Ratings & Comments

I would still argue that Gilman isn't wrong in anything, but takes different definitions and reaches different conclusions.

So far, I have not addressed his math, which I will now check here:

Here is what he wrote on the page:

Standard diagonal (SD) directions, the diagonal of the standard square-cell geometry formed by simultaneously moving equal numbers of orthogonal steps at right angles have steps of length root-2. Nonstandard diagonal (ND) directions include the diagonals outside a 2d plane in the cubic geometry, colloquially called triagonal, and the diagonals of the 2d hex board. I use the same term for both as both have steps of length root-3

I now understand that by root-2 and root-3 he means the square roots of 2 and 3. For square and cubic boards, these values check out. The diagonal distance between the midpoints of two diagonally adjacent squares is the same length as the diagonal of the square itself. This is the hypotenuse of a right triangle with two of the sides. Taking each side to have a length of 1, the Pythagorean theorem gives it a length equal to the square root of 2. Likewise, the distance between the midpoint of two cubes sharing a single corner is the same as the distance between the two opposite corners of a cube. When I looked this up, I learned that it is equal to the square root of 3 times the length of one side. Giving the side a length of 1, this diagonal has a length of root-3.

Let's now consider the hexagon. When the side is equal to 1, the distance between two opposite corners will be 2, as a hexagon can be divided into six equilateral triangles sharing a common point. See How many equilateral triangles are there in a regular hexagon?. However, since there is a gap between two diagonally adjacent hexagons, this has to be added in. With a side length of 1, this gap is 1, making the total distance a Bishop travels between two diagonally adjacent hexagons as 3. Well, 3 is not its square root. So this doesn't give us the same value.

But when we start with a side length of 1, the distance a Rook moves between two adjacent squares is not 1, which is already a problem. To calculate this distance, I will refer to this Regular Hexagon Cheatsheet:

The first thing I'll do is calculate the distance between two opposite sides when a side has a length of 1. For this, we want to look at the Inradius formula for R_i. With a side length of 1, a is 1, and R_i is half the square root of 3. Since R_i is half the length we're looking for, the distance between two opposite sides is root-3, which is 1.732050807568877. So, with a side length of 1, a Rook in hexagonal Chess travels the same distance as a Unicorn in 3D Chess.

To better harmonize the distances that pieces travel on hexagonal and 3D boards, I figured it would be best to give the Rook a distance of 1 on a hexagonal board. This allows Rooks to start out traveling the same distance on each board. Since the distance between two opposite corners is twice the length of one side, I need to figure out the length of one side when the distance between opposite sides is 1. I can do this with a ratio, since I already know three of the values and can solve for the fourth. So, x is to a length of 1 as 1 is to the square root of 3. As an equation, it looks like this:

x/1 = 1/root-3

So, x = 1/root-3

Since we want three times that, we get 3 divided by its square root, which is its square root. So the math checks out.

While the math does check out, I consider this a reductio ad absurdum against his idea of defining piece movement in terms of distance. A piece that moves in all three dimensions, as a Unicorn does, cannot have any legal moves on a 2D board. Although the hexagonal Bishop moves the same distance as a 3D Unicorn, the distance a piece moves is not the same as the manner in which it moves. I conceive of the Bishop as going through 2D diagonals of polygons and of the Unicorn as going through the 3D diagonals of polyhedrons. If you change the shapes of the spaces, you are going to get different measurements, but the idea behind 2D diagonal movement and 3D diagonal movement will remain the same. In circular or spherical Chess, the measurements will even vary between different spaces on the same board, but the idea of 2D diagonal movement will still make sense when it is understood as going through the corners of spaces.

OK, I see. And projecting a 3d Rook would give a null-move-capable 2d Rook.

But the diagonal slice through the cubic lattice does not produce a board of hexagons. If you extend the plane outside the cube you showed in the drawing, you get triangles bordering the hexagons.

If you take the slice parallel to the cubic lattice coordinate planes, you get a normal chess board, and 3d bishops project to normal Bishops on it. As do Unicorns.

The way I'm thinking about it, when you project along one of the 3d orthogonal directions, you do get a normal chess board, but 3d bishops become 2d queens, and unicorns become 2d bishops. E.g., projecting along the z direction, a 3d bishop sliding along (+1,0,+1) projects to a rook movement (+1,0).

The connection between the 2d and 3d moves is a projection, right? You make a 2d slice through the 3d lattice, and then projet teh trajectory of the 3d piece on it. If you take the slice parallel to the cubic lattice coordinate planes, you get a normal chess board, and 3d bishops project to normal Bishops on it. As do Unicorns. The interesting quastion is what would happen if you do that with the haxagonal slice. Neither Bishops nor Unicorns move in the slice.

Apologies for the confusion; I had misremembered what I worked out years ago, and didn't spend much time checking how my thoughts aligned with the Red Blob site I linked.

When I've done this before, I project down along one triagonal, and the result is a hexagonal grid, but not with the "cubic coordinates" of the Red Blob site. Instead, you get coordinates whose sum is 0, 1, or 2 (any other sum in the cube coordinates get projected down to one of these, depending on its remainder mod 3), and those correspond to the 3-coloring of the hex grid. Under this projection, if my doodling just now is correct, cubic rooks are hex rooks; cubic bishops are hex rook+bishop; cubic unicorns are hex dabbabah+null-move.

That aside, I would still argue that Gilman isn't wrong in anything, but takes different definitions and reaches different conclusions. We've had games here on non-regular tesselations, and differences of opinions arise when trying to think more-topologically or more-geometrically.

The relationship between hexagonal and cubic grids runs deeper, and in that sense hex bishops do correspond to unicorns. See e.g. https://www.redblobgames.com/grids/hexagons/#coordinates-cube

The article you linked to describes how hexes on a hexagonal grid could be described with three coordinates, because the six sides of a hexagon give it three axes. It's because of this that there is a choice when it comes to which two axes will be used to represent spaces for a particular hexagonal game, and different games have represented coordinates differently. Anyway, this article tries to make a correspondence between hexagonal grids and 3D grids, because each can be described with three coordinates. That's interesting as far as it goes, but the question remains whether there will be any correspondence between how pieces move on a 3D board and how they move on a hexagonal board. Since the article is simply about geometry and not about Chess variants, it does not cover that.

On a 3D cubic board, a Unicorn should move an equal distance in all three dimensions. So, if it starts out at (0,0,0), it could move to spaces with coordinates like (1,1,1), (2,2,2), (-1,1,-1), and so on. Looking at the diagram for axial coordinates on a hexagonal grid, the spaces that a Bishop could move to from (0,0,0) are all at a distance of 1 in two dimensions and a distance of 2 in the other dimension. These distances do not match how a Unicorn moves on a 3D board, and in fact, the grid does not contain a single hex with coordinates matching any space a Unicorn could move to from (0,0,0) on a 3D board.

I will also note that the spaces a hexagonal Bishop could reach from (0,0,0) do not match those that a 3D Bishop could reach. A 3D Bishop would be able to move to a space that is the same distance away in two dimensions but a distance of zero in the other dimension. Looking at the Axial diagram, these are on the orthogonal rows passing through (0,0,0). In other words, the way a Rook moves on a hexagonal board corresponds with the way a Bishop moves on a 3D board, at least in terms of how the coordinates change.

On a 3D board, a Knight move should adjust one coordinate by 1, one by 2, and leave the third the same. Looking at the coordinates on the Axial grid, there are no coordinates in that relation to (0,0,0). Every coordinate with no change in one coordinate has the other two at equal distances from (0,0,0). These all match the spaces a hexagonal Rook could move to.

So, in terms of Chess piece movements, I do not see much correspondence between hexagonal boards and 3D cubic boards. The changes in coordinates that Chess pieces are capable of will not be the same on each board. And even with three coordinates, the hexagonal board is missing many spaces from a 3D board and does not seem to support the sorts of coordinate changes that Rooks, Knights, and Unicorns are capable of in 3D Chess. While it supports the sort of coordinate changes Bishops are capable of, these belong to the hexagonal Rook, which can reach every space on the board.

Given all this, I do not see any way to support the idea that hexagonal Bishops correspond to 3D Unicorns. My takeaway from this is that piece movement should be defined in terms of geometric relations and not in terms of numeric coordinate changes. By defining it in terms of geometric relations, I can use the same definitions for Chess pieces on square, cubic, and hexagonal boards, and the pieces will move appropriately on each board. But if I try to define them in terms of coordinate changes, six-sided polygons (hexes) and six-sided polyhedrons (cubes) will not work the same way.

In 3d geometry one distinguishes body-diagonals and face diagonals.

BTW, isn't it possible to cut through a cubic lattice so that you get hexagons (in a plane perpendicular to a body diagonal)?

The Unicorn was invented for 3D games, and I disagree with the claim that the diagonals on a hexagonal board are the Unicorn's diagonals rather than the Bishop's. In various hexagonal adaptations of Chess or Shogi, the Bishop has been understood to move along lines of spaces that are the same color.

His reasoning for this begins with how he defined standard diagonals:

Standard diagonal (SD) directions, the diagonal of the standard square-cell geometry formed by simultaneously moving equal numbers of orthogonal steps at right angles have steps of length root-2.

I would disagree with this, defining diagonals instead as lines formed by passing through the opposite corners of spaces. This fits with the etymology of diagonal, for dia- means through, and -gonal refers to the corner. Similarly, a polygon literally means many-angled. If the shapes are not squares, then diagonal movement will not reach the same spaces as you could reach by making an equal number of moves in two directions at right angles to each other. It is merely a coincidence that it works this way on a square board, but it is not the definition of diagonal.*

The problem comes in when we start thinking about how diagonals should work on 3D boards. Since the Bishop is a colorbound piece on the 2D board, the decision was made to make it a colorbound piece on the 3D board too. But to do this, it was given a move that does not pass through the corners of a cubic space. Thinking of each space in a 3D game as a cube, the level-to-level movement of a Bishop in 3D Chess goes from one cube edge to the edge that is opposite in all three dimensions. However, this can be thought of as diagonal movement if we think of each vertical slice of a 3D board as a 2D board of squares. In fact, this is not so different from how we think of movement on the same level. We easily see the usual Bishop's moves as diagonal on one level of a 3D board, because we view it as a 2D board of squares. But if we thought of each space as a cube, the movement would no longer appear diagonal. This is easier to see, because of the way 3D boards are constructed. But mathematically, the board can be divided into vertical slices just as easily as it can be divided into horizontal slices. (Note that these vertical slices come in same-rank varieties and same-file vartities.) If we view the Bishop's movement as 2D diagonal movement through a single vertical or horizontal slice of the board, it can still be understood as diagonal in a 3D game.

Therefore, I would distinguish between 2D diagonals and 3D diagonals, and not between standard and nonstandard diagonals. Bishops move along 2D diagonals, even on 3D boards, and Unicorns move along 3D diagonals, which just don't exist on 2D boards, including 2D hexagonal boards. So it is wrong to say that the diagonal movement on a hexagonal board is how a Unicorn rather than a Bishop would move. The 3D Unicorn simply has no legal moves on a 2D board of any kind.

* As an afterthought, his definition of Standard diagonal could be tweaked for spaces of other shapes than square. On a hexagonal board, you can reach spaces along the diagonal though pairs of alternating orthogonal moves that are 60 degrees apart from each other. This works, because each angle of a hexagonal space is 60 degrees, not 90 degrees like it is for a square.