Non-Congruent Tiling - 11

This has been a lengthy series - mainly due to experts having produced a series of results on the basic theme of tiling the plane with mutually non-congruent tiles which nevertheless share common properties. The common properties intensively studied are area and perimeter. The latest results (known to me) are in https://www.sciencedirect.com/science/article/abs/pii/S0097316521000601 ("the Euclidean plane can be tiled by mutually non-congruent convex quadrangles all of same area and perimeter").

Before putting up the main content of this post, let me put links to earlier instalments in this series (there could be some overlap in content!):
- Parts 1 to 6
- Part 7
- Part 8
- Part 9
- Part 10

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Let me restate a Broad Question that was put up in some form or other earlier on these pages: Apart from {area, perimeter}, what could one say if the mutually non-congruent tiles have to share, say, {area, diameter} or {area, least width} or any such pair of global properties?

And one can ask if in addition to area and perimeter if one more quantity, say, diameter or least width needs to be equal among the non-congruent tiles, can the plane be tiled with quadrangles. Another possibility is: all non-congruent pieces ought to have some specified centralness coefficient (its value ranges from 1/2 for all triangles to 1 for centrally symmetric convex regions)
The following is still more speculative:

Moving on from tiling to covering and packing, one think of covering or packing the plane with a set of mutually non-congruent polygons with shared properties. Are there global results such as "if pieces have to be mutually non-congruent with same area and perimeter, the resulting packing/covering is always inferior to the best packing/covering with congruent copies of any tile that has the same area and perimeter."

And let me repeat, all of the above questions can be transplanted onto the hyperbolic plane! -

2 Questions on Rep-tiles

A rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape.

Question 1: Are there any convex pentagons that are also rep-tiles?

Remarks: 15 convex pentagonal tiles of the plane are known and none of them appears to be a rep-tile. Assuming this observation is right, one can invoke a proof given in 2017 by Michel Rao - that these 15 are the only convex pentagonal tiles possible - to answer our question in the negative. However, I don't know if Rao's proof has been validated and if there is a simpler (elementary) proof that there are say no convex pentagonal rep-tiles. Basically, one is asking for a simpler proof for a weaker claim.

Question 2: Let us define a *multi-way rep-tile* as a polygon *P* with the property: if *P1* and *P2* are larger scaled up copies of *P* and *P1* can be tiled with *m* units of *P* and *P2* can be tiled with *n* units of *P* with *n*> *m*, then, a layout of *n* *P*-units can form *P_2* without *m* of the units in the layout together forming a *P_1*. As shown on this page: https://en.wikipedia.org/wiki/Rep-tile, there are isosceles trapeziums with angles 60 and 120 degrees with multi-way property (with *m*= 4 and *n* = 9). Are there other convex polygons with this multi-way rep-tile property?

Note: A square is obviously a rep-tile but is not multi-way.

'Centralness Coefficient' Etc.

There has been a shift of focus to MathOverflow: https://mathoverflow.net/users/142600/nandakumar-r

However, some discussions there seemed worth collecting in this post.

1. The recent encounter with centralness coefficient was important enough to copy here as well.
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https://mathoverflow.net/questions/396579/a-center-of-convex-planar-regions-based-on-chords
A point P in the interior of a planar convex region C divides every chord of C that passes thru it into 2 segments. Consider, for each chord thru P, the ratio between the length of the shorter segment and length of the longer segment.

For every P, this 'chord length ratio' has the maximum value 1 (for every P, there is at least one chord of C for which P is the mid point) but its least value varies with P. That position of P where the least value of this ratio is maximum (in other words, position of P where the values of the chord length ratio are within narrowest bounds) can be called a center of C and the highest value of the least chord length ratio over all interior points can be called the centralness coefficient of C.

It can be shown that the centralness coefficient of any convex figure cannot be less than 1/2 (this value holds for all triangles) and at least 3 chords pass thru a center of C with chord length ratio equal to the centralness coefficient (e.g. the medians of a triangle).

Question: Is the center of any convex region a unique point?
Remarks: If C is centrally symmetric, the center is unique. It appears to coincide with the center of mass even for regular polygons with odd number of sides. One can also ask about the relationship of this center with other special points such as center of mass — how far apart they could possibly be etc. How does one algorithmically determine the center(s) of a convex n-gon?
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2. The centralness coefficient is unlike area or perimeter or diameter a quantity that does not change with scaling - it is about the shape of the convex region. So, one could naturally bring it in as among the quantities to be equalized when a convex region is partitioned into n pieces and see the implications.
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3. Another question that was recently pondered was: how many quantities need to be equalized among the convex faces of a convex polyhedron to guarantee that all the faces are mutually congruent? It has been argued here: https://mathoverflow.net/questions/395013/convex-polyhedra-with-non-congruent-faces that with only area and perimeter equal among faces, all faces of a convex polyhedron can be mutually non-congruent.

However, the question can be raised: can one constrain a few more quantities to be equal among faces to guarantee, say, "at least a pair of faces will be congruent" or "all faces are necessarily congruent"?
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There should be updates to this post as I learn more about these questions...