List of Figures.

• Unicorn. [Vignette.]
• Knight (Nullhorn), Unicorn, Giraffe (Zweihorn). [Vignette.]
• Fig. 1. Cubic Chess Space, Four-Board.
• Fig. 2. Cubic Chess Space, Eight-Board.
• Fig. 3. Square Chess Cell.
• Fig. 4. Cubic Chess Cell.
• Fig. 5. Movement of the 3 Basic Pieces.
• Fig. 6. Movement of the Knight.
• Fig. 7. The 8 Raumschach Pieces of Practical Play.
• Fig. 8. Playing Field with Square Notation, Five-Board.
• Fig. 9. Movement diagram of all pieces of the 1st–3rd Shell, in the Seven-Board.
• Fig. 10. Initial arrangement of pieces in the Four-Board.
• Fig. 11. The same in the Five-Board.
• Fig. 12. The same (with 4 Unicorns).
• Fig. 13. The same in the Seven-Board.
• Fig. 14. The same in the Five-Board (with forward pawns).
• Fig. 15. Four-color playing model.

Foreword.

The World War is over. The world Raumschach game — world-space chess game — finished. Lost! At first, in the development phase of the struggle, it seemed as though White, the Germans, would win the game. The opening period was brilliant. Then, in the middlegame, we would gladly have agreed to a draw. But it went on. The endgame brought the collapse. Black scraped through to promotion and won. This time. The pieces now lie jumbled again in the box. And await their resurrection. When will the “return match” come? When will White move again?? — But by then the rules of the game will have changed. The law of the conservation of force (of the pieces) in closed space leads to the democratization of the hitherto aristocratic game of chess. One will play without a king. Choose a different objective. One emancipates oneself from the board. Raumschach! . . . .

During the war I often endeavored to publish something on Raumschach. It failed. Manuscripts piled up in the desk; ideas in the head. And awaited peace. Now it is here. But in what state?! . . .

Since then: upheaval of circumstances, reorientation of perspectives — rearrangement of manuscripts, overturning of plans . . . .

Result: to publish extensive works all at once is now impossible. But one can nonetheless keep the greater goal firmly in sight, work toward it, and for the time being bring out here and there the smaller pieces. Out of sequence, yet following a far-sighted plan. Gradually joining into the whole . . . .

But one thing must be emphasized again and again: Raumschach, too, needs time. It could not stand complete in a single stroke. It has growth, has development, needs improvements. The idea is not always the main thing. Ideas are often as cheap as blackberries. Execution, translation into action, practice! That is what matters . . . .

With action I began 12 years ago. May other friends of Raumschach continue to help and lend a hand to the ideal work. So it will become perfect: our Raumschach! . . . .

But in order for Raumschach to gradually perfect itself more and more, the call to gather had to be sounded. To that end I issued the following appeal, dated Hamburg, 19 January 1919:

German Raumschach Society.

This appeal has already met with good success. May the same fortune attend the present first issue of our Raumschach work! Then further issues will soon be able to follow.

The Editor.
Easter 1919.

First Section.
Preliminary Remarks and Explanations.

1. The Nature of Chess.

Chess — including board chess — is a space problem.

The purpose and goal of a chess game is to “checkmate” the enemy “king.” Now since the K commands 3×3−1 squares on the board and 3×3×3−1 squares in space, checkmate amounts to cutting off all the K’s flight squares by means of one’s own and the opponent’s pieces — smothering him spatially, so to speak. The K is mobile, and with him his center of gravity, consisting of his available flight squares at any given moment. The K therefore serves merely as a guiding piece for a mobile spatial zone of 9 or 27 squares respectively, which one must seek to bring under one’s control. Whichever of the two players first achieves complete mastery of the opponent’s spacemate has won the game. The K is caught in the meshes of the “mating net.”

Chess is not an arbitrary, artificial game, but a natural, scientifically necessary, mathematical-mechanical game. The laws of chess have a logico-mathematical character. Artistic — that is, individual, personally arbitrary — is solely the handling and manner of applying the natural laws of chess.

Chess is therefore a skillful contest for space according to natural laws.

Through our sense organs, a space of three dimensions is naturally given to us human beings — not artificially constructed. It is therefore entirely self-evident that the pieces of so “royal” a game of movement as chess must also move in all possible directions and not merely on a plane. Board chess is — for historical and mathematical reasons — to be regarded only as a reduced form of space chess. The “primal chess” was three-dimensional; at least in conception, even if perhaps not in execution.

2. The Chess Cell.

The mathematical foundation of chess, and hence the starting point of chess thinking in general, is the chess cell. Everything depends on its form. Not only the structure of the chess space and the playing field, but also the movement, the gait, and the number of the chess pieces.

For the seamless, regular filling of space, board chess generally uses the square; Raumschach, by contrast, uses the cube (hexahedron). One can, to be sure, fill space regularly in other ways as well. In the second dimension by triangles, rhombuses, hexagons, etc. In the third dimension by cuboctahedra, rhombic dodecahedra, hexagonal prisms, etc. But the square and the cube are the simplest and most convenient spatial units — “squares” or “cells.”

Board chess is therefore built on a square basis. The structure of Raumschach is cubic, tesseral, cube-shaped, hexahedral.^*)

Draw a square. It consists of 4 sides and 4 corners. It is therefore bounded by only two kinds of geometric elements: points and lines.

Take a cube, by contrast. It is bounded by 6 faces, 12 edges, and 8 corners. Here we are dealing with three kinds of geometric elements: faces, lines, points.

A four-dimensional chess cell, a so-called tesseract, is bounded — not surrounded, but bounded — not only by 16 points and 32 lines and 24 faces, but also by 8 “spaces.”

3. The Chess Space.

The playing space, or scaccarium, of board chess conventionally consists of 8×8=64 squares, although other variants also occur here.

The terrain of Raumschach always consists of n³ cells, where n can be 4, 5, 6, 7, 8, 9, 10 … n. (Cf. Figs. 1 and 2.)

The color of the squares alternates in the third dimension as well.

Although the “Four-Board” — that is, a chess space of 4×4×4=64 cells (symbolically: S^III₄) — holds certain advantages for the beginner and can also produce very attractive games, the excessively confined terrain, the movements of the knight, and other factors bring corresponding disadvantages in their wake.

The true domain of the Raumschach player is the “Five-Board” — a chess space of 5×5×5=125 cells. It is neither too small nor too large. Its symmetry and the central square facilitate orientation. The knight can develop fully. A game on the “Five-Board” lasts on average no longer than a game on the [two-dimensional] board. In what follows we shall concern ourselves primarily with S^III₅. This will make the presentation simpler and more transparent for the beginner.

Higher demands are satisfied by the “Seven-Board” with 7×7×7=343 cells. Naturally, Seven-Board games are more difficult and longer in duration.

Playing on the “Six-Board” or “Eight-Board” is not recommended at first.

4. The Chess Pieces.

On the two-dimensional board chess, the pieces move forward and backward, right and left. In three-dimensional Raumschach, by contrast, they also move up and down — that is, in all directions.

The nature and movement of the pieces depend on the geometric structure of the chess cell. (The movement of the pieces is a function of the chess cell.)

To the familiar 6 board-chess pieces (rook, bishop; knight; queen, king, pawn), 2 new Raumschach pieces are added (for mathematical reasons): the cunning “unicorn” and the mighty “zweihorn.”

The zweihorn, or “giraffe,” finds use only in the “Seven-Board,” so that for now we have to deal with only one new piece: the unicorn.

Imagine a chess piece standing at the center M of a square cell. (Cf. Fig. 3.) Since the square possesses only lines and points as its boundaries, the piece can leave its square either through one of the 4 sides or through one of the 4 corners. If the (as yet undifferentiated) piece leaves the square through a side, it thereby becomes p. 10 the “rook” (R). If it leaves through a corner, it becomes the “bishop” (B). In board chess there are therefore two “basic pieces”: R and B. All other pieces — queen (Q), king (K), pawn (P), and knight (N) — are, or can be regarded as, combinations of these two basic pieces.

Now imagine a chess piece standing at the center of a cubic cell. (Cf. Fig. 4.) Since the cube possesses faces, lines, and points as its boundaries, the piece can now leave its square not merely in two ways but in three ways: either through one of the 6 faces, or through one of the 12 edges, or through one of the 8 corners. If the piece leaves through a face, it thereby becomes the “rook” (R). If it leaves through an edge, it thereby becomes the “bishop” (B). If finally it leaves through a corner, it thereby becomes the “unicorn” (U).

The unicorn is therefore a new chess piece mathematically conditioned and necessitated by the third dimension. Characteristic of this new piece is that it cannot move on its current level — it is inactive there. This circumstance lends the unicorn an interesting cunning and treachery not found among board pieces. Since the unicorn can only exist in three-dimensional space, it is, so to speak, the symbol of Raumschach. (Cf. title vignette.)

In Raumschach there are therefore not merely two but three basic pieces: R, B, and U. All other space pieces that we shall come to know are combinations of these three basic pieces.

As on the board, so too in space: all basic pieces move outward in straight rays as far as they wish or, p. 11 according to the rules of chess, are able. The basic pieces move along lines. They are therefore multi-step, long-range linear pieces.

Connect in your mind the center of a cube with the midpoint of a face, the midpoint of an edge, and a corner point (Fig. 4), and make clear to yourself the difference among the 3 directions. The beginner easily confuses the corner move (e) — that is, the move through corners — with the edge move (k). The face move (f) is the simplest.

Chess (both board chess and Raumschach) is therefore not only a problem of space (the positions of the pieces as energy-points in space; statics) and not only a problem of movement (dynamics), but above all a problem of direction.

Each newly added dimension gives rise to new directions. In linear chess (cell = line segment) there would be 2 directions; in planar chess or board chess (cell = square) there are 8 directions; in space chess (cell = cube) there are 26 directions; in four-dimensional chess (cell = tesseract) there are 80 directions; and so on.

In accordance with the cell structure, the rook advances first, and the basic pieces that exit through corners follow in succession: B, U, “balloon,” etc.

In four-dimensional chess, S^IV, there are therefore four basic pieces: R moves through the “spaces” bounding the tesseract, B through faces, U through edges, and the balloon through corners.

But this excursion into the fourth dimension need not detain us further here and is intended only to provide perspective and insight.^*)

After the basic pieces we come to the direct derivation of the knights. Specifically, we wish to derive them not through combination of the basic-piece steps, but “centrally.”^**)

Let us imagine that a chess piece stands at the center of a cubic cell. If we call this home cell the “kernel,” then the kernel is surrounded on all sides in the chess space by 3×3×3−1=26 cubic cells. Of these, the piece commands 6 as a rook, 12 as a bishop, and 8 as a unicorn. These 26 cells form the first shell. It is therefore completely filled by the moves of the basic pieces. (Cf. Fig. 5.)

Around this first shell lies a second shell containing 5×5×5−27=98 cells. The 3 basic pieces again command 26 of the 98 cells. There remain therefore as a residue 98−26=72 cells. These 72 cells are divided among three different knights, each with 24 moves: namely the familiar knight (N), the “zebra” (Z), and the “antelope” (A). The zebra and antelope are of purely theoretical, p. 12 mathematical interest! In practice, only the ordinary knight comes into consideration as a second-shell piece. The second shell is therefore the “geometric locus” of N.

Around the second lies a third shell. It consists of 7×7×7−125=218 cells. Of these, the 3 basic pieces again command 26 cells. There remain as a residue 192 cells. These are divided among four further knights: the “giraffe” (G), or zweihorn, with 72 moves; the “okapi” (O) with 72 moves; the “stag” (H) with 24 moves; and the “roe deer” (Rd) with 24 moves.^*) 72+72+24+24=192; 192+26=218. In practice only the giraffe comes into consideration! It is a knight raised to a higher power. A chess polyp with 72 enormous tentacles, which extend not over five planes as with the ordinary knight but over seven! The giraffe is very easy to handle in practice: a knight move followed by a unicorn or corner move — landing, nota bene, in the third shell as its geometric locus. But as easy as it is to play with it, so difficult it is to play against it. That is, the opponent’s giraffes are hard to see through, which enormously heightens the appeal of a game on the Seven-Board (for only there is the giraffe usable).

The board-chess player will come to know, to his surprise and delight, something absolutely new in the unicorn and the zweihorn (giraffe). The problem enthusiast is beckoned here by the highest pleasures of chess thinking!

The further (higher) shells [extending to infinity], with ever new chess pieces — all varieties of “knight” — are of mathematical interest only.

From all shells the 3 basic pieces cut out 26 squares each. In all remaining gaps roams that mathematical world of “knights.” —

To the basic pieces and the knights there are finally added the queen and king, which combine the movements of the basic p. 14 pieces, and the pawn. R, B, U are primary pieces, as are all knights. But Q, K, and P are secondary pieces, since they can only be derived through combinations. They therefore appear at the back in Fig. 7.

Q moves like R or B or U, and does so over multiple steps.
K likewise; but only one step.

According to Philidor, the pawns are the soul of board chess. I hold them to be the cross of Raumschach! We shall have to occupy ourselves at length with their movement and arrangement. —

For practical play, therefore, only the eight pieces come into consideration: R, B, U; N; G; Q, K, P. Since G can only operate in S^III₇, in S^III₅ we have to deal with only a single new piece: the unicorn.

After these general preliminary remarks and explanations we now turn to practical play proper. In doing so we shall return in closer detail to what is most important: the specific movement of the pieces.

Second Section.
Practical Play.

1. The Playing Field.

In Raumschach, just as each individual cell has a cubic form, so too does the entire playing space in theory. (Figs. 1 and 2.) In practical play the cube becomes a prism. The terrain is drawn apart from bottom to top so that the individual planes can be surveyed, the pieces placed between them, and moved by hand. In the process the vertical cubic cell edges disappear entirely. The squares of the planes represent the remaining base faces of the cubic cells. (Fig. 8.) The pieces are to be imagined between the planes at the center of the cell-prisms. The prismatic form of the model does not hinder play in any way!

For scientific purposes [spherical, crystal, projection, and numerical chess; knight’s tours, etc.; also for the composition and solution of problems, tasks, and studies in space] one makes advantageous use instead of cubic research models.

In cubic models one “sees” more correctly and more quickly, to be sure. But grasping and moving the pieces is impeded. Instead of horizontal planes, it is therefore better to choose a model made of rods (a grid model). But this too must be relatively large. One advantage of cubic grid models is that one is not unwittingly led to favor the planes.

For ordinary play the prismatic plane model is entirely adequate.

Anyone can easily make such models themselves.^*) For the Four-Board it suffices to cut a chess board of 64 squares into 4 parts and to fix the 4 planes one above the other at a distance of 10–12 cm in some tiered arrangement.

The Five-Board model I prefer to use has the following dimensions:

Size of individual square cells: 4×4 cm². Size of individual planes (plus border): 24×24 cm². Distance between individual planes: 12 cm. Total height of the model from the table surface: 52 cm. Thickness of the base board: 2½ cm. Thickness of the cardboard planes: 4 mm.

At the 4 corners of the base board, 4 rods of 5 mm thick galvanized iron wire are erected. The planes are held apart by brass sleeves. The entire model can therefore be dismantled and laid in a box.^**)

The question of the model is naturally of the greatest importance for players. Numerous proposals and experiments have therefore been made, concerning material (glass), method of fastening (retort stands), distance between planes (unequal spacing), overall form of the chess space (cubic, parallelepiped), etc. To discuss all proposals here^***) would go too far, especially since the model in Fig. 8 has proven itself most satisfactorily.

The main thing is that the models are stable and stand firm.

The squares can be drilled through for peg-based pieces, or one uses pieces with a broad base, with a lead and cloth underlay.

The notation of the squares is the same as in board chess, except that the designation of the plane is added, so that we obtain a triple index instead of the double index used on the board. (Fig. 8.)

The planes are designated from bottom to top by the Greek letters α (alpha), β (beta), γ (gamma), δ (delta), ε (epsilon), ζ (zeta), η (eta), ϑ (theta) … Designating the planes with I, II, III, IV … is not recommended for practical reasons.

This algebraic notation suffices for practical play. Raumschach science, by contrast, employs a coordinate notation in which the central square serves as the origin point.

When a position is to be represented diagrammatically, the simplest method is to display the various planes side by side.

Other methods (projection methods) are of scientific interest only.

In a diagram, the U can be indicated by an inverted N; the G by an N laid on its side — unless one prefers new type pieces designed as unicorn and zweihorn.

2. Movement of the Raumschach Pieces.

One can make the missing pieces oneself just as easily as the models. The knights (nullhorns) are already at hand. From the knights of another set one makes unicorns and zweihorns (giraffes). In any case, U, N, and G are inwardly closely related to one another, which justifies their common knight form.

We now wish to summarize once more, in clear overview, the number and movement of the pieces:

a) Number of Pieces.

In practice we are dealing in total with 8 pieces: R, B, U; N; G; Q, K, P.

Each player has, in the:

For complete coverage of the space one can manage with 1 rook. But one already needs 2 bishops. And only with 4 unicorns p. 18 can one cover all squares! (The initial placement of the U in S^III₅ is governed accordingly.)

Like the bishop, the giraffe is also a “color-bound” piece. One therefore needs 2 giraffes. Color-boundness greatly facilitates operating with the giraffes.

b) Movement of the Pieces (“combinatorial”).

f means: “The piece moves through the faces of the cells”; k means “through the edges”; e means “through the corners.”

I. Pieces through all shells.
R = f.  6 directions
B = k. 12 directions
U = e.  8 directions
Q = f or k or e. 26 directions.

II. Pieces of the first shell.
K = f or k or e. 26 directions. Cf. Fig. 5.
P^*) moves f; but only through the forward, upper, and lower faces (not backward and not to the sides); captures K; but only through the four forward edges, i.e., right and left, up and down (not through the remaining 8 edges). Thus P moves in 3 directions and captures in 4 directions. If P were to move in all 6 directions and capture in all 12 directions, it would become far too powerful.

III. Pieces of the second shell.
N = f + k. 24 directions.^**)

IV. Pieces of the third shell.
G = f + k + e. 72 directions.

The giraffe is a “super-knight.” It always leaps over two (horizontal or vertical) planes and can therefore jump over two pieces.^***) The knight always leaps over one (horizontal or vertical) plane. The unicorn leaps over no plane. It must, however, change its home plane.

Since the movement of the knight in space sometimes causes difficulties for the beginner, the following experiment is recommended. Following the formula “the knight moves either one square straight ahead plus two squares sideways, or two squares straight ahead plus one square sideways,” mark the 8 knight squares on the chessboard, i.e., on the horizontal plane. Then stand the plane upright on the table: 1) frontally, i.e., from left to right, and 2) sagittally, i.e., from front to back. In this way one can directly visualize all 3×8=24 knight squares and recognize that the knight in space naturally moves upward and downward in exactly the same way as on the board.

Specialized movement of the Raumschach pieces.
Assuming that the respective pieces stand on the central square γc3 of S^III₅, the following squares in the “Five-Board” are commanded from that position by the

Rook (6×2 squares):

Bishop (12×2 squares):

Unicorn (8×2 squares):

Knight (24×1 squares):

This pawn movement, the “old” one formerly always played, is designated A.

For the sake of completeness, the moves of the giraffe in the Seven-Board, S^III₇, from the central square δd4, shall also be given individually. The 24 moves of the “small” G are marked with an asterisk.

Giraffe (72 squares):

Finally, the following diagram provides a consolidated overview of the movement of all primary pieces of the first three shells, in the Seven-Board, S^III₇.

Queen (26×2 squares): like rook or bishop or unicorn.

King (26×1 squares):

White Pawn (3 or 4×1 squares):

Black Pawn (3 or 4×1 squares):

This pawn movement, the “old” one formerly always played, is designated A.

Key to the Movement Diagram.

3. Positions at the Start of the Game.

Frontal positions are customary, i.e., the white pieces stand in the frontal plane nearest to White — always in vertical plane 1 — and the black pieces stand in the frontal plane nearest to Black: in vertical plane 4 for S^III₄, in plane 5 for S^III₅, and in plane 7 for S^III₇.

Accordingly we obtain the following initial positions:

The following further remarks apply to S^III₄:

Initially the game was played with 8 pawns each, so that 4 additional white pawns stood on δ and 4 additional black pawns on α. In this arrangement, however, the power of the pawns in the small space is too great.

In Fig. 10 the white Q stands on white, following the principle “regina regnat ad colorem.” However, if a “K-side” and “Q-side” are to be formed — that is, if the polarity principle is to be maintained — then for Black the K and Q must exchange their places.

The piece N can (by agreement between the players) be handled in various ways in the Four-Board: 1) as a pure knight; 2) unicorns may be chosen instead of knights; 3) one may take one knight and one unicorn each; 4) N may be regarded in the Four-Board as a mixed piece, i.e., on the plane = N, and in space (from one horizontal plane to another) = U. The last option in particular has proven itself well in practice.

Earlier published games are to be assessed accordingly. Later we abandoned the NU (Knight Unicorn) completely.

This initial position I call the “old” arrangement, because we formerly always played this way, and designate it initial position A.

The position has only the one disadvantage that each player has only 2 unicorns p. 24 instead of 4, which are required to cover all squares with the unicorn. Attempts to place 2 additional unicorns on each side have so far failed. If, however, one were to proceed from the mathematically very sound principle of using only and exactly as many pieces of each type as suffice to reach all squares with each piece-type — namely 1 R, 2 B, 4 U, 1 N, plus Q, K, and pawns — then the following initial arrangement would very satisfactorily solve the 4-unicorn problem in the Five-Board:

The pairing of pieces, which is to be regarded as a vestige of board chess, is thus overcome here. The 4 unicorns are differently localized — that is, they can never meet one another. The preferred unicorn stands on αe1. (Cf. p. 31.) The position has only the one disadvantage that U αe1 and U εa5 are en prise from the outset. This could be prevented by placing “forward pawns” on rank β2 (and possibly also α2).

This initial arrangement “with four unicorns” I designate Position B.

The 4 U thus share coverage of the total space, just as the 2 B do. To that extent they form, as it were, a coherent body or “organism.” Since this organism is disturbed as soon as even 1 U is captured, one may as well forgo its integrity from the outset and manage with 2 U. Added to this, 4 U each will lead to many vexing en-prise situations. With 2 U one already has enough to watch! For this reason the beginner does well to begin with Position A combined with pawn movement A. This is the quickest way to the goal.

The Four-Color Playing Model.

Originally all squares in board chess were colorless — that is, white. Color is of no inherent consequence. But it facilitates orientation on the terrain and aids play. On account of the “color-bound” pieces it is of great advantage. Each player has a “white” and a p. 25 “black” bishop, as well as a “white” and a “black” giraffe.

Since there are now four different unicorns, a four-color space model would consequently be required — or at least very practical. It would facilitate the operation of the unicorns even when only 2 each are used. In such a model the U would also be color-bound. Accordingly there is a white, black, green, and red unicorn.

I have used such a colorful model to greatest advantage in the construction of cubic knight’s tours.^*) The 4 unicorns constitute the best guide-pieces for this purpose. At the same time this furnishes proof of the inner kinship between U and N.

The coloring of the squares is accordingly arranged as follows:

Unfortunately the colored model has the great disadvantage that the bishops now lose their color-boundness! The “white” bishop has become white-black and white-green and white-red; and the “black” bishop has become black-white and black-green and black-red. This corresponds to the 3 mutually perpendicular planes in space. This color-change of the bishop manifests itself, however, only when it takes a single step. As soon as it skips a square, it skips the three other colors and remains what it is — white, black, green, or red.

Nevertheless the bishop’s disadvantage is greater than the unicorn’s advantage.

One can, however, combine both models by affixing the unicorn colors in small round discs to the squares of the two-color model. —

The number of colors in the scaccarium is governed by the corner-movers of the dimension in use. The corner-movers are: in S^I the rook (1 basic piece: R); in S^II the bishop (2 basic pieces: R, B); in S^III the unicorn (3 basic pieces: R, B, U); in S^IV the balloon (4 basic pieces: R, B, U, Bl); etc. The number of colors follows the formula 2⁽ⁿ⁻¹⁾, where n = dimension: S^I: 2⁰=1-color; S^II: 2¹=2-color; S^III: 2²=4-color; S^IV: 2³=8-color; etc.

It remains to give the initial position in the Seven-Board:

The possibilities for initial positions are by no means exhausted by these. One could, for example, choose the body diagonal of the scaccarium as the maximum distance and place the pieces in the opposite corners. In that case the pawns would have to make corner moves. Or one could place White at the center of the chess space and Black at the periphery. And so on — there is still a large field here for independent activity and personal reflection.

Finally it should be mentioned that various attempts have been made to circumvent the difficulties of a satisfactory initial position.

One begins with tabula rasa — that is, with an empty playing space. The players then alternately place their pieces into the model move by move, or operate with those already present according to agreed rules.

The future may perhaps belong to this method of introducing pieces successively from outside! In particular, one could very effectively combine the “exogenous p. 27 arrangement” with the “recruitment principle,”^*) incorporating captured pieces into one’s not-yet-deployed reserves and bringing them back into play as needed. The exogenous method does, to be sure, eliminate all calculation and makes the game more of an arbitrary affair.

Another such attempt is the following: instead of first choosing the space (S^III₄, S^III₅, S^III₇) and arranging the pieces accordingly, one chooses first the pieces one wishes to play with and arranges the space (possibly non-cubic, i.e., parallelepiped) to suit them. . . . .

Further details on this can be found in the previously published Mitteilungen über Raumschach, nos. 1–7.

We adhere, however, on principle to the cubic space S^III₅ and reject configurations such as 8×8×3 squares. The true Raumschach player is a “cubist” — preferably in the model as well.

4. General Remarks on Rules of Play, Tactics, Pawns, Value of the Pieces, etc.

a) Rules of Play.

The rules of play in Raumschach are in general the same as in board chess.

Raumschach is likewise a “two-player game.” A separate game is not played on each plane; rather, both opponents command all planes simultaneously.

The goal here too is to checkmate the enemy king — that is, to cut off all remaining terrain to which he might still escape. The tendency — of board chess as well — is therefore a “spatial” one. Board chess too is — reduced — “Raumschach.”

The frontal initial position has the consequence that the opponents immediately face each other openly from the very start of the game. All pieces, including all officers, can strike immediately — not, as in board chess, merely the knights. This gives the opening phase of the game, the development of the pieces, an entirely different character in space than on the board.^**)

Check can be given immediately, with the very first move — often many times in succession: 10, 15 times or more,^***) provided the enemy king does not evade “asymmetrically.” This “pre- p. 28 mature” check-giving is not yet aimed at checkmating the king at all, but serves solely to “develop” one’s own pieces while the opponent is forced to make nothing but compelled moves. It also makes him nervous and uncertain, since it is much easier to give check than to defend against it. The offensive in space is therefore very sharp.

The initial double pawn step and castling do not exist in space. Such aids to development are unnecessary here.

It is, however, expedient to retain pawn promotion. The black pawns promote on rank α1; the white pawns on the diagonally opposite rank — that is, on δ4 in S^III₄, on ε5 in S^III₅, and on η7 in S^III₇.

Thorough knowledge of the moves is of course the first requirement for play. Never confuse the edge move and the corner move! Watch out for the enemy unicorns!! Even from their starting position! Always monitor the positional relationship of K to Q, so that N and G cannot suddenly announce “check-gardez”! For N has an action radius spanning 5 planes, G even 7 planes! With G, color-boundness protects against surprises. Guard your own pawns and avoid, with G except in special cases, exchanging queens. Try to drive the enemy K toward boundary faces, edges, and into the 8 corners of the space. Note the weak points αb2 and εd4 respectively.

Precise knowledge of the terrain is also very important. The alternating color of the squares in conjunction with the color-boundness of certain pieces (B, G) facilitates orientation just as much as the perfect symmetry of the odd-celled spaces [planes, axes, and center of symmetry; edges and corners of the octahedron, tetragonal double pyramid, tetrahedron, and cuboctahedron inscribed in the space]. There are in space (mathematico-crystallographic) praedilection points — “lines” along which the contest preferentially plays out, which one seeks to reach and hold with the queen and to avoid with the king. Practice and experience soon teach one to do the right thing here.

Begin playing immediately and let nothing discourage you! The game in space is much easier and simpler than one first thinks. The initial effort will richly repay every chess enthusiast!

Even though the principle of Raumschach is the same as that of board chess, one must be warned against carrying over without further thought the knowledge acquired in board play p. 29 to Raumschach. On the contrary! The more one frees oneself from board-chess conceptions, the more sharply one thinks stereometrically from the outset instead of planimetrically, the better. The stronger a person’s spatial intuition, the better and more accurately he will play. Raumschach is an entirely independent game for which knowledge of board chess is not required. Away from the board! Into free space! Up into the chess aether! Let that be the watchword.

It might even be advisable to leave behind all earth-bound thoughts and board-heavy conceptions entirely — for instance: to dispense with all combination pieces (queen, king, pawn); to prevent knights in space from leaping over pieces;^*) to remove pawns altogether, or at least to omit pawn promotion; not to eliminate pieces by “capturing” them but to retain them in play on the opponent’s side; to choose a different conclusion to the game than checkmate of the king; and so on.

There are therefore without doubt many revolutionary ideas contained in Raumschach, whose realization is not yet timely for the moment.

b) Pawns.

If one regards the two promotion ranks as the maximum distance of the opposing pieces, then the objection frequently raised appears justified: that the white pawns should not be permitted to move and capture downward again, nor the black pawns upward again. This, it is argued, is an inadmissible “backward” movement. Thus a white P on γc1 should be permitted to move only forward to γc2, upward to δc2, and downward to βc1; and to capture only forward to γb2 and γd2, and upward to δc2 — but not downward to βc2. A black P conversely in the opposite directions. This “reduced” movement we shall designate B.

Against this the following must be urged:

1) The opposing pieces stand in frontal planes. The movement of the opponents from bottom to top and vice versa is therefore neither “forward” nor “backward,” but a neutral movement. (This is especially true when promotion planes are used.)

2) If the pawns were permitted to move in only 2 directions and capture in only 3, instead of moving in 3 and capturing in 4, they would be too weak in space. Especially since there are few pawns to begin with, and in my opinion the game does not go as well with a second pawn rank (i.e., 8 pawns each in S^III₅ and 10 pawns each in S^III₇) above p. 30 or below the first rank, or with “forward pawns” in front of the pieces. The latter drag out the game.

3) Twelve years of playing experience have shown that the 3-fold move direction and 4-fold capture direction of the pawns is best. The forces are well balanced in space under this arrangement.

4) In any case, the so-called backward movement of the pawns is rarely made use of (in the interest of promotion possibilities). In the majority of cases the reduced movement is in practice the only one that occurs. The extended movement has, to be sure, influenced many moves in conception without ever having made its appearance.

5) One cannot warn emphatically enough against carrying over the conceptions and rules of board chess directly to space chess. Raumschach is a game in its own right! It requires stereometric thinking. No translations and transpositions from the plane! I shall show in the “History of Raumschach” how greatly the development of Raumschach was hampered by board-chess conceptions, and how difficult it has been to emancipate oneself from the board gradually. For this reason it might also be advisable to dispense with a “promotion rank” altogether. That would resolve the backward question by itself.

6) If it must be conceded that certain chess problems can only be solved when pawn movement A is altered, then one would simply have to make a change for the sake of the problems, while retaining the extended movement A for the game itself. Or else one must simply not transfer problem ideas from the board to the space! Especially since there is an extravagant abundance of new material here for the problem composer! For these counter-reasons a so-called backward movement of the pawns by agreement seems to me permissible. (For still other movement types, see below.)

c) Piece Values.

The value of the pieces in space is substantially different from that on the board — for geometric reasons, since the movement of the pieces depends on the structure of the cubic cells. The general scale is: Q — G — N — B — R — U — P.

It is not correct to base the assessment of value solely on the number of squares a piece can reach, or on the number of its directions of movement. This is why we have also ranked G below Q and U below R. But the circumstance that a piece can reach all squares must not be the sole criterion either. We have nonetheless placed R rather far down. Playing experience has shown that the rook is cumbersome — especially at the start of the game. In the endgame, however, R has very great value for preventing pawn p. 31 promotion. Composers of Raumschach problems are admittedly inclined to rate R more highly and rank it above N.

Also speaking against the rook in space is the following interesting calculation, which we take from the Deutsche Schachzeitung of 1894, page 93: “If one places the knight in turn on each of the 64 squares, it commands 2 squares from a1, 3 squares from b1, etc.; total for all 64 squares = 336. If one then places the bishop on each of the 64 squares, it commands 7 squares from a1, 7 squares from b1; total for all 64 squares = 560. If one finally does the same with the rook, one easily sees that its domain is 14 squares everywhere; total for all 64 squares therefore 64×14 = 896. But knight and bishop together also command 336 + 560 = 896 squares.”

Performing the analogous calculation in space, one obtains:

With 64 horizontally arranged squares (board chess, S^II) the R therefore commands as many squares as B and N together. By contrast, with 64 cubically arranged squares (chess space, S^III₄) the R commands only as many squares as N alone. In space, then, the power and value of R decreases, while that of N and B increases.

If for S^III₅ only this calculation is applied, one obtains the scale for S^III₅: B — R — N — U.

It is interesting that among the 4 U there is a “preferred unicorn.” Each U commands exactly ¼ of the terrain only in odd-celled chess spaces (S^III₃, S^III₅ …). In odd-numbered spaces (S^III_E, S^III₇ …) one U — namely the one that can pass through the central square — commands a slight surplus of squares. In the Five-Board the ratio is 30+30+30+35=125; in the Seven-Board 84+84+84+91=343.

Incidentally, in odd-celled spaces there is also a “preferred bishop,” which likewise passes through the central square. In S^III₅ the ratio is 63:62; in S^III₇ it is 172:171.

The question of the ratio of pieces to squares also belongs to the problem of piece values.

The board-chess ratio of 1:2 changes completely in space. One goes into the game here with relatively few pieces, because the power of the pieces increases greatly in space. For it is less the absolute number of squares that matters than the number of exits from them. In S^II₈ the 64 squares together have 420 exits; in S^III₅ by contrast 936!

With 420 board exits and 32 board pieces we obtain the ratio 420 : 32 = c. 13.

With 936 space exits and the ratio 13, the result would be 936 : 13 = 72 pieces! This is absurd, since there are only 64 squares.

This example thus proves that the greatest mistake in Raumschach is to model oneself on board chess. Raumschach is an entirely independent game with its own laws and rules. This cannot be emphasized often enough.

For mathematical reasons it would be most correct to use only the 7 pieces that also belong together chess-crystallographically: R (f), B (k), U (e); N (f + k), Z (f + e), A (k + e); G (f + k + e) (large giraffe). [Thus not: Q, K, P.]

In this case R, B, U would have to function only as short-steppers. U moves through the corner into the 1st shell; Z through corner + rook move into the 2nd shell; A through corner + bishop move into the 2nd shell; G through corner + knight move into the 3rd shell.

5. Game Openings.

A. Some Opening Lines and Practice Moves in the Four-Board.

The following sequences illustrate typical opening courses in S^III₄.

B. Main Openings in the Five-Board.

Opening theory is “the study of the purposeful development of forces at the start of the game.” The following survey does not claim to withstand rigorous analysis of the kind that has gradually come about for board-chess openings through the efforts of many outstanding chess players. Rather, our presentation is intended only to give the beginner some opening moves for the game on the Five-Board.

We divide the openings into three classes:

A. Pawn openings;
B. Officer openings;
C. Mixed openings.

In class C, White begins with a pawn or an officer and Black may follow with an officer or a pawn. Since the moves listed under A and B will in most cases recur, we leave C aside here.

A. Pawn Openings.

^*) It is better to advance N, B, or Q at this point, in order to forestall Black’s B or Q.

^*) Or a move with P, N, or R.

^*) Or K εc5 – δd4.   ^**) Or Q βc1 – εc1 †

^*) If the K evades, check follows immediately again.

^*) The black Q will not give check on αb5, because doing so would cost it greatly in symmetric positional strength.

^*) Strongest opening.   ^**) Or interpose a N, or evade asymmetrically with K at once.   ^***) K evading is best. Possibly interpose B δb5 – δc4 first. Caution! For P γc2 – γc3 follows. If B × P, then B × B †. If B evades, discovered check P γc3 – δc3 † and P δc3 × U εc4.

B. Officer Openings.

^*) These two important pawn moves keep presenting themselves; whether one begins the game with them or lets them follow soon after is fairly immaterial.

^*) Popular strong opening.   ^**) Or interpose a N, or evade asymmetrically with K at once.

^*) N threatens K and the unprotected R from αd3. Important threat!   ^**) White does better to move a N.

An interesting four-knight game.

6. Some Played Games.

Since it is intended to publish a separate issue of Raumschach games at a later date,^*) only a few specimen games are communicated here. Reference should also be made to the games contained in previous publications:^**)

1908 Das Schachraumspiel. (Three-dimensional chess.) A new, practically interesting and theoretically important extension of the two-dimensional board game. With figures and diagrams. Potsdam. A. Stein. (1.20.)
1908 Anleitung zum Raumschach. (Three-dimensional chess.) Hamburg. Published by the author. (1.00.)
1909 Mitteilungen über Raumschach, wissenschaftliche Schachforschung und verwandte raumwissenschaftliche Probleme. 7 issues. 1911 Hamburg. Published by the author. (5.00.)
1913 Spielregeln zum Raumschach. Berlin. Staub & Co. (0.30.)

Game I.

S^III₅ — (Played in the “Hamburg Raumschach Club” on 17 May 1912.)

An interesting checking tour of the queen.

The Q stands protected on the central square and simultaneously attacks 2 N, 2 R, Q, and K.

Just as White is about to promote a pawn to queen, a remarkable checkmate overtakes him.

Game II.

S^III₅ — (Played in the “Hamburg Raumschach Club” on 31 October 1913.)

Black stands decidedly worse than White, quite apart from the loss of the B and N. The threat now is Q γd2 × N δd3 †. Instead of evading with the K, Black forestalls this move by giving check. An unbroken sequence of nine checks leads directly to victory by means of a rook sacrifice, after some tense so-called “check-gardez” positions have arisen in the meantime. Black thereby risks forgoing the queen exchange, which would have led to a draw. But active queen exchange would have been better for White.

White is compelled to accept the rook sacrifice, since the remaining 15 squares to which the king might escape are already commanded by enemy pieces.

Q γb1 – γb3 checkmate.

Game III.

S^III₇ — (Correspondence game, 10 February – 13 March 1915.)

Black already commands the central square fourfold, with G, U, N, Q.

Black evidently intends to bring the Q to δa4, in order to win the R by check.

With 7 pieces developed (a different piece every move!), White is considerably better developed. White plans G βc1 – εd3 to cover the threat Q βd2 – ζd6 †.

This should not happen in correspondence games. For:

White has now developed 9 pieces, Black only 3 (Q, U, N).

In 10 moves White has developed 10 pieces. View the chess space from the side.

The black K can escape to only 8 squares. U εd4 is unprotected. If K evades to ζd5, εe5, or δd5, an interesting giraffe maneuver follows in each case. In case I, White plays G δd3 – εg5 [giving check-gardez and covering the U]; in case II, White plays G δd3 – ζa2 [giving check and covering U]; in case III, White plays G δd3 – δg4 [giving check, covering U]. This example proves the fabulous agility and great power of the giraffe.

For the second time Black overlooks the treacherous U!

The threatened next move is B ηf6 – ηd4 with check-gardez, where B ηd4 is protected by G δd3.

Black resigns, as further material losses are imminent.

7. Endgames.^*)

The following win:

1) always:

K + Q vs. K

(e.g. White: K γc3, Q βc3.   Black: K αc3 checkmate.)

K + 2 B vs. K

(e.g. White: K γd2, B βd1, B βe1.   Black: K αe1 checkmate.)

2) conditionally:

K + P vs. K

provided the P can promote to queen.

K + 2 R vs. K

in certain positions. Whether these positions can be forced, however, remains open to question.

(e.g. the following typical position: White: K βb3, R βd1, R γd1.   Black: K αc1.)

White wins by:

A.

3) never:

K + 1 R;   K + R + B;   K + 4 U.

The knights require further and more precise investigation.

8. New Methods of Play in the Five-Board.

The greatest difficulties and liveliest controversies for the Raumschach player have always arisen in answering two questions: how should the pawns move, and how should the pieces be arranged at the start of the game?

Every possible proposal has been made and tried in practice. The principal ones have been discussed in this issue. To ventilate all of them here would go too far. It is a pity that they (like so much else!) have not found their historical record in a continuation of our Mitteilungen.

The simplest and most radical solution is of course to dispense with pawns and with an initial position altogether.

More recently, attempts to compose Raumschach problems have revealed the necessity of a pawn movement other than A and B. With reference to the distal promotion ranks, all “backward movement” of the pawns is accordingly forbidden, while all “forward movements” — including lateral (!) capturing — are required. (The f-k pawn movement is retained; the theoretically possible f-e and k-e pawns continue as before to be of no practical consideration.)

In order to channel the development phase of the game into calmer waters and prevent the heavy initial check-giving,^*) the use of forward pawns has proven practical.

Leaving open for the moment whether both innovations are not still too redolent of board-chess memories, the following new method of play in the Five-Board with pawn movement C now emerges.

I. Pawn Movement.

Summary overview:

II. The Initial Position (C).

The advantage of this arrangement lies in the calmer development of the game and in the advancing pawn reserve.

If one wished to proceed entirely consistently and surround the officers completely with a protective pawn wall, pawns would have to stand on α 2, β 2, γ 2, and γ 1 — that is, 20 pawns each in total. Since that is too many, one must limit oneself somehow in practice. It might even be advisable to take only 6 pawns each instead of 8, p. 42 i.e., to remove the edge pawns in front of the B and U, so that only the 6 P on each side that are strictly necessary for the protection of the K remain.

Accordingly we designate position C with 20 P = C¹, with 15 P = C², with 10 P = C³, with 8 P = C⁴, with 6 P = C⁵, with 5 P = C⁶ = A.

The pawns impede development. To that extent they serve “time.” They thereby increase the initial tension of the game. One can now play a game of greater or lesser tension with more or fewer pawns. Initial position A is a low-tension one with little intervening pawn resistance.

The more pawns one uses, the longer the game lasts in general. For this reason too, position A is to be recommended for the beginner.

The “old” method of play thus follows arrangement A and pawn movement A; the “new” follows arrangement C and pawn movement C. Which of the two has the definitive advantage must first be determined by further experience. It is also expedient to combine arrangement A with pawn movement C.

Incidentally, pawn movement C is “new” only with respect to the Five-Board. In itself it is not only not “new” but is in fact the one originally specified first by me (Das Schachraumspiel, p. 15). It dates from the first Raumschach period, when we still played on the Eight-Board and the pieces stood on the base plane α. When we later moved to the Four-Board and Five-Board and to the frontal diagonal arrangement, and no longer regarded the space (erroneously) as an “extension” of the board, we changed movement C to A.

To summarize once more briefly, the old method of play has the following disadvantages:

1. Regarding the arrangement: White as the moving party has too great an advantage and must win with correct play; there is no proper opening period — we have only middlegame and endgame without a quiet development phase, which is made possible only by the obstacle of the forward pawns; pawn play is absent; the γ-pawns are “passed pawns,” i.e., they can promote unimpeded by the opposing pawns (provided the latter are not permitted to move “backward”); the K is unprotected.

2. Regarding pawn movement: The pawns move backward with respect to the promotion rank; equality of directions toward promotion is not guaranteed; a precise endgame and certain problem compositions are impossible.

3. Regarding the model: The two-color-only model makes the operation of the unicorns very difficult. A four-color model is better. The colors must be graduated in tone. Excessively garish colors irritate the eye.

The new method of play (which is enthusiastically advocated by perceptive and unprejudiced strong players and problem composers well known in the official chess world — Messrs. Hans Klüver, Wilhelm Roese, and others) seeks to avoid these disadvantages.

I designate the new method as “normal play.” From this norm, deviations are now permitted in certain cases (by agreement between the players), which we have discussed in this issue.

The “old” arrangement is an open one; the “new” a more closed one. Beginners will prefer the open; more experienced players the closed, because it admits of more chess subtleties.

III. Game Openings.

1. The Queen Game.

In the old arrangement a P stood above the white Q. After 1. P γc1 – γc2, the strongest opening was 2. Q βc1 – εc1. (Cf. p. 34, VI.)

The new arrangement leaves square γc1 free, so that 1. Q εc1 can be played immediately.

Since this strong move in favour of the first player is evidently a vital question for the new method of play, I take the following information from a letter of Mr. Klüver dated 18 May 1919:

“After 1. Q εc1 there are three defences for Black:

The first line of play (a) shows that the initial developing move of the Q, obvious as it is, is nevertheless not as strong as initially assumed. The Q is driven from εc1. This in turn makes it possible for Black to develop the Q to αc5. Moreover the two black N on α stand very strongly. They cannot be driven away by a P and threaten squares lying behind the white pawns, which are therefore weak from the outset. This analysis seems to me to prove that 1. Q εc1 is a strategic mistake — even though on tactical grounds it sometimes succeeds!

The second line of play (b) is appropriate when one wishes to steer quickly into the endgame. At first it seems as though after

Black loses a tempo by moving the B twice. But this is not the case; for the white Q also moves twice. If one examines the position after the above p. 44 three moves, White has developed one piece and Black likewise. White to move. There can therefore be no question of a loss of tempo.

“. . . . . The more I occupy myself with Raumschach and Raumschach questions, the more I come to love the three-dimensional game. I cannot understand how people of chess renown are unable and unwilling to emancipate themselves from the board. I can already say now: I prefer a Raumschach game to a board game by far.”

2. Of Other Openings

only brief mention need be made: the knight games; the unicorn game: 1. U αa2, U×U, 2. B×U, B εc4; the pawn game: 1. P βb3, P δd3, which frees the path for the B and Q and allows the (very noteworthy) weak points αb2 and εd4 respectively to be better protected; rook games and bishop games are not so much to be recommended.

IV. Played Game.

(Consultation game in the H.R.S.C. on 24 May 1919.)

White has: 1 R, 1 B, 1 U, 2 P.   Black has: 2 R, 4 P (!).   After a prolonged struggle, draw.

Notes by H. Klüver and W. Roese.

Note by Dr. Maack.

In my opinion Black can win. Position after move 42: White: K αc2, R αa5, B αa3, U εb4, P αb2, P βe2, P δa3. — Black: K βd4, R δa4, R εa3, P δd3, P δd4, P γc3, P εc3, P εd4. —

Mr. Roese writes to me further: “This game is probably the best yet played in attack and defence, and shows the characteristic features of the new method of play with full clarity.” It is therefore recommended for careful replay with attention to the notes, which have been kept to a minimum. One thereby comes to know the differences between the (old) “historical” and the (new) “normal” methods of play. The arrangement of the officers has remained the same. Only the arrangement and movement of the pawns has changed.

Conclusion.

One final request: do not judge Raumschach too hastily! Do not criticize and find fault too precipitately with details and minor points of terminology, pieces, moves, arrangements, rules of play, etc. Everything has been thought over and reflected upon a thousand times, discussed with friends of Raumschach, and practically tried out and played through. Certainly much can still be improved. But on the whole Raumschach now stands firmly founded on a logico-mathematical basis and is equal to all attacks and aspersions.

Some of our friends — who are steadily growing in number — are of the opinion that one must offer the beginner something ready-made: “This is how you shall play!” Others say: a practical introduction to play must not be a fixed commitment! In this issue we have held the middle ground. We have presented much that is ready-made. But we have also not committed ourselves for all future time; rather we have prepared a promising path for changes and improvements. It is not the least attraction of Raumschach that every chess thinker finds here a field for independent collaboration, while board chess has long since become ossified and every further development appears excluded.

All the more striking it is, therefore, that — with laudable exceptions — the official chess world has not yet occupied itself to any greater degree with the rich, chess-theoretically so fruitful material of Raumschach, but instead, in keeping with conservative inertia, still conducts itself more or less dismissively toward our innovations. For this reason I felt it necessary to make the present issue comprehensible to non-chess-players as well.

Raumschach neither wishes nor intends to displace and replace board chess! But if, according to Rudolf von Bilguer, “the destiny of man lies in the investigation of chess­truths,” then Raumschach surely brings us a good step closer to that noble destiny. For both S^II and S^III are ultimately only special cases of the general Sⁿ, in which scientific chess research finds its fullest expression.

In this spirit we can now formulate our task conclusively as follows:

We wish
1. to apply space to chess. But not merely two-dimensional space, which leads to “board chess”; and not merely three-dimensional space, which leads to “Raumschach” (in the narrower sense); but space of every dimension — absolute n-dimensional space — which leads to general Raumschach. In this way we arrive at an exact spatial science of chess.

We wish
2. to apply chess to space. To examine and set forth spatial forms, relationships, and proportions in chess terms. Not only to spatialize chess, but also to chessify space. In this way we arrive at a chess-based science of space. It will be seen that in this endeavour number plays the greatest role — and not only ordinary quantitative numerical ratios, but also the quality of number, the position of number in space.

We wish,
mindful of the ancient saying of the Bible: Thou hast ordered all things in measure (space), number (time), and weight (force, matter, movement) (harmoniously),
3. to raise chess, in conjunction with space and number, to a philosophical height; with the aid of space and number to create a true philosophy of chess that is based not, as hitherto, solely on board chess, but on an absolute science of chess.

We hope to be able to approach the fulfilment of these great tasks gradually, with the help of old and new collaborators.

Outline of the Complete Work.

A. History of Raumschach.

B. Art / Cubic Chess.

C. Science:

I. Spherical Chess.

II. Crystal Chess. Symmetry of the chess space. Symmetry centre and central square. The 13 symmetry axes of the regular crystal system correspond to the 26 directions of the 8 basic pieces. Symmetry planes and fundamental domain. — The angles of the chess pieces. — Chess-space crystals. (Octahedron. Pyramid. Tetrahedron. Cuboctahedron.) Chess-piece crystals. α-, β-, γ-crystals. Kinship and transitions of the chess crystals. The hexakisoctahedron as giraffe crystal. The giraffe as the most general basic piece. Diagrammatization of crystals. The regular polyhedra and star polyhedra. — Crystal structure. Face nets and space lattices. Regular sphere and point systems.

III. Projection Chess. Parallel projection. Primal chess. Central projection. Projection model. Diagrams of pieces and positions. — The Raumschach board.

IV. Numerical Chess. Magic squares and cubes. Board-chess and Raumschach moves in the magic square. Spatial analysis of magic squares. (Resolution of magic surfaces into spatial knight moves.) Magic structure of the fundamental domain. Magic squares and cubes of irrational numbers (roots) in the fundamental domain. Octogram and natural square. Significance of the magic square for practical play.

V. Knight’s Tours. Planar and spatial traversals. The preferred unicorn. Spatial control of the pieces. — Queen’s dance. — Planar and cubic knight’s tours. Rhombohedral structure of the perfect, closed cubic knight’s tour. — The octogram (star octagon). — Knight (steed), zebra, antelope, and giraffe leaps. — Knight, zebra, antelope, and giraffe nets and lattices.

VI. Higher-Order Raumschach. Infinite, hyperbolic, multidimensional, hexagonal Raumschach.

VII. Other Mathematical Raumschach Problems. Regular multi-rays. — Kinship between nullhorn (knight), unicorn, and zweihorn (giraffe); between bishop and unicorn. — Negative, imaginary chess pieces. — Spatial structures of higher order analogous and homologous to the octogram.

D. Raumschach as Philosophy. Philosophy of chess. Arithmosophy and stereosophy (philosophy of number and of space). Space as the essence of things —

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