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🕸Fergus Duniho wrote on Sun, Jan 12 02:15 PM UTC in reply to Ben Reiniger from Sat Jan 11 06:06 PM:

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 Ri. With a side length of 1, a is 1, and Ri is half the square root of 3. Since Ri 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.


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