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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.

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, 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, 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
in
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
through
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
with
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.

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:
R
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.

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
cells.

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.

Jared: Ah! I think I see (why you're using an order-4 octahedron). Very
timely!
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.

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.)

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.

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.

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, 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.

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.

Very nice game. I like the idea of a non-cubic 3D chess variant.

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.

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.''

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.

Tetrahedral is one of the four geometries Gilman's system explores within ground-breaking M&Bxx. They are squares, cubic, tetrahedral and hex-prism, three of the four being 3-D. Pioneered by Mark Thompson in 2002, Tetrahedral is tetrahedron, pyramid, turned at first in unaccustomed angle. Mark Thompson also wrote ''Defining the Abstract'' that is still easily locatable on web from defunct site Games Journal. Gilman explains to Thompson 18.January.2004 that ''Yes I did mean square of distances. The base-36 was simply a way to represent every distance as a single character.'' The squares are in fact rhombic dodecahedra, he and McComb enlighten. Gilman uses base 36, instead of 10 or 2 or 16, for convenience to get leap lengths. I had the discussion down once but have to go over it again. Oh for bygone days of quality before these degenerate times.

Thanks! 'Defining the Abstract' is also on my own defunct website at flash.net/~markthom/html/game_thoughts.html .

How is tetrahedral different from cubes and hex-prism? Connectivity. How is it the same? Being 3-D. From player's standpoint, Rook always commands 18 squares without blocks. Check Rooks' corners' initial positions and count 6+6+6, and whereever else Rook is there are always potential 18, like 8x8 14 and 5x5x5 12. King one-steps colourswitching and there are four colours. King has to peer at the 12 directions. If King moves on own level, there is one available colour, and if he changes level, there are the other two. Knight is as if ''not King,'' going away two King steps and mandatorily colourswitching, and it means it's a jump.

Beyond question Tetrahedral is real 3-D object -- known to antiquity -- neither optical illusion nor impossible object perceived yet irreproducible. Get used to it, because 'Man&Beastsxx' generalize from cubes, hex-prism, and tetrahedral. Cubes have 26 directions and tetrahedral 12 only, easing acclimatization. Notice the one notation -- out of many possible -- has only a1-a7 in order from level VII to level I. Not so doing for b1-b7 etc. It's for convenience having each of a1, b1, c1, d1, e1, f1 and g1 ''start'' on their own level radiating severally as needed. Full notation has level and then the square dual, like 'VIIg7' for one corner cell end-level. Such as 'c4' is not full descriptor since there are all of IIc4, IVc4 and VIc4 on different levels. For example, Rook Ia7-IIa6 and Rook Ib6-IIa6 -- adjacent Rooks optionally going to the same square one away -- are both legal moves, and they would be following different planes to get there, i.e., different directions within different planes. Use edge lines, faces, as well as the four colours to see legitimacies in whatever ways work. Knight is suitable to use colours for shorter ranges. In Level 1 and Level 7 (hey there is a science fiction classic 'Level 7') one plane eviscerates to a line, simplifying. All the corners mid-levels are more intuitive reference frames. The 12 directions are those of the 6 edges. Mentally involute, if you are still able within these widespread degenerate times, as used to be standard practice during past centuries.

Depth vs. Clarity, Drama vs. Decisiveness. ''Anthropologists from another planet who wanted to study the way humans think would do well to study our abstract strategy games.'' Written 10 years ago by inventor of Tetrahedral is classic 'Defining the Abstract'.
http://www.thegamesjournal.com/articles/DefiningtheAbstract.shtml

I like the creativity in creating a colorbound piece on a board that doesn't lend itself easily to non-rook moves. Very simple, yet complex enough to offer a challenge to players. I've been halfheartedly toying with the idea of tetrahedral chess myself for a few months, and this is the first version I've seen.

Pritchard's CECV lists a game "Xyrixa Chess" by David Samuel c.1980 played on this same board (provided I'm reading correctly). It's not clear at all how the pieces move or whether the inventor knew about any of the geometry facts discussed in the other comments here.

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