Talk:Heesch's problem
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I think that the definitions in this article need to be rewritten. The definition of corona refers to "the" set of tiles ... What set of tiles? A base shape may have various valid coronas. The definition also omits the necessary condition that a corona surround the previous corona. The introductory paragraph mentions this condition, but the formal definition does not.
This condition must be defined precisely. For example, if the starting shape is a square and we adjoin four squares to it, edge to edge, do they surround it? Later the text mentions a modified definition, in which the tiles of a corona must be "simply connected." This cannot be right. "Simply connected" is meaningful only for a single region, not a set of regions. In every example that I have seen, a corona (except for Corona 0) is multiply connected, because it has a hole, the hole that contains the previous corona.
Mann's definition requires that no point of the surrounded shape may be visible in the plane from the exterior of the corona. I think that this definition is adequate, provided that "exterior" is taken to mean the complement of the tiling with respect to the plane, including any holes that may remain among the tiles of the corona. Mann explicitly disallows holes in a corona. This too is essential to the definition. Sicherman (talk) 12:08, 28 June 2020 (UTC)
- When it defines a corona, it is with respect to a specific already-given tiling. So, re your question "what set of tiles?" The tiles in the given tiling. A base shape has only one corona in that given tiling. Because these definitions assume a given tiling of the entire plane (which may for instance have tiles of other shapes, including a big tile containing all but a finite area of the plane), the condition that it surround the previous corona follows automatically. Therefore there is no need to include complicated attempts to define "surround" in the actual definition of a corona. As for your complaint about holes, look again at the definition of the zeroth corona and at the statement that the zeroth through kth coronas form (as their union, to spell out the obvious but unstated detail) a simply connected region. —David Eppstein (talk) 16:36, 28 June 2020 (UTC)
Wouldn't it make more sense to define Heesch's problem in such a way that the middle tile (zeroth corona) with its coronas must form a closed topological disk? Then the problem with the holes in the outer corona would have been dealt with.Mikematics (talk) 12:03, 30 April 2021 (UTC)
Heesch numbers in 3D edit
Here are some examples of solids with finite Heesch numbers: First, I found this example by Jadie Adams et al. It also includes a 2D example with Heesch number 2.
https://www.sci.utah.edu/~jadie/3DHeesch_Poster.pdf
These are my own creations. The images show that they have at least Heesch numbers 1, 2 or 3. The first example is a 3D analog to the example by Anne Fontaine.
The second and third tiles are analogous to the example by Casey Mann and Bojan Bašić. Since the tiles are "unbalanced", they cannot tile 3D space, so the Heesch number must be finite. Theoretically, six of the "pointy ends" can be arranged to form a cube, so the argument with the imbalanced tiles does not hold. It is highly unlikely that such a configuration would form a better tiling.
Jens G. Maier (talk) 22:58, 16 August 2023 (UTC)