Comments/Ratings for a Single Item

OK, unless I've missed something, I think this one's ready to go.

Can I assume the corner cells are deleted so you don't have to work out what to do with the diagonals there?

If a pawn or spear find themselves on an Open face, there are two (or all four, on faces 6 and 19!) directions that are "toward" the enemy Home face; how do they move then?

Playing on the 2d surface has the nicety of rook lines still actually restricting the enemy king into one side or the other. What does mating material look like here?

I tried to work out (but without paper) how many squares a rook attacks on an empty board. There are 12 faces that it reaches in each direction, but those overlap, I think four faces in common? So it should be 5*12*2-4-1=115 (that last being the rook's current cell)? What about the bishop, or nightrider?... Oh, I guess bishops aren't colorbound?

Is there a reasonable way to flatten this for displaying on a table/screen? (I suspect not, because of the forking of paths.)

Does the setup section's use of "clockwise" actually make sense?

Why alternate ordinary and berolina pawns? Doesn't that hurt pawn structures? (Does it just not matter on this wacky board?)

Can I assume the corner cells are deleted so you don't have to work out what to do with the diagonals there?

Yes, you can, and correctly so -- especially with the leapers! (I've seen others try to manage that mess, and those were just cubical boards. On a tesseract? No, thank you!)

If a pawn or spear find themselves on an Open face, there are two (or all four, on faces 6 and 19!) directions that are "toward" the enemy Home face; how do they move then?

That would be based on where on the face they are. If they're at the midpoint, it's player's choice (I probably should include that in the text).

Playing on the 2d surface has the nicety of rook lines still actually restricting the enemy king into one side or the other. What does mating material look like here?

I'm not sure I understand the question.

I tried to work out (but without paper) how many squares a rook attacks on an empty board. There are 12 faces that it reaches in each direction, but those overlap, I think four faces in common? So it should be 5*12*2-4-1=115 (that last being the rook's current cell)? What about the bishop, or nightrider?... Oh, I guess bishops aren't colorbound?

That math/geometry is a bit wild, and it's a bit late right now (for me).

But yes, the Bishops are not colorbound, strictly speaking. It's not possible, on a cube's corner (much less a tesseract's), to have a colorbound check pattern. I didn't realize that when I put four on each side, but then I decided that it wasn't that big of a deal; switching is a trick that requires rounding a corner.

Is there a reasonable way to flatten this for displaying on a table/screen? (I suspect not, because of the forking of paths.)

You're correct, and that's only one reason; the nature of a tesseract makes it hard in general to lay it out in two dimensions. Someone could probably do a diagram showing the Home Faces and how they connect to the Territory Faces, and the latter to each other, for each side; how to extend that to the entire tesseract is another question entirely.

That said, I have seen people program 4D and even 5D Rubik's cubes, so it's probably possible.

Does the setup section's use of "clockwise" actually make sense?

Clockwise as seen by the player on the 2D board.

Now that I consider it, I probably should set up some kind of system where two corners are blacked-out and two are whited-out, to keep track of which corners are connected. That won't always be obvious.

Why alternate ordinary and berolina pawns? Doesn't that hurt pawn structures? (Does it just not matter on this wacky board?)

They should only run into each other in their early moves, and even then only if they make single moves at the start instead of double moves.

Personally, I think it doesn't so much hurt "pawn structures" as just make for different ones that would have to be worked out in play. That said, I don't think I'd do it that way on a board without this structure's unique flavors.

Fixes just done:

• Created a couple of illustrations showing the connections among the Home and Territory Faces on each side.
• Changed two corners of the Faces, to help keep track of how the pieces move from board to board.
• Explained in the text how Pawns and Spears can find their way off an Open Space.

Playing on the 2d surface has the nicety of rook lines still actually restricting the enemy king into one side or the other. What does mating material look like here?

I'm not sure I understand the question.

For starters, can two rooks force mate against a lone king? If the corners weren't removed I think they could. Two queens probably suffice? Maybe it's best to see if this has been resolved on the surface of a 3D cube first...

But yes, the Bishops are not colorbound, strictly speaking. It's not possible, on a cube's corner (much less a tesseract's), to have a colorbound check pattern. I didn't realize that when I put four on each side, but then I decided that it wasn't that big of a deal; switching is a trick that requires rounding a corner.

I'd like to try to work out a bishop's path (and number of attacked squares) partly for this reason: it might be possible to effectively wrap around corners by going the long-long way around?...

Clockwise as seen by the player on the 2D board.

But I'm not sure that makes sense, in the same way that clockwise isn't unambiguous in 3D: it depends on from which side of the surface you're looking.

For starters, can two rooks force mate against a lone king? If the corners weren't removed I think they could. Two queens probably suffice? Maybe it's best to see if this has been resolved on the surface of a 3D cube first...

I don't think that two Rooks could force mate against a lone King without a board's edge to work with, whether it's a tesseract, cube, toroid, or whatever. Granted, the blanked-out corners might give a little extra leverage if this were a cube, but I'm pretty sure the split seams of the tesseract would always allow the King an escape.

On the other hand, two Queens could probably sandwich a King into the middle of a Face to force a mate -- at least a stalemate, if not checkmate. I think a Queen probably could do it with the Chancellor, though I'm not sure about the Archbishop.

I'd like to try to work out a bishop's path (and number of attacked squares) partly for this reason: it might be possible to effectively wrap around corners by going the long-long way around?...

Well, I can follow at least one path of a Bishop using the flat diagram: on an empty structure, the one that starts at 1.c5 can proceed to 1.c5, 2.d1, 2.e2, 4.a3, 4.b4, 4.c5, etc. (or the same numbering on Faces 7 and 9); or, continuing from 2.e2, to 14.c5, 14.b4, 14.a3; or, from 7.e2, to 14.e3, 14.d2, 14.c1...

Without mapping out the whole thing, I'm pretty sure just about any position allows a Bishop to cover eight of the 21 spaces on each side -- either a2, a4, b1, b5, d1, d5, e2, and e4; or a3, b2, b4, c1, c5, d2, d4, and e3 -- plus up to two additional spaces on the Face where it starts. The exception is the center square, where it's limited to one space in each direction.

"Wrapping the corner" would not be possible in a single move.

But I'm not sure that makes sense, in the same way that clockwise isn't unambiguous in 3D: it depends on from which side of the surface you're looking.

Well, there's an inside and an outside. I'm pretty sure the Faces are arranged so that they're all viewed from the tesseract's outside (I can check on that, though).

This might be the most ambitious 4D variant I have come across. It's the sort of game I wouldn't want to try without a computer interface. The PDFs provide some help, but it would be better to see the actual boards with coordinates mapped out.

I don't think I'd want to play it without a computer interface either; using physical boards would probably require three tables, with one for each of the two Home/Territory combinations shown and the third with all of the Open Faces. As I say in the text, this is less a proposal for an actual game than an illustration to help others visualize four-dimensional space.

I'm not sure what sort of further illustrations would be helpful, but I'll see if I can come up with something like diagrams with multiple Faces together showing moves (especailly diagonal and oblique moves) from one to another. (Well, I knew when I started that there'd be special challenges illustrating four dimensions using only two....)

I've added an illustration that I hope is helpful for visualization. If nothing else, it's prompted me to look closely at that PDF to make sure that all of the faces are illustrated as seen from the outside of the tesseract.

(I haven't done that yet, but I will at some point in the next few days.)

I've added an illustration that I hope is helpful for visualization.

It's not enough. I'm still not visualizing how this game is supposed to work.

I'm still not visualizing how this game is supposed to work.

After I've proof-checked the PDF, I'll see if I can make another with square pages that could be used for an actual board layout, and provide an explanation of how they could be arranged (theoretically) for a physical game. (With a note, of course, that it would require three tables.)

I think that just numbering the faces is not good enough. The diagrams at the top are woefully inadequate, because they don't even make it clear what number each face has. So, I propose something more systematic. The tesseract image shows a smaller cube inside of a larger cube with lines between their corresponding corners. The faces of these two cubes do not count as faces of the tesseract, but the quadrilateral shapes that extend between the larger and smaller cubes are the faces of the tesseract. For each pair of parallel cube faces, the tesseract has four faces, and since each cube has six faces, we get six sets of four faces, which is a total of 24 faces. So, I propose naming each face with a pair of numbers. Use 1-6 to designate which cube faces the tesseract face is between. Let's assume it's like a die with opposite numbers adding up to 7.

Then use 1-4 to distinguish the faces on one side of the tesseract. Ideally, we could use the same number for faces that share the same cube corners, though I haven't worked it out to check whether this is possible. I think it is these two that you are saying a piece has a choice between moving to one or the other.

Let me try to work this out. Suppose 1-1 pairs with 5-1, 1-2 with 2-2, 1-3 with 3-3, and 1-4 with 4-4. With 2-2 paired with 1-2,

As I was trying to work this out, I got an idea that should prove easier. Each face of the tesseract abuts two different sides of each cube. One of these is primary, and the other one just indicates which of the four faces extending from the primary cube face is indicated. Numbering the cube like a die, the faces of the tesseract are 1-2, 1-3, 1-4, 1-5, 2-1, 2-3, 2-4, 2-6, 3-1, 3-2, 3-5, 3-6, 4-1, 4-2, 4-5, 4-6, 5-1, 5-3, 5-4, 5-6, 6-2, 6-3, 6-4, and 6-5. Pairs with the same number (1-1, 2-2, etc.) and pairs with numbers adding up to 7 (1-6, 2-5, etc.) are excluded. You could remove the hyphens and just use them as coded number pairs rather than true numerals. Using these makes visualization easier, because faces with the same numbers in the opposite order pair up, and neighboring faces will always share a number.

The faces of these two cubes do not count as faces of the tesseract, but the quadrilateral shapes that extend between the larger and smaller cubes are the faces of the tesseract

I think this is incorrect. The faces of the two cubes are 2d faces of the tesseract. E.g. the faces numbered 1 and 2 in the first diagram (the slant of the font helps identify where the numbers are supposed to be.)

Numbering the cube like a die, the faces of the tesseract are 1-2, 1-3, 1-4, 1-5, 2-1, 2-3, 2-4, 2-6, 3-1, 3-2, 3-5, 3-6, 4-1, 4-2, 4-5, 4-6, 5-1, 5-3, 5-4, 5-6, 6-2, 6-3, 6-4, and 6-5.

I'm not positive I understand what you're describing here, but I think 1-4 and 4-1 are the same face. And the ones you're missing are two copies (one from the large and one from the small cube) of say "1" (1-1? 1-0?) through "6". (While 1-6 indeed doesn't exist.)

Ben is correct, in that the inner and outer cubes are included; in fact, the setup pieces are put there (on the nearer face of the "large" one, and the far face of the "small" one). That's a total of 24 faces.

And I'm even more confused than Ben about what you're suggesting with the numbered pairs.

I'll give a further example of how they're connected. A piece leaving Face 1 going "up" (as seen in the cubic illustration) has a choice of going to Face 2 (the top of the big cube) or Face 7 (the slanted face leaning down from the top of 1). If it's sliding orthogonally, then it can contiunue across Face 2 to either Face 6 (opposite Face 1) or Face 15 (which slants down from the top of Face 6), or across Face 7 to either Face 19 (the small-cube face "behind" Face 1) or Face 20 (the top of the small cube).

Then, as if that wasn't murky enough for folks limited to three dimensions: if the piece continues from there, it can then go to Faces 5 or 18 from 6, 20 or 24 from 15, 10 or 23 from 19, or 15 or 24 from 20 (that last being the first of probably several errors in the PDF document, which claims 17 or 24).

In order to help that, I suppose I could call the Home Faces White Zero (W0) and Black Zero (B0), and the ones adjacent to them White/Black 1-8 (or 1-4 a/b). The remaining six faces (currently the "Open Faces": 6, 11-14, and 19) could then be Grey 1-6.

I'd be afraid that that might be even more confusing, though.

The (solid) tesseract can be described as the set of points (x,y,z,w) with every coordinate between 0 and 1. The 2d faces then are characterized by setting exactly two coordinates to either 0 or 1 (and letting the other two coordinates range between 0 and 1).

Subdividing those faces into a 5x5 checkerboard, the centers of the squares are points with exactly two coordinates equal to 0 or 1 and the other two among {0.1, 0.3, 0.5, 0.7, 0.9}. Perhaps then a usable coordinate system just replaces those decimals by a,b,c,d,e respectively.

In that system, one rook path would start
01cc, 01dc, 01ec,
at which point the rook hits the edge point "011c" (not a playable square) and the two continuation paths must move off the extreme value of either of the first two coordinates:
, a11c, b11c, c11c, ...
, 0e1c, 0d1c, 0c1c, ...

I think this is incorrect. The faces of the two cubes are 2d faces of the tesseract.

Okay, I was trying to make sense of this in a way that added up to 24 faces. Assuming that what I called 1-2 and 2-1 are the same face and so on, the faces I counted as 24 reduce to 12, there being one for each edge of a cube. Then the remaining faces would be the six faces of each cube. That makes more sense.

However, simply numbering them has not been a helpful way of keeping track of them, and the diagrams have not made it clear which numbers match which faces. So, I recommend going with something more systematic that can better convey relations between faces.

Ideally, the method should be neither redundant nor arbitrary. But I'll start by describing a redundant method. This would start by numbering the inner cube like a die. Using one side of the inner cube as a reference point, we could identify up to six different faces. For example, starting with 1, 1-0 would be the inner face, 1-2, 1-3, 1-4, and 1-5 would be neighboring faces, and 1-1 would be the outer face parallel with 1-0. This would be redundant, because 1-2 would be the same as 2-1, etc.

To make this non-redundant, I propose writing each designation with the lower digit first. So, then we would have, dispensing with the hyphen, 01, 02, 03, 04, 05, and 06 for the inner cube faces, 11, 22, 33, 44, 55, and 66 for the outer cube faces, and 12, 13, 14, 15, 23, 24, 26, 35, 36, 45, 46 and 56 for the other faces.

To make this non-redundant, I propose writing each designation with the lower digit first. So, then we would have, dispensing with the hyphen, 01, 02, 03, 04, 05, and 06 for the inner cube faces, 11, 22, 33, 44, 55, and 66 for the outer cube faces, and 12, 13, 14, 15, 23, 24, 26, 35, 36, 45, 46 and 56 for the other faces.

That might be doable. Let me ponder it for a while, and see what it looks like in implementation.

In the figure above, White sets up on and around Face 1 (the front of the large cube), while Black sets up on and around Face 24 (the back of the small cube)

These would be what I'm calling 11 and 06.

Adjacent to each side of the Home Face are two more Faces; these eight are the Territory Faces.

So, you're saying that each side is individually adjacent to two more faces, and since each face has four sides, it has eight adjacent faces.

For White, these are Faces 2, 3, 4, 5, 7, 8, 9, and 10;

These would be what I'm calling 12, 13, 14, 15, 22, 33, 44, and 55. However, I'm not sure how these labels would match yours.

for Black, they're 15, 16, 17, 18, 20, 21, 22, and 23.)

These would be what I'm calling 02, 03, 04, 05, 26, 36, 46, and 56. Again, I don't know the exact correspondence with your labels.

On the edge of the Face adjacent to the Home Face,

I think you mean "On the adjacent edge of each Face that is adjacent to the Home Face,".

In the figure above, White sets up on and around Face 1 (the front of the large cube), while Black sets up on and around Face 24 (the back of the small cube)

These would be what I'm calling 11 and 06.

Adjacent to each side of the Home Face are two more Faces; these eight are the Territory Faces.

So, you're saying that each side is individually adjacent to two more faces, and since each face has four sides, it has eight adjacent faces.

Correct. That's the basic geometry of a tesseract.

For White, these are Faces 2, 3, 4, 5, 7, 8, 9, and 10;

These would be what I'm calling 12, 13, 14, 15, 22, 33, 44, and 55. However, I'm not sure how these labels would match yours.

for Black, they're 15, 16, 17, 18, 20, 21, 22, and 23.)

These would be what I'm calling 02, 03, 04, 05, 26, 36, 46, and 56. Again, I don't know the exact correspondence with your labels.

I'm not sure either. In case this part isn't clear: 2, 3, 4, and 5 are on the "outer" cube; 7, 8, 9, and 10 extend from the "outer" cube to the "inner" one; 20, 21, 22, and 23 are on the "inner" cube; and 15, 16, 17, and 18 extend from the "inner" cube to the "outer" one.

Perhaps if I broke down the tesseract image to just the faces on each side, showing their relationship to Faces 1 and 24, things might be a bit clearer. (A third image showing just the Open Faces would also be included.)

On the edge of the Face adjacent to the Home Face,

I think you mean "On the adjacent edge of each Face that is adjacent to the Home Face,".

Yes, I was perhaps a bit overly concise in that phrase. :)

If you were to start White on the inner cube and Black on the outer cube, then White's Home Face could be 01 and Black's could be 66, which would be the lowest and highest numbers respectively, like 1 and 24 currently are. Then White's adjacent faces would be 02, 03, 04, 05, 12, 13, 14, and 15, which are all low, while Black's would be 22, 26, 33, 36, 44, 46, 55, and 56, which are all higher.

If you were to start White on the inner cube and Black on the outer cube...

I actually have it the other way around, with White on the "outer" cube (Face 1) and Black on the "inner" cube (Face 24), but the suggestion still works.

These are the illos I made from the tesseract. White's Home and Territory Faces:

White sets up the main pieces on Face 1, and the Pawns and Spears go on the other eight.

Black's Home and Territory:

Black sets up main pieces on 24, and the Pawns and Spears on the other eight Faces. (This one may need some revisiting, since the numbers are so cramped on the "inner" cube.)

And the Open Faces:

I probably will give a different name to 6 and 19, as they have a special quality of both being on the opposite sides of a cube from the Home Faces. Something like Far Faces, Remote Faces, Opposing Faces... I'm currently undecided.

As for the renumbering suggestion, I'm still mulling it over. I'll put my thoughts on it into a separate Comment, probably later today.

If you were to start White on the inner cube and Black on the outer cube...

I actually have it the other way around, with White on the "outer" cube (Face 1) and Black on the "inner" cube (Face 24),

I know. That's why I used the subjunctive mood to express a counterfactual within the antecedent of a conditional.

While these diagrams provide some help, they don't address the arbitrariness of just numbering the faces sequentially. With the system I propose, it is immediately obvious which neighbors of a face border on the same edge. For example, 12 and 22 share the same border with 11, as do 13 and 33, 14 and 44, and 15 and 55. Knowing that the numbers 1-6 are arranged like on a die, other relations can also be known without consulting a diagram. For example, 12 is adjacent to 13 and 14 but not to 15. However, it is purely arbitrary, using your iterative system, that 2 and 7 share a border with 1 or that 7 is adjacent to 8 and 9 but not to 10. When something is purely arbitrary, it must be committed to memory and is easier to forget.

Hi Bob,
an interesting approach, multi-layered and engaging.

But one thing interests me: can you play it? I don't think I'm capable of it!

In my opinion, it's a matter of mathematical models that defy human understanding. Beautiful, complex and hardly playable. Who wants to dig into the subject?

But it is certainly an interesting construct.

It's not really "purely arbitrary," but the pattern it uses isn't really related to the 4D shape so it might as well be.

I'll see what I can do to make the numbers relevant to their position, using a system similar to yours.