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This item is a game information page
It belongs to categories: Orthodox chess, 
It was last modified on: 2002-11-01
 By Mark  Thompson. Tetrahedral Chess. Three dimensional variant with board in form of tetrahedron. (x7, Cells: 84) [All Comments] [Add Comment or Rating]
George Duke wrote on 2008-07-02 UTC
This is the neatly-connected CV we recognize at Chessboard Math and that inspires our 84-cell variation, more or less from tetrahedral numbers Mark Thompson talks about here. Actually, there instead, square numbers 1 and 4 are also tetrahedral numbers, and the total 84 is tetrahedral number, also being 1+9+25+49, four levels.

George Duke wrote on 2007-11-07 UTC
Ralph Betza's 500 Comments to mid-2003, when he left, are labelled 'gnohmon'. Betza's nom de plume 'gnohmon' as 'gnomon' means Basic Unit in mathematics. The gnomon is the Piece needed to add to a figurate number to get the next bigger one. A figurate number, including tetrahedral numbers, can be represented by geometric pattern. For example, dots show 'triangular numbers' 1,3,6,10,15 and so on, as the triangle of dots enlarges. Also figurate, square numbers are 1,4,9,16... So the particular gnomon varies with the pattern. In square numbers, 'gnomon' is L-shaped: just wrap a 1x9 bent once 90 degrees into an L-shape, the gnomon, to get 25, 5x5, the next square number. In Tetrahedral Chess, remarkably ''the Pawn rows completely enclose the pieces in the starting position,'' unlike ordinary cubic 3-D, ''so that play begins with pawn development, as in usual chess.''

George Duke wrote on 2007-11-07 UTCExcellent ★★★★★
Tetrahedra have 4 faces, 6 edges. This pyramid stands on edge instead of face. Two edge-pair's midpoints determine one line, and 2 such lines determine a plane (the 4x4 here). That cross-section is a rectangle not usually square. So are all cross-sections of all other parallel planes intersecting. Cleverly and arbitrarily, Mark Thompson chooses 5 such planes and 2 more planes fashioned out of two edges, totalling 7, in order to get notional-3D 84 spaces(year 2002 84-square contest). They then divide conveniently into two(1x7), two (2x6), two (3x5), and one (4x4) making 84. How is that related also to 84 as tetrahedral number? 84 as tetrahedral number(think sphere-stacking oranges) sequences one(1), supported by 3 making four(4), supported by 6 making ten(10)[this is experimental too], then 10 making twenty(20): 1,4,10,20,35,56,84,120... For the five piece-type differentiation, the six edges each have two directions for 12 altogether. Contrast these 12 to the 26 directions necessary for complete interpretation in awkward standard cubic chesses (6 orthogonal, 12 diagonal and 8 triagonal or trigonal). See in the tripartite diagram the distinguishment between King and Rook.

Todor Tchervenkov wrote on 2005-11-23 UTCExcellent ★★★★★
Very nice game. I like the idea of a non-cubic 3D chess variant.

Charles Gilman wrote on 2004-03-29 UTCExcellent ★★★★★
Observing that the faces of this board are hex boards, I notice that your usage of King and Knight is identical to that of S. Wellisch in the first hex variant. This shared usage is now recorded (complete with link to this page) in my piece article Constitutional Characters.

Mark Thompson wrote on 2004-01-24 UTC
Charles, after reading your latest about the rhombic dodecahedral grid, I
thought to look up 'rhombic dodecahedron' in the invaluable Penguin
Dictionary of Curious and Interesting Geometry, where I found the
following tidbits you might find interesting:

'rhombic dodecahedron: Take a three-dimensional cross formed by placing
six cubes on the faces of a seventh. Join the centres of the outer cubes
to the vertices of the central cube. The result is a rhombic dodecahedron.
... From the original method of construction, it follows that rhombic
dodecahedra are space-filling.' [etc.]

Indeed, if you imagine space filled with alternating black and white
cubes, and perform the construction by dividing up the white cubes into
six pyramids apiece and affixing them onto their black cube neighbors, you
get the r. d. grid, and this supports your observation that the grid is
conceptually identical to the cubic grid with the white cubes removed.

Charles Gilman wrote on 2004-01-19 UTCGood ★★★★
At last, I can see it all. It started when I noticed that on cubic 3d board, a plane including two standard (root 2) diagonals not at right angles (a) automatically includes a third, (b) comprises squares of only one Bishop colour but all four Unicorn colours, and (c) resembles a hex board. The 3d orthogonal and root 3 diagonal are lost but the root 2 diagonal becomes a hex orthogonal, the 2:1:1 (root 6) oblique a hex root 3 diagonal, and the 3:2:1 (root 14) oblique a root 7 hex oblique. Then I realised what the Tetrahedral Chess pattern of 3 diagonals mutually at right angles and 6 orthogonals each at right angles to one other and a diagonal reminded me of. It was the cubic board's orthogonals and diagonals swapped over! Yes, the Tetrahedral board can also be seen as a cubic-cell board with cells of one Bishop colour missing, and the cell notation shown actually fits with this interpretation!

Charles Gilman wrote on 2004-01-18 UTCGood ★★★★
My last comment was a quick correction/explanation of what I had said before. Now that I have had some time to think offline I can comment on the idea of splitting the the orthogonals into two groups of three. The truth is, any such division is arbitrary. They naturally divide into three groups of two - those in the horizontal plane, and those in the vertical plane through each horizontal diagonal. In fact these planes are interchangeable, each having 2 orthgonals of its own and two diagonals each shared with one of the other two. However limiting pieces to those pairs of orthogonals binds them to single planes, and even limiting them to the two orthgonals in a plane and the diagonal outside it would bind them to alternate planes. The oddity that the root-three (hex) diagonal is not colourbound on this board is matched by the root-two (square) diagonal is not colourbound on a hex-prism board.

Charles Gilman wrote on 2004-01-18 UTC
Yes, I did mean squares of distances. The base-36 was simply a way to
represent every distance as a single character.
Regarding Jared McComb's comments, the cells in this game ARE rhombic
dodecahedrons, I just thought that that was too much information to
include in my own comment, and the orthogonals are indeed at right angles
to the boundaries they pass through. The view presented uses one of the
square cross-sections, but an alternative view of the same game would be a
pyramid with a single cell with progressively larger triangles of cells
viewed in hex cross-section below.

Jared McComb wrote on 2004-01-12 UTC
Oh, don't worry, I've got something in mind.  *wink*

Also, in an effort to remain on-topic here, I think it would be a very bad
idea to make that sort of distinction between 'rook' and 'bishop'
moves. since they are all topologically the same move.  Rather, you could
replace the 'rooks' and 'bishops' with differently defined pieces
altogether.  (For example, see D. Nalls' pages on the Zig-Zag, etc.
pieces in the Piececlopedia.)

Mark Thompson wrote on 2004-01-11 UTC
Jared: Ah! I think I see (why you're using an order-4 octahedron). Very

But opposite faces will have only space for 10 pieces, and the armies are
already only separated by 2 layers, if I'm imagining it right. That would
mean rather small armies for the space available.

Jared McComb wrote on 2004-01-11 UTC
I was going to use an order-4 octahedron for my variant, with armies on opposite faces. See if you can figure out why I'm using that shape.

Mark Thompson wrote on 2004-01-10 UTC
Jared: Are you still going to have the two armies start on opposite edges
of the board? That was what prompted me to orient it as I did in my
diagrams, rather than the usual idea of a tetrahedron resting on its base.
I look forward to seeing your variant.

One could also use the basic rhombic dodecahedron grid as a playing space
with something other than a tetrahedron as the overall shape of the board.
For example one could chop off the corners and make either an octahedral
board, or (by chopping smaller pieces) a board with 4 hexagonal and 4
triangular sides. I calculate that an order-6 octahedron would have 146

Jared McComb wrote on 2004-01-10 UTC
Here's the deal.  The boards are topologically identical, but I find the
directions easier to visualize when the board is reoriented like that,
since it is easier to see that the orthogonal directions are parallel to
the edges of the board.  I do not have any RT software at the moment, but
I'm working on a variant using this setup myself, so you may see an MS
Paint interpretation sometime soon.

Here's a quick'n'dirty diagram of what I mean, on a 10-cell board:

 Y G    B
R B R  G Y  R

(Each layer is centered on top of the previous one.)

As you may be able to see, when the board is reoriented in this way, each
layer has a four-color tiling that makes Dabbabante (spelling?) moves
about ten times easier to see, and it opens itself up to interesting
interpretations of 'triagonal' movement.  For example, the two green
cells in the example above could be considered 'triagonally' adjacent. 
If you use a Glinski interpretation of a bishop, and extend it into all
four 'hexagonal' planes that come out of a single cell, you get a
non-colorbound piece.

The problem with this setup is that it muddles your interpretation of pawn
moves a bit, since 'forward' is in a totally different direction.

If I'm not making any sense here, don't mind me.  I came up with this a
couple years ago on graph paper, and had been thinking about it a while
before the 84SC, but I'm only now realizing the parallells.

Mark Thompson wrote on 2004-01-09 UTC
Larry, your idea of showing the cells as points where color-coded lines of
movement intersect works well with another idea I've been turning over
my mind. I've never been quite satisfied with the 'Dababantes'
that I used as Bishops in this game -- they're color-bound, but that's
about the only way they resemble chess Bishops.

What I've been thinking of is to designate three of the six lines
each cell as 'rook lines' and the other three as 'bishop
lines'. This would make rooks weaker than they were in Tetrahedral Chess,
and Bishops would have really equal power to Rooks. In your xiangqi-style
board representation, the rook lines might be colored red and the bishop
lines blue.

If the seven squares of level I where the White pieces begin are
considered to be in an 'east-west' row and the seven squares of
level VII are in a 'north-south' row, then I would make
north-south and two of the vertical edges 'rook lines,' and
east-west and the other two vertical edges 'bishop lines.'
Neither the rook line edges nor the bishop line edges would make a
triangle on the surface of the tetrahedron; they would be symmetrical
one another. And then, I would arrange the Black pieces differently from
the White pieces, putting rooks in place of bishops and vice versa,
because the orientation of the levels on which the two sides begin would
in effect 'turn a rook into a bishop,' if you see what I mean.
(Sorry, it's hard to describe without a diagram.)

But this is just thinking out loud in public, I haven't tried any of it
out yet.

Mark Thompson wrote on 2004-01-09 UTC
Jared, I believe the cells of the board shown here are topologically
connected in the same way as the rhombic dodecahedron tiling you mention.
Only the topological form of the board is relevant to play, so I wouldn't
think that the translated rules would be enlightening ... if I'm
visualizing correctly what you have in mind, I think it would be far
harder to understand what the game is about. The trouble is that in any
diagram I can imagine, you can only see a cross-section of each level,
which prevents the full geometric form of the 3D cell from being seen. If
you have 3D raytracer software you might be able to demonstrate it. I'd
be interested in seeing that too. The ideal thing would be a virtual
reality board, that players would see by donning those goggles that
present stereoscopic 3D images that you can see all sides of by moving
your head. When those become commonplace I predict a lot of wonderful 3D
games will get implemented on them. I still haven't seen that technology,
but I hope someday to use them to play Renju on a 'tetrahedral' board of
order 13 or so.

Charles, I'm reading your post for about the tenth time and am starting
to figure out what you're talking about. You say 'square roots' but I
believe you mean 'squares.' The base 36 business was confusing to me but
you're really just doing it for compactness, so you can indicate each
distance (or its square root) by a single character. And your use of
'coprime' doesn't seem to match the meaning I understand by that word.
But I'm interested to see that the cells to which a knight at your origin
can move are all labelled as distance sqrt(3) from the origin - well, that
would make sense, just as a FIDE knight's moves are all sqrt(5) in
length. Okay, I'm starting to follow your arithmetic - and I'm
surprised, I wouldn't have guessed that the centers of cells in a rhombic
dodecahedral grid would have distances whose squares are integers - though
now that you point it out, I don't see why not.

I'm not sure how playable your proposals for Unicorns and Nightriders
would be on this grid -- it seems to me that to give them sufficient scope
to practice their powers the board would have to be considerably larger
and so have a huge number of cells, and a IMO game whose board has too
many cells becomes too complicated to be interesting, because the moves
have so many consequences no human player can foresee them; hence, it
turns into a game of chance rather than skill. However, many people
disagree with me, and I would be glad to see other game developers try
their hand at this grid. If you're inspired, go for it!

Larry Smith wrote on 2004-01-09 UTCExcellent ★★★★★
A very nice game.  Interesting playing field, pieces and rules.

As to the idea of using other shapes to denote cells in non-cubic fields
within a 2D medium, a simpler form might be to have points with colorized
radiating lines noting the various directions.  Like Chinese Chess, pieces
would move from point to point.

Although this might also be confusing as lines would criss-cross one
another without actually intersecting.  But this is where colorization
would come in handy.

But I don't think it is necessary to change the graphics for this
particular game as it is quite understandable in its current form.

Jared McComb wrote on 2004-01-08 UTCGood ★★★★
It seems to me that a true tetrahedral form of chess would have 'cells' which, in three dimensions, would take the form of rhombic dodecahedrons, which would allow the board to be pyramidial with 'hexagonal' tiled layers. (Rhombic dodecahedrons tesselate space quite nicely, you know, and naturally lend themselves to making tetrahedra with.) Does the current setup of this game allow for such an analogue? The board can be easily translated, complete with cell coloration and the same twelve directions, but can the rules be translated as easily? I'd love to see an attempt.

Charles Gilman wrote on 2004-01-06 UTC
Having analysed this arrangement of cells I have worked out the distances
between cells. Below is a sample of square roots of distances from the
cell marked 0 on alternate layers of 6x5 and 5x6, shown as digits in base
36 (A=10, B=11,... Z=35). Thus moves can be traced for Rook (0149G...
or successive levels), Bishop (028I... same or alternate levels), Unicorn
(03C... successive or alternate levels), and Nightrider (05K... same or
successive levels). Note however that the 9 on the level adjoining the
starting one is NOT part of a Rook move, but a coprime move like 2:2:1 on
a cubic-cell board. Note also that the number 7 represents exactly the
commonest oblique piece of hexagonal-cell variants.

2125AH	3359F
10149G	1137D	4347CJ	779DJ
2125AH	1137D	3236BI	557BH	A9ADIO
5458DK	3359F	4347CJ	557BH	989CHN

Charles Gilman wrote on 2003-06-28 UTC
The problem with your idea of reducing the colours to two is that four colours really do reflect best the real relationship between the squares, as the rotations in my previous comment illustrate. In a game that does not use the diagonal move it makes sense for two of the Pawn's orthogonal moves to be capturing and two non-capturing, but which are which is entirely arbitrary. Associating one pair of these moves with the orthogonal same-level move and the other pair with the diagonal same-level move seems oddly asymmetric. Introduces Shogi generals is a great idea but you do need to use the true inter-level diagonal - which on closer examination I notice is the vertical move up or down two levels!

Mark Thompson wrote on 2003-06-21 UTC
I've been thinking lately that the 84-cell tetrahedral board might adapt
better to a 3D Shogi. My reasoning is that Shogi pieces are less powerful
than Chess pieces, getting much of their value from being parachutable
once captured, and the difficulty of visualizing moves on this board might
be lessened for less powerful pieces. I'm considering replacing the Rooks
with Lances that can only move orthogonally 'forward', the Dabbabantes
with some kind of Silvers and Golds that can only move to a subset of the
adjacent cells, and having a Horse (a Knight, but only with its
forwardmost moves) that automatically drops into the King's starting
square whenever the King first vacates it. I don't think I'd include any
pieces like the Shogi Bishop or Rook. The board's colors could be reduced
to two. Pawns might move to the forward cells of the same color, or of
opposite color, or there might be two kinds of Pawn. A Silver and Gold
would move to any of the four forward cells, or to the adjacent lateral or
rear cells that are the opposite color (Silver) or the same color (Gold,
considering the same-levels cells 'diagonally' adjacent as adjacent for
this purpose).

It appeals to me that the board is also nearly the same size as a
conventional Shogi board. These armies would be a bit smaller, but I think
they're also a bit stronger.

Charles Gilman wrote on 2003-06-21 UTCGood ★★★★
There is a sensible diagonal move on your board, although I can see why you
did you not use it as it is complex. On one level (literally rather than
the usual metaphorically!) the diagonal move is self-evident, along path
constant in colour and also in either letter or number. What a diagonal
move between levels means is determined by observing that the board can be
rotated into five other positions in the same frame (and reflected into
another six), which your noation recognises as they split evenly into your
levels I-VII, red-blue and green-yellow levels 1-7, and red-yellow and
blue-green levels a-g, revealing such diagonals as all the c4 squares.
Temperature goes out of the window (metaphorically rather than the usual
literally!), but is hardly needed once a diagonal capture has been found
for the Pawn.
	As well as this lot the board can be viewed with any corner as a
hexagonal cell at the top and the rest of it as six progressively larger
triangles of such cells down to 28 at the bottom. Each hex level has all
four colours and the diagonal move described in my first paragraph
requires a change of hex level.

Mark Thompson wrote on 2002-12-01 UTC
As of November 30, 2002, there is a new and corrected version of the ZRF
available for download. If you downloaded the ZRF before that date, the
version you have has a bug (sorry!), which causes it to allow the
Dababante to move past an enemy piece on a square that it could have
captured. As described above, every line of squares on this board
alternates between two colors, and the normal move for the Dababante is to
those squares that share the color of its starting square; and it can
reach those squares even if a piece (other than a Pawn) intervenes on one
of the squares of the other color. BUT, it CANNOT continue past a piece
occupying a square of the same color as its starting square -- a piece on
a square where its own motion would 'touch down' (possibly to capture the
piece). This is what the earlier, incorrect version of the ZRF allowed.

My thanks to Dan Troyka for figuring out how to fix this error in the

By the way, the new version also has a modified board image, making it
look more like the 3-D levels are separated by struts instead of attached
to upright wooden planks. Dan and I both prefer the new image.

Mark Thompson wrote on 2002-10-06 UTC
Have you actually built a board? I haven't done that yet myself, so yours would probably be the first one in existence. I'm inclining toward plexiglas levels, held up by threaded metal rods (with nuts to hold the boards in place), and a wooden base, probably made from a round cutting-board. I might want to make a set of squat chessmen somehow too, since standard chessmen seem too tall for a convenient 3-D game. They force the levels too far apart.

LCC wrote on 2002-09-28 UTCExcellent ★★★★★
This has perhaps the most interesting geometry in a chess variant ever.
And if you take the time to actually make a board of the thing (paper,
tape and barbecue sticks spring to mind), the gameplay isn't even as hard
as in a geometrically coherent cubic chess.

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