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The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. It is well known that in Euclidean geometry, all triangles and all quadrilaterals tessellate the plane. A natural question would be: In non-Euclidean geometry, are there triangles and quadrilaterals that cannot tile the hyperbolic plane?

  • There indeed are triangles and quadrilaterals that cannot tile the hyperbolic plane (example given in note 3 below). So, one can ask: which triangle(quadrilateral) gives worst(least) packing fraction for the hyperbolic plane?

  • Which convex region gives worst packing in hyperbolic plane?

Note 1: All these questions can be asked in elliptic geometry as well. One can also ask about the worst polygons for covering in the two geometries.

Note 2: If one defines a parallelogram as a quad where pairs of lines that contain opposite sides do not intersect, then, in hyperbolic geometry, opposite sides of a parallelogram need not be congruent, so even the usual 2D periodic lattices of Euclidean plane do not have obvious analogs.

Note 3: Here is a family of non-regular polygons that, I gathered, can tile the hyperbolic plane: Triangles with angle measures: pi/p, pi/ q, pi/r where p,q,r integers are non-regular polygons that tile the hyperbolic plane. If p, q, r are irrational, such triangles do not tile. But it is not clear which triangle is worst for tiling the hyperbolic plane.

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    $\begingroup$ I think it should be clear that an equiangular triangle whose every angle is $59^\circ$ is incapable of tessellating the hyperbolic plane. $\endgroup$
    – Lubin
    Commented Jul 23, 2021 at 20:15
  • $\begingroup$ Thank you. I am convinced now that not all triangles (or quads) can tessellate the hyperbolic plane. So, the only issue is the bounds on how bad they can be for tessellation. $\endgroup$
    – Nandakumar R
    Commented Jul 23, 2021 at 20:33
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    $\begingroup$ Not all quadrilaterals tesselate the plane. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 24, 2021 at 0:54
  • $\begingroup$ Thanks. In euclidean geometry, all non-self intersecting quadrilaterals tessellate the plane. The question did not explicitly say so but it concerns only non-self intersecting, simple polygons. $\endgroup$
    – Nandakumar R
    Commented Jul 24, 2021 at 9:38
  • $\begingroup$ It is unclear what do you mean by worst triangles for tiling. $\endgroup$
    – Moishe Kohan
    Commented Jul 25, 2021 at 20:48

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