'Mixed' Packing of the Plane
The basic question: to pack the plane efficiently - without overlaps - using more than one type of unit. In particular let us consider packings with 2 types of units.
Packing the plane with two types of units - call them C1 and C2 - seem to belong to two categories - where the relative numbers of the 2 types of shapes are equal and where they are not. 'equal number' is used here in the sense that the plane ought to be filled with an 'alloy' which consists of C1 and C2 units in equal proportion. One feels that the equal numbers case is more interesting.
Example: We need to pack the plane with maximum coverage using unit disks and unit squares both in equal number.
A simple approach which might be optimal: pack a half plane optimally with unit circles alone (Note: the hexagonal lattice is known to be the best layout with unit circles if the entire plane is to be packed. It is very likely the best way for a half plane as well) and the other half plane can be tiled with unit squares. One needs to be careful as to how the two half-plane lattices meet at their interface.
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We could also think of homogeneous layouts using both disks and squares. By homogeneous, one could mean: for any sufficiently large real number r1 and any point on the plane P, if a circle with radius r1 centered at P contains or intersects more disks than squares then, there is some other real number r2 >r1 such that a circle with radius r2 centered at P intersects more squares than disks. And if the same condition holds for any center P and also with 'disk' and 'square' interchanged, then the layout is homogeneous. Homogeneity of a 2-unit layout formed by C1 and C2 can also be viewed as the property that every C1 unit has among its neighbors at least as many C2 units as C1 units and vice versa.
Questions: Which homogeneous layout of unit disks and squares gives maximum coverage of the plane? (On the other hand, which homogeneous layout of disks and squares gives the thinnest rigid pack of the plane?)
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On a more general note: Consider any two convex regions C1 and C2 satisfying some additional properties such as say, equal area or equal area and perimeter (unit disk and unit square are not of equal area). Can one have some 'sweeping' results such as (say): "the max coverage pack of the plane using C1 and C2 in equal numbers is necessarily formed of two half planes one packed with C1 and the other with C2. IOW, any kind of mixing of the units leads to reduced coverage"? Oppositely, are there pairs of convex equal area shapes for which the best plane packing is a homogeneous one? What could one say about periodicity / aperiodicity of such optimal mixed packings?
Obviously, similar questions can be asked for covering the plane and for packing in higher dimensions.
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Further Question 1: Are there pairs of convex shapes of equal area such that the layout of maximum packing coverage using both (with no homogeneity requirement) necessarily has them in a ratio other than 1:1?
Here is an example
That is a perfect pack of the plane with 'chopped hexagons' (all these heptagons are identical although shown in 3 different colors) and equilateral triangles (all in yellow) with their relative populations in the ratio 3:1. No other layout - including the ones where the two shapes don't mix - appears to yield a perfect pack.
One can ask for pair of convex shapes with both shapes being of equal area and/or that pack the plane optimally when they are in other population ratios, say, 7:3.
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2. This is a variant of the Heesch problem (https://en.wikipedia.org/wiki/Heesch%27s_problem) - Given two convex regions C1 and C2 that together do not tile the plane. Starting with a single C1 unit, how many gap-free layers of tiles can be built around it using both C1 and C2 units, at least using one C2 unit in every layer - and without further constraints on their relative numbers? The question can just as well begin with a central C2 unit.
Obviously, with C1 and C2 being a square and a disk, we wont progress at all from the central unit. There must be {C1, C2} pairs such that the number of layers possible around a central C1 and the number layers around a central C2 are different. And then there is the 'wall' variant of the Heesch problem - https://arxiv.org/abs/1605.09203!
Packing the plane with two types of units - call them C1 and C2 - seem to belong to two categories - where the relative numbers of the 2 types of shapes are equal and where they are not. 'equal number' is used here in the sense that the plane ought to be filled with an 'alloy' which consists of C1 and C2 units in equal proportion. One feels that the equal numbers case is more interesting.
Example: We need to pack the plane with maximum coverage using unit disks and unit squares both in equal number.
A simple approach which might be optimal: pack a half plane optimally with unit circles alone (Note: the hexagonal lattice is known to be the best layout with unit circles if the entire plane is to be packed. It is very likely the best way for a half plane as well) and the other half plane can be tiled with unit squares. One needs to be careful as to how the two half-plane lattices meet at their interface.
--------
We could also think of homogeneous layouts using both disks and squares. By homogeneous, one could mean: for any sufficiently large real number r1 and any point on the plane P, if a circle with radius r1 centered at P contains or intersects more disks than squares then, there is some other real number r2 >r1 such that a circle with radius r2 centered at P intersects more squares than disks. And if the same condition holds for any center P and also with 'disk' and 'square' interchanged, then the layout is homogeneous. Homogeneity of a 2-unit layout formed by C1 and C2 can also be viewed as the property that every C1 unit has among its neighbors at least as many C2 units as C1 units and vice versa.
Questions: Which homogeneous layout of unit disks and squares gives maximum coverage of the plane? (On the other hand, which homogeneous layout of disks and squares gives the thinnest rigid pack of the plane?)
-----------
On a more general note: Consider any two convex regions C1 and C2 satisfying some additional properties such as say, equal area or equal area and perimeter (unit disk and unit square are not of equal area). Can one have some 'sweeping' results such as (say): "the max coverage pack of the plane using C1 and C2 in equal numbers is necessarily formed of two half planes one packed with C1 and the other with C2. IOW, any kind of mixing of the units leads to reduced coverage"? Oppositely, are there pairs of convex equal area shapes for which the best plane packing is a homogeneous one? What could one say about periodicity / aperiodicity of such optimal mixed packings?
Obviously, similar questions can be asked for covering the plane and for packing in higher dimensions.
----------
Further Question 1: Are there pairs of convex shapes of equal area such that the layout of maximum packing coverage using both (with no homogeneity requirement) necessarily has them in a ratio other than 1:1?
Here is an example
That is a perfect pack of the plane with 'chopped hexagons' (all these heptagons are identical although shown in 3 different colors) and equilateral triangles (all in yellow) with their relative populations in the ratio 3:1. No other layout - including the ones where the two shapes don't mix - appears to yield a perfect pack.
One can ask for pair of convex shapes with both shapes being of equal area and/or that pack the plane optimally when they are in other population ratios, say, 7:3.
-----------------
2. This is a variant of the Heesch problem (https://en.wikipedia.org/wiki/Heesch%27s_problem) - Given two convex regions C1 and C2 that together do not tile the plane. Starting with a single C1 unit, how many gap-free layers of tiles can be built around it using both C1 and C2 units, at least using one C2 unit in every layer - and without further constraints on their relative numbers? The question can just as well begin with a central C2 unit.
Obviously, with C1 and C2 being a square and a disk, we wont progress at all from the central unit. There must be {C1, C2} pairs such that the number of layers possible around a central C1 and the number layers around a central C2 are different. And then there is the 'wall' variant of the Heesch problem - https://arxiv.org/abs/1605.09203!
posted by R.Nandakumar @ 11:54 PM
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