TECH-MUSINGS

Thoughts On Algorithms, Geometry etc...

Saturday, December 16, 2023

Non-congruent tilings- 20: Rational triangles

The previous episode of this lengthy series is here .

Ref: this mathoverflow discussion .

Broad Question: to tile the plane into rational triangles (all side lengths rational) all mutually non-congruent.

Additional constraints: all triangles should have same area or same perimeter or...

A construction (tiling with mutually non-congruent triangles all of equal area with rationality of side lengths not insisted) presented in detail by Stan Wagon here , it appears, can also be made to yield a tiling of the plane with rational triangles with perimeter unbounded - it appears to give only mutually non-congruent rational triangles, no equality among their areas. A further result obtained by Frettloh (https://arxiv.org/pdf/1603.09132.pdf, more specifically figure 4 therein) indicates that tiling with rational triangles with bounded perimeter also can be got - again with no guarantee on equality of areas.

So, what we could ask for here is for a tiling of the plane into mutually non-congruent rational triangles all with (1)same area or (2) same perimeter or whatever.

An earlier discussion on mathoverflow is here .
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Variant: Rationality of triangles could also be defined as: all angles are irrational fractions of pi. What could one do with them?

Further question: It appears that any polygon with all angles rational can be cut into some finite number of rational angled triangles. Will the question of dividing an n-gon with rational angles into the least number of rational angled triangles have interesting optimization features?

Wednesday, December 06, 2023

A locus problem

This wasn't received well at Mathoverflow. So here goes:
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Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB is a right angle, then the locus of points is the circle with the diameter AB. Ref: https://mathpages.com/home/kmath173/kmath173.htm#:~:text=Loci%20of%20Equi%2Dangular%20Points&text=Given%20a%20line%20segment%20AB,circle%20with%20the%20diameter%20AB.

Question: Given two line segments AB and CD lying on same plane, what is the locus of points P such that the sum of the angles APB and CPD is a constant? What are the qualitative features of this locus and how does this locus depend upon the relative position, length and orientation of the two segments and the value of the angle sum?

Note: Numerical calculations indicate the following: when AB and CD are kept fixed near to each other or intersecting and the angle sum varied, (1) if the angle sum is small, the locus is a connected curve that lies far away from the two line segments and surrounding them and shows concavities; (2) for larger values of angle sum, the locus lies closer to the two segments and appears convex. And in some cases, for still larger values of the angle sum, the locus probably breaks into two closed curves. Basically, we seem to have a one-parameter family of curves that go from convex to non-convexr>

An analogous question in 3D (with 2 line segments - that could be either skew or coplanar - and an angle sum) could give surfaces as loci of P.
Picture below contours of the angle sum measured with respect to a pair of short and intersecting line segments near the origin

Tuesday, October 03, 2023

More on packing and covering with circles

Reference: Erich's packing center.

We continue from these mathoverflow pages:(1) https://mathoverflow.net/questions/455365/bounds-for-the-dispersal-problem-in-convex-regions and (2) https://mathoverflow.net/questions/455309/bounds-for-minimax-facility-location-in-a-convex-region. Thanks to Prof. Roman Karasev for his comments.

1. Basic question: Given a convex region R, to pack n circles in it such that the smallest among the circles has max radius.

It is clear from Erich's packing center that even when R is a circular disk, all the n circles in an optimal layout (see n =8, for example) need not have the same radius. Then, the natural further question is to pack n circles in R that (1) maximize the least radius and (2) maximize the total perimeter of the packed circles (studied with only sum of perimeter maximized here) OR maximize the total area of the packed circles OR maximize the total number of different radii among them.

If R is a v long and thin isosceles triangle, all circles in a layout of n circles with smallest among them having max radius seem to be of different radii.
A theorem like: "When R is a disk, for any n, there can only be at most some constant number of different radii among the disks when the least radius is maximized" would be cool.

2. To cover R with n circles such that the largest among the circles has the least possible radius.

Here I don't know any R and n such that the n covering disks that minimize the maximum radius among them could have different radii. If such an example is found, the natural questions would be to find layouts that (1) minimize the max radius among covering disks and (2) minimize the total area/perimeter of all disks OR maximize the number of different radii among them or...

Friday, August 04, 2023

On wrapping solid bodies with planar regions

Given a sheet of paper P which is some planar region that cannot be stretched but can be folded or wrinkled at will. We want to find the 3d solid Q of largest volume that can be wrapped with P. We say "P wraps Q" if any line coming from far away has to pierce P at least once before it hits Q.

Can one think of general results (in Euclidean geometry) like this:

- whatever be the shape of P, the max volume Q it can wrap is necessarily non-convex.

- for any given convex 3d body, the P of least area that wraps it is necessarily non convex.

Can things be different in non-Euclidean geometry?

Remark:

I just saw that if the planar region is a long and very thin strip, the max volume 3d body it can wrap would be a sphere - just winding around it many times - which is convex. So, the question of finding least area wrapper for any 3d body might be meaningful only if we disallow such multiple windings - else a thin and v long strip with area equal to the surface area of the 3d body would be the answer.

However, the other question of finding the max volume a specified planar region can wrap around seems ok.

Thursday, July 13, 2023

Enclosing and Embedded isosceles triangles for a triangle - orientations

In this paper: https://arxiv.org/pdf/2205.11637.pdf, the following questions are answered:

- Given a triangle, how to find the smallest area(perimeter) isosceles triangle that contains it?

- Given a triangle, how to find the largest area(perimeter) isosceles triangle contained in it?

Let me just record a pair of associated questions for which the answer might be readily obtained from the above paper:

Which is the triangle for which the smallest area (perimeter) isosceles container and largest area (perimeter) isosceles containee have angular difference between their orientations is maximum? The orientation of an isosceles triangle is given treating the triangle as an arrowhead - from the midpoint of its base towards the apex.

A further pair of questions:
Which is the convex polygon for which the smallest area (perimeter) rectangle that contains it and the largest area (perimeter) rectangle contained in it have the angular difference between their orientations a maximum? The orientation of a rectangle is naturally its length or width - for a square, one could take either as orientation.

The answers for the above should be available soon. I shall update this post when they reach me.

Monday, July 10, 2023

Wrapping a 2D lamina with paper

Basic question: to wrap a given planar region with a convex sheet (such that every point on both sides of the lamina has at least one layer of paper covering it) with the wrapping convex sheet being of least area/perimeter. An n-fold wrap is a wrap such that to reach any point on the lamina from outside, a 'needle' will have to cut through at least n layers of paper.

It is not difficult to cook up convex 2D regions such that the least area wrapper and least perimeter wrapper are different. Here is a pentagon for which the least area wrapper is a 7-gon and the least perimeter wrapper appears to be a rectangle.



Note that above pic is a rough one. The x length of the pentagon being wrapped ought to be considerably more than the y height (say, twice). the slope of the near horizontal tilted edges should be much less than the pic indicates. Indeed, the pentagonal least area wrapper is only very marginally different from a rectangle.
Overall, the 7-gon wrapper (least area) is almost a rectangle with dimensions say 20 X 5 and the rectangular wrapper (least perimeter) has dimensions 10X11 approximately.

Questions:
- Which planar convex region of unit area has the least area wrapper having max area? Is it a disk?
- Which planar convex region of unit perimeter has least perimeter wrapper with max perimeter?
- Same questions as above with the wrapping generalized to an n-fold one.

Thursday, June 29, 2023

Lines segmenting convex planar regions - some questions

A pair of claims on area bisectors and perimeter bisectors of convex planar regions were posted at mathoverflow here and a further question is here .

Let us define a width of a planar convex region C as the distance between two parallel lines that just touch C. A width bisector is aline that is parallel to a pair of parallel lines that are tangent to C and is at same distance from both these lines - basically, it divides a width into two equal parts.

Claim: All width bisectors of a convex planar C are concurrent if an only if C is centrally symmetric.

Question: What can be said about families of lines that divide all widths in ratio t:1-t where t is between 0 and 1/2?

Guess: If we are looking at families of lines that segment area(equally, perimeter or widths) in some specified ratio, for no convex planar region can families of these dividing lines be concurrent if the specified ratio is other than 1:1.

Just like the envelopes of families of lines that segment area - these have been discussed by Fuchs and Tabachnikov - one could ask for properties of the envelopes of the other families of segmenting lines too.
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The ratio between areas (or perimeters, least widths or diameters) of these envelopes (or the convex hulls of these envelopes) and those of C could serve a measure of how 'uncentered' C is. Yaglom and Boltyanski have talked about 'centralness' based on chords and proved that it has the least value for triangles. It seems likely that the envelopes of the segmenting lines with reference to the different quantities mentioned above (area etc) would be quite different for the same C. The centralness measures based on these ratios too could have intresting implications.
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Higher dimensional equivalents of these questions are obvious.