Non-congruent tiling - 17

A pointer to the previous instalment of this series.

We consider tiling with equal area, mutually non-congruent tiles:

Question: Can the plane be tiled with equal area triangles all of which are pair-wise non-congruent and with every edge in the layout having unique length?

Remark 1: If we require the pair-wise non-congruent equal area triangles all to have every angle unique, it seems possible based on the arguments given in https://arxiv.org/pdf/1603.09132.pdf.

Further question: The above question (both unique angles and unique edges variants) can be asked with 'perimeter' replacing 'area'. Note: It has been proved that the plane cannot be tiled by non-congruent triangles all with same area and same perimeter (https://arxiv.org/abs/1711.04504).

Here is the mathoverflow page with above question
-------
We now move to the hyperbolic plane:

1. Can the hyperbolic plane be tiled by mutually non-congruent equal area triangles?

2. Can the hyperbolic plane be tiled by mutually non-congruent equal area triangles with the further constraint that every angle is unique?

Equipartitions of surfaces of convex spatial regions

Let S be the surface of a 3D convex region. Let S' be a subset of S. We shall refer to S' as geodesically convex wrto S if the following condition holds: If A and B are two points in S', the shortest path in S between A and B lies entirely in S'.

Question: It is easy to see that if S is the surface of a sphere and n is any integer, S cen be partitioned into n equal area pieces that are also geodesically convex in S - these pieces are separated by great circle arcs between 2 antipodal points on S. Is the sphere the only convex surface (surface of a convex 3D region) for which such an equipartition exists for any n? If not, how does one characterize such surfaces? Prolate and oblate spheroids too seem to allow such equipartitions. Beyond that, not sure.

If S is the surface of a cube, I don't see how it can be cut into 3 geodesically convex pieces all of same area.

Approximating planar convex sets by n-gons

Let me add a little bit to the last post here - on approximating triangles with ellipses (and viceversa)

In chapter 2 of 'Combinatorial Geometry', Pach and Agarwal discuss approximating convex compact planar sets by n-gons. They treat 2 cases: min area circumscribed n-gons and max area inscribed n-gons.

A third possible approximation is by n-gons that are closest to the given planar set in terms of hausdorff distance and that seems unexplored. And I don't even know about approximations by min perimeter circumscribed and max perimeter inscribed n-gons.

A bit more has been added in this post at mathoverflow:

(We can consider) other measures of distance between planar convex regions C1 and C2, for ex: the average least distance of a boundary point on C1 from a boundary point on C2. So, one could ask:

Question: Given a general triangle T, find that circle C (ellipse E) such that the average over boundary points of C(E) of closest distance from that point to a boundary point of T is minimized.

'Approximating' triangles by ellipses

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.

Question: Given a general triangle T, to find the ellipse E that minimizes the Hausdorff distance from T. It is not even clear to me if E is unique.

Remarks: One can ask how this ellipse relates to the Steiner ellipses of T, whether the area/perimeter of E(assuming it is unique) is related to area/perimeter of T and so on..

Further questions: if E is somehow found, then is T that triangle (or one of the triangles) that minimizes Hausdorff distance from E? Is there a mapping from ellipses to triangles using this closeness criterion - ie. given any ellipse, is it the closest ellipse to some triangle(s)? This same question can be asked the other way around as well.

Another question: to find closest triangle to a given convex n-gon. And (not) finally, one can replace triangles with quadrilaterals/rectangles and so on!
Guess: If one uses the Frechet distance rather than Hausdorff to formulate the above question, the answer(s) are not affected.
--------------
Update (Jan 18th, 2023): The mathoverflow page for this question is here . It has been pointed out there that the special case of approximating a triangle with a circle is itself of non-trivial interest.

Update (Feb 5th 2023): A further mathoverflow page on the same theme is here

Axiality of planar convex regions - alternative definitions

We continue the the previous post:

Here is how axiality of a planar convex region is defined in Wiki:
Axiality is a measure of how much axial symmetry a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the whole shape. Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape (with any orientation).
A shape that is itself axially symmetric, such as an isosceles triangle, will have an axiality of exactly one, whereas an asymmetric shape, such as a scalene triangle, will have axiality less than one. Lassak (2002) showed that every convex set has axiality at least 2/3. This result improved a previous lower bound of 5/8 by Krakowski (1963). The best upper bound known is given by a particular convex quadrilateral, found through a computer search, whose axiality is less than 0.816.

Alternative definitions for axiality:
1. In the above relation, replace "largest axisymmetric subset" by "smallest axisymmetric superset". No idea what can then be said about the shape with least axiality with this new definition. Replacing "area" by "perimeter" could also be considered.
2. In the spirit of Yaglom and Boltyanski, one can define an 'axiality ratio' for every chord L of a planar convex region C as the minimum of the ratio between length of the smaller to the larger segment into which L cuts any chord of C that is perpendicular to itself. Now, the best axis is that chord that maximizes this axiality ratio and that value of the ratio could be called the axiality of C itself.
Question: What shapes minimize axiality under this new definition(s)?

Update (Jan 17th, 2023). Question 2 above has a very simple answer as given here: https://mathoverflow.net/questions/438451/on-axiality-of-planar-convex-regions. Thanks to Prof. Anton Petrunin

Centralness of convex planar regions - alternative definitions

Centralness as defined by Yaglom and Boltyanskii:
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).
-------
A pair of alternative definitions for centralness of C:
Centralness is the ratio between the area of the largest area centrally symmetric figure that is contained by C to that of C.
Alternatively, centralness can also be defined as the ratio of the area of the smallest centrally symmetric figure that contains C to that of C.
There is no reason for the same shape of C to extremize both these ratios.
Note: We can also consider using perimeter instead of area to define centralness.
----------