When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of the 8 cardinal directions to another tile is the same length. (not necessarily euclidean). It would also be nice to minimise holonomy.
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$\begingroup$ what is a direction in a non-Euclidean geometry ? $\endgroup$– Selim GCommented Dec 11, 2020 at 11:11
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1$\begingroup$ This perhaps? en.wikipedia.org/wiki/Octagonal_tiling $\endgroup$– WojowuCommented Dec 11, 2020 at 11:37
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$\begingroup$ @Selim G the same could be said about euclidean geometry. In this case i just mean that the cardinal north always points up no matter the rotation $\endgroup$– YEp dCommented Dec 11, 2020 at 11:58
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$\begingroup$ In Euclidean geometry, directions are well-defined because since the curvature vanishes, tangent spaces are canonically isometric via the holonomy of Levi-Civitta. This ceases to be true in non-zero curvature. What I mean by that is that it really makes sense to compare directions at any two points in Euclidean geometry, whereas it does not in other geometries. $\endgroup$– Selim GCommented Dec 12, 2020 at 17:31
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