The Book
of
3DChess


developed by
Björn Karlson, Ralph Puchta, Steffen Herrmann
submitted and presented at
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 threedimensional 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 3Dchess 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 3Dchess.
Nürnberg, March 1992 The authors

2. Basic concept
2.1 Chessboard

The
chessboard for 3Dchess ideally consists of a cube with a sidelength of
eight measuring units. The spaces that can be occupied by the chess pieces
are 8x8x8=512 cubes inside the chessboard with a sidelength 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 3Dchess one has to realize that
such a chessboard can not only be represented by the arrangement of eight
regular boards stacked vertically (in zdirection), but just as well by
eight regular boards stacked horizontally (either in x or in ydirection).
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 wellknown rules of 2Dchess.
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 3Dchessboard actually being a cubeshaped 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 3Dchess are a logical extension of the moves in 2Dchess.
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/yplane. (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 alphaplane, 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 ycoordinate value of 8.

3. Definitions
3.1 Notation

The system of cartesian coordinates of the conventional chessboardplane
(with xaxis from A to H and yaxis from 1 to 8) has been extended by a
zaxis; 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 2Dchess is allowed. This may be independent
from the opponent's positioning and his chosen projectionplane.
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:
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 "forwardvector" and its "altusvector",
the forwardvector being the basis vector which points from the officers'
rank to the pawns' rank in the projection of the initial position. The
altusvector is the basis vector which is orthogonal to the chosen projection
plane. Altus and forwardvector 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 forwardvector or its altusvector. In case of
the forwardvector the coefficient of the linear combination must be positive,
in case of the altusvector 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 2Dchess, 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 2Dchess, 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 forwardvector.
3.4.3 Others
Like in 2Dchess 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 enpassantrule can be dropped.
Furthermore practice has shown, that due to the high increase of available
space enpassantsituations 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 3Dchess 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.

1 wooden board, 40 x 40 cm, 2 cm thick

7 transparent acrylic boards, 37 x 37 cm, 0,5 cm thick

4 threaded rods 8 mm metric x 1 m (or comparable)

36 nuts 8 mm metric (or comparable, depending on the rods)

1 hotglue gun (or comparably strong glue)

1 magic marker, not watersoluble (preferably black)
either

per player several sheets of rigid paper or thin cardboard, a different
color for each player

scissors

glue for paper
or

per player a 2 x 2 cm piece of wood, 35 cm long (preferably no hardwood)

a saw

per player a different paint for colouring the cubes

another paint, nontransparent and clearly visible on a background of the
above paints

several long needles to prop up the cubes for painting

a piece of styrofoam (about 1 cm thick and 50 cm long)
4.2 The chessboard
Parts: wooden board, acrylic boards, threaded rods, nuts, glue, marker

First, all the boards need to be drilled in the four corners in such a
way that the threaded rods may fit through when the eight boards are put
squarely on top of each other. It is important to drill all the way through
the acrylic boards but to create only tapped holes of a depth of 15 mm
in the wooden board.

Then a chessboard
pattern is drawn on each board by leaving the white squares blank and using
a light hatch for the "black" squares. It is best to begin with the wooden
board, which is used as the bottom level, because care has to be taken
that squares lying on top of each other always alternate in color.

After drawing the chessboard pattern on all the boards they are marked
with the greek letters alpha through theta (with the wooden board being
alpha). Then one baseline on the wooden board is marked with the latin
letters A through H and the other baseline with the numbers 1 through 8.

Now the four tapped holes of the wooden board, are treated with a generous
dab of hotglue and the four rods are glued into the holes. Then the board
is put on the floor.

Now a nut is screwed onto each of the rods so that it rests about 5 cm
above the wooden board, and the first acrylic board is placed on top. This
procedure is repeated for the remaining six boards, while taking care to
always put alternating colors on top of each other. The topmost board is
fixed with an additional set of nuts on top.
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 making chess cubes out of paper or cardboard a template is used. The
cube template is shown at the end of this text. It has
to be copied onto a piece of cardboard.

For each player the material for 16 cubes of the same color can be cut
out of rigid paper or cardboard by following the outline of the template.

Now the squares not marked with an X can be decorated with the symbols
of the respective piece. This can be done by simply drawing the symbols
on, or by copying them out of books and glueing them on. The pawns remain
unmarked.

Four additional cuts must be made along the strongly marked lines on the
template, the remaining lines are marks for folding.

The cube is assembled so that that all the unmarked squares (which may
be showing symbols) are on the outside. So, using a glue for paper, the
squares marked with X are being glued to the underside of the symbol squares,
thus stabilizing the cube.
for b.

For making chess cubes out of wood, one of the described pieces of 2 x
2 cm wood is needed for every player.

The wood is cut into cubes using the saw.

The cubes are painted. For this a needle is pushed into a cut face of every
cube (because the cut faces are usually softer than the outside faces of
the original piece of wood). Now the cubes can be painted and then stuck
into the styrofoam to dry.

After the paint has dried, the symbols of the pieces can be drawn or glued
onto the six sides of the cubes. The pawns remain unmarked.
When the pieces are finished you have a complete 3Dchess in front of
you, and can start playing.
5. Outlook
During our work on the rules of 3Dchess we realized that the vast gain
of playing space over 2Dchess, 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:

Field marshal
It may move like the king, but may also move a distance of two squares.

Skateboarder
It may move like the queen, but may roll over (capture) any number
of pieces in his path, even those of his colour. The position after his
move is the position of the last captured piece.

Alchemist
It may move like the king, but captures by blowing up enemy pieces
at a distance of three squares.

Secret agent
It may move like the knight, except for the factors of linear combination
for its vector of movement having the value 2 and 3. After capturing he
assumes the type of the captured piece.

Fairy
It may move like the queen, but may move a distance of four squares
at most. However it may also remain on its position and turn into another
piece.

Medic
It may move like the pawn, but may neither capture nor be captured.
On reaching a position of promotion, it "saves" a captured piece of his
choice.
Certainly, other possibilities of further development exist:

Extension of the rules to the 4., 5., ..., n. dimension

Playing with three players, since 2Dchess is played by two players

Playing with even more players

Playing with a double set of pieces (with the second king being replaced
by another piece, e.g. by one of the above suggestions)

Playing with three sets of pawns to protect the higher ranks in all directions.
It just goes to show that the old chess, the game of kings, doesn't have
to be oldfashioned, 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
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.
© WWWversion 1996 by Steffen
Herrmann
English version © 1999 by Björn
Karlson
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