'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...
View ArticleApproximating 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...
View ArticleEquipartitions 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...
View ArticleNon-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...
View Article'Spectrums' of convex regions of same area and perimeter
This is a bit of speculation that takes off from this discussion. Consider all planar convex regions of same area and perimeter. As was noted over several mathoverflow discussions, one can define the...
View ArticleSome questions on partitioning into Triangles
1. For any n, can any triangle be cut into n non-degenerate triangles all of same diameter? 2. If the answer to (1) is yes, can any convex m-gon be cut into some finite number of triangles all of same...
View ArticleTrapeziums - Non Congruent Tiling (18) and Oriented Containers
This post marks a meeting of two tracks we have been pursuing - Oriented containers and Non-congruent tiling. The previous posts in those series are here and here. We also build on this earlier post....
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Some Fair Partition Extensions
Some raw claims: 1. For a circular disk, for any n, the only convex fair partition is the one into n sectors. 2. For any convex region, there are at least some values of n for which there is only one...
View ArticleCutting n-gons into triangles and quadrilaterals
Basically, we are trying to push the envelope beyond Monsky's theorem which states: a square cannot be cut into any odd number of equal area triangles. Question: Given a convex n-gon. It is needed to...
View ArticleA deleted Mathoverflow post - stored
A post at mathoverflow can sometimes get deleted by a bot. The following was: https://mathoverflow.net/questions/416530/thinnest-3-fold-and-n-fold-coverings-of-the-plane-by-congruent-convex-shapes. So,...
View ArticleInside out dissections - contd.
This post continues not an earlier post here but a query posted at mathoverflow in June 2021: https://mathoverflow.net/questions/394823/further-queries-on-inside-out-polygonal-dissections...
View ArticleConvex partitions - averages of quantities
We continue from the following posts: - https://nandacumar.blogspot.com/2022/03/max-of-min-and-min-of-max-3.html - https://nandacumar.blogspot.com/2021/04/convex-partitions-max-of-min-and-min-of.html...
View ArticleNon-regular tilings of Hyperbolic plane
We add a bit to https://mathoverflow.net/questions/398191/which-polygons-tessellate-the-hyperbolic-plane. Background: It is known that there is a tiling of the hyperbolic plane by regular hyperbolic...
View ArticleOriented Containers (again) - Biconvex Lenses
We add one more link to the chain of Oriented containers, the latest of which was here . And there are two arix preprints on this topic - both done some years ago: this and this Definition: A biconvex...
View ArticleAnother Question on congruent partitions
A few days back, I put up a question at mathoverflow. Definition: A perfect congruent partition of a planar region C is a partition of it into some finite number n of pieces that are all mutually...
View ArticleNon Congruent Tilings - 19
Here is the latest episode in this series. And here is a mathoverflow discussion. The answers show tilings of the plane with mutually similar right triangles of unbounded size. An old instalment of...
View ArticleCutting and Covering - a cluster of questions
Two recent posts at mathoverflow are here and here. We record some further questions: -------- - Given an integer n, to cut n equal area isosceles triangles from the unit square that leave out as...
View ArticleAffine Geometry - 3 questions
1. Can any convex polygon C be partitioned into some finite number m of quadrilaterals that are mutually affine-equivalent? If the answer is "yes", how does one do it for a given n -gon efficiently,...
View ArticleLines 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...
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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...
View ArticleEnclosing 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...
View ArticleOn 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...
View ArticleMore 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)...
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A locus problem
This wasn't received well at Mathoverflow. So here goes: ----------- 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...
View ArticleNon-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...
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