. Likewise, a king is checkmated when it cannot be saved from this threat by a lawful move.

    And of course, like in 2D-chess, a player may not capture his own pieces.

    3.4.2 Pawn

    The pawn may capture like the king is allowed to move, except that it may not capture where it itself may move (cf. 3.3.7) and also not in any direction with a component in the negative direction of its forward-vector.

    3.4.3 Others

    Like in 2D-chess the vector of capture follows the same rules as the vector of movement of the respective piece.

    3.4.4 Capturing en passant

    Since the pawns assume a new tactical role through their gained powers and are not just pieces of protection and sacrifice in front of the king, the en-passant-rule can be dropped.

    Furthermore practice has shown, that due to the high increase of available space en-passant-situations almost never occur.

    3.5 Castling

    With some restrictions a definition of castling in three dimensions could be found, however it does not appear to make sense, since the main objective - to protect the king behind pawns - cannot be fulfilled adequately in space.

     

    4. Construction

    The following shall give an instruction for building a 3D-chess in all its detail all by yourself.

    4.1 Components

    For a correct assembly the following parts are needed. All listed parts are usually available at your local hardware store. Most parts need additional work.
    either or

    4.2 The chessboard 

    Parts: wooden board, acrylic boards, threaded rods, nuts, glue, marker
    In the end the eight boards should form an evenly constructed tower.

    4.3 The chess cubes

    Parts: sheets of paper/cardboard and glue, or 2 x 2 cm pieces of wood and saw, paints, needles, styrofoam

     

     
     
     
     
     
     
     
     
     

    There are two ways to make the chess cubes:
    a. chess cubes of paper/cardboard
    b. chess cubes of wood

    for a.

    for b. When the pieces are finished you have a complete 3D-chess in front of you, and can start playing.

    5. Outlook

    During our work on the rules of 3D-chess we realized that the vast gain of playing space over 2D-chess, allows not only for interesting moves and combinations within the above described rules, but also lends itself to the invention of new rules.

     

     
     
     
     
     
     
     
     
     

    When confronted with a chessboard of 512 positions, one of the first thoughts may be the invention of additional pieces. Here are a few ideas for pieces that crossed our minds - some sensible ones as well as some not so serious ones:


    Certainly, other possibilities of further development exist:

    It just goes to show that the old chess, the game of kings, doesn't have to be old-fashioned, despite its old tradition. Whatever becomes of a game of chess, or indeed what becomes of the game of chess itself, lies solely within us, the players.

    6. Cube template

    Würfelschablone
    Cut along the outline and along the four strongly marked lines.

    The squares not marked with an X are used for the pieces' symbols.
    When copying or printing the template, pay attention to the correct size:
    The assembled cubes must not be larger than the squares on the chessboard.

    © WWW-version 1996 by Steffen Herrmann
    English version © 1999 by Björn Karlson

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