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Bn Em wrote on Mon, Aug 2, 2021 05:04 PM UTC:

While on its own the original diagram on this page was a bit obscure, in conjunction with Fergus' circular diagrams it really clarifies the rationale behind Nadvorney's interpretation of the diagonal move; a bit of thought also reveals why Miller's reasoning in keeping the bishop on its own colour is flawed: if the bishop stays on its own colour you would expect a rook stepping over the pole to change colour as it does on a normal square‐cell board, whereas here (or on any spherical/Klein‐bottle‐shaped board with a multiple of four files) it doesn't. On e.g. a 10‐file board, Miller's reasoning would line up with Nadvorney's.

As for Chess on the Dot, the change in the diagonal's handedness at the poles also keeps it on one colour (on a board of this parity), but isn't stirctly necessary for a closed loop: Nadvorney's version (as can be seen on its diagram) does it just as well, and even Miller's manages, albeit via a much more circuitous route.

Fwiw, here's the original diagram as salvaged from the Internet Archive:

c7  d7  e7  f7  g7  h7  a7  b7  c7  d7  e7  f7
c8  d8  e8  f8  g8  h8  a8  b8  c8  d8  e8  f8
g8  h8  a8  b8  c8  d8  e8  f8  g8  h8  a8  b8
g7  h7  a7  b7  c7  d7  e7  f7  g7  h7  a7  b7
g6  h6  a6  b6  c6  d6  e6  f6  g6  h6  a6  b6
g5  h5  a5  b5  c5  d5  e5  f5  g5  h5  a5  b5
g4  h4  a4  b4  c4  d4  e4  f4  g4  h4  a4  b4
g3  h3  a3  b3  c3  d3  e3  f3  g3  h3  a3  b3
g2  h2  a2  b2  c2  d2  e2  f2  g2  h2  a2  b2
g1  h1  a1  b1  c1  d1  e1  f1  g1  h1  a1  b1
c1  d1  e1  f1  g1  h1  a1  b1  c1  d1  e1  f1
c2  d2  e2  f2  g2  h2  a2  b2  c2  d2  e2  f2

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