Putting N points on a Spherical Surface

One could think of several ways to optimally put N points (call these "vertices") on a spherical surface:
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1. Such that the distance from any point on the spherical surface to the closest vertex to it (geodesic distance measured along the surface) is least ( this is basically the same as optimal covering).

2. such that the distance between any pair of vertices is maximized (same as packing).

3. such that the volume of the N vertex polyhedron formed by the vertices has max volume.

4. such that the polyhedron has the max surface area.

5. such that the polyhedron maximises the sum of edge lengths.

6. such that the arithmetic mean of the NC2 distances (measured along the surface) is maximum.

7. such that the geometric mean of the NC2 distances is max.

8. such that the harmonic mean of the NC2 distances is max.

Right now, let us stop with these 8.
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one can ask several questions now:
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1. For what lowest value of N are the arrangments achieving each optimum necessarily different (two arrangments of points are different even if N-1 of the points are identically placed but 1 is 'off')?

2. Is there any pair of questions above for which for all N, the same configuration of N vertices can be optimal? This looks unlikely.

3. Is there any pair (or triplet) of questions above for which for *infinitely many N*, the same arrangment of the N vertices can give optimal result?

4. Is there any pair of questions above for which one can have the same optimal arrangment of N values only for finitely many values of N? If so, are all pairs such?

And so on...
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One can also go from spherical surface to *any convex closed smooth surface* and ask analogous questions.

Non- congruent tiling - one more

Seeking a little clarification on https://arxiv.org/pdf/2004.01034

The paper establishes that there are tilings of the plane by convex quadrilaterals all mutually non congruent and with same area and perimeter.

QUESTIONS
1. given any convex quadrilateral Q can we form tilings with Q and choosing other quads of same perimeter and area as Q and all mutually non congruent?

2. If the answer is “not in general”, how will we characterise Qs which can function as ‘seed tiles’?

And can such tilings be achieved selecting from convex quads having any possible {area, perimeter} pair of values?

More soon on this hopefully...

Oriented containers - kites

This continues an old thread on oriented containers with latest episode here

How does one find the least area convex kite that contains a given convex polygon P? What about the least perimeter kite container?Isosceles triangles may be treated as degenerate kites

Which is the convex shape that maximises the difference in orientation (measured by angle between the lines of symmetry of the two containers) between the smallest area and smallest perimeter kite containers?

Note 1: Unlike in the case of the smallest area containing rectangle for a convex P, the smallest enclosing kite might not share an edge with the polygon - if P is a thin rectangle, a containing kite that shares an edge with it appears suboptimal. well not sure!

Note 2: one can also ask about the largest area/perimeter kite contained inside a given convex polygon.



if the smallest area kite container too shares an edge with P, one can ask if there is any type of container where the optimal container need not share an edge with P.

One more non congruent tiling thought

Question:

Is there any set of polygons that have the same angle set with each member individually incapable of tiling the plane but together can manage to tile?

Obviously, the polygons in the set should be pentagons or hexagons...

A further constraint would be: elements of the polygon set should not only possess the same angle set but be pairwise non-congruent.

And what if we constrain "angle set" above to "angle sequence"?

Packing - some more thoughts

1. Which convex hexagon is worst for packing (ie leaves largest fraction of plane open)?

2. it has been conjectured that the regular heptagon is the worst convex region for packing. Is it known if regularity is significant? Indeed, for hexagons, regularity gives a perfect pack.

So, one can ask: among all convex heptagons, is the regular one the worst for packing?

Further, is there any reason to believe that the worst packing region has SOME rotational symmetry?

And what is the connection, if any, between a body being bad for packing and bad for COVERING?

Stretching Fair Partitions - 2

We add another very speculative claim to those in the last post.

"If a convex planar region C has the property for any positive integer n that at least one convex fair partition of C into n convex pieces has at least 2 (or maybe 3, say) of the pieces mutually congruent, then for any n, C allows a convex fair partition into n pieces that are all mutually congruent - and further, C is either a sector of a disk or parallelogram."

Stretching the Fair Partition question

Can one make claims of the following type?
"If some convex region C allows partition into n convex pieces all of equal area, perimeter and one more quantity, say diameter or least width for all values of n (or infinitely many values of n), then all pieces are necessarily congruent."

If the above is true, one can stretch things a bit and guess: "If all pieces are congruent for all n, C is necessarily a sector of a disk (with the full disk as a limiting case) or a parallelogram (including rectangles). If 'for all n' is relaxed to 'infinitely many values of n', one also has the case of C being a triangle." This latter guess was once posted at mathoverflow.

Note: perimeter and diameter can be nonzero even when a polygon is degenerate but not area or least width. Basically the question/claim is about 3 quantities being equal among pieces (with 2 of the quatities being like perimeter and one like area or vice versa). If 3 quantities being equal isnt enough for the congruence claim to hold, consider 4!