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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

7 votes
0 answers
156 views

Thales Style Level Sets

Encouraged by Joseph O'Rourke ( and inspired by the discussion at Thales' semicircle theorem in higher dimensions ), I ask about level sets in three dimensional space occuring from considering how bi …
The Masked Avenger's user avatar
6 votes
1 answer
236 views

Pigeonholing Polygons: Can two rigid regions fit in twice the space needed?

This is a tweak of Henry Segerman's question Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1? , but restricted to the point of possibly having a proof in the lite …
The Masked Avenger's user avatar
3 votes

How many unit simplices are needed to cover a unit $n$-cube?

I'm now thinking the answer is 22, or very close to it. Put tetrahedral caps on each corner of the unit cube, and reflect about the base to place a corresponding interior tetrahedron, accounting for …
The Masked Avenger's user avatar
2 votes

Diameter of random segment intersection graph?

Here is a way of thinking. It may turn out to be misleading, but it may also suggest a way to think properly about this problem. Lets take a random instance of a matching and reconstruct it edge by …
The Masked Avenger's user avatar
2 votes

what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

I think the easiest route to a lower bound is to pick four states such as Hawaii, Alaska, Rhode Island, and Florida, and show that any geodesics cutting them leave too many states uncovered, or are fi …
The Masked Avenger's user avatar
2 votes

Inside-out polygonal dissections

Riffing off the tiling comment to another answer, imagine a square penny packing of circles, and then translate the tiling so that a circle is in the center of four other circles. Now replace each of …
The Masked Avenger's user avatar
2 votes

Lightray trapped between two mirror disks: Computation formulation?

Consider the following approach. For the case of two circles, it is clear that there is a stuck ray that "lives" on the line between the two centers. Pick a point Q near this line and shoot a ray fro …
The Masked Avenger's user avatar
2 votes

Planar linkage that traces a circle from its exterior?

I don't have a facility with graphics; hopefully the verbal description below will work. There is a linkage that magnifies: place a tracing stylus at point P, a marking stylus at point Q, and this li …
The Masked Avenger's user avatar
1 vote

Thales' semicircle theorem in higher dimensions

Having seen the analyses presented in the other answers and Douglas Zare's comment about difficulties in visualization, I offer a view on the problem that shows how to arrive at a qualitative result ( …
The Masked Avenger's user avatar
1 vote

Is there always a maximum anti-rectangle with a corner square?

Here is an approach which is incomplete, but might be combined with Nick Gill's approach to yield something nice. I 'll let others do the routine of formalizing the notions of cover, orthogonal cover …
The Masked Avenger's user avatar
1 vote

Equipartition of the circle

Not quite an answer, but food for thought. Consider a curve with N (let's say distinct) distinguished points on it. Call two such C and D PR (or C PR D, for projectively related) if there is a point …
The Masked Avenger's user avatar
1 vote

Lightray trapped between two mirror disks: Computation formulation?

Here is a separate idea, which may give you a better intuition. Let's give clay pigeons a break and call it shooting at a moving plate. Setup: pistol P, moving target C parameterized by angle r of …
The Masked Avenger's user avatar
1 vote

Sampling uniformly from all possible line segments of a given length that fit inside a conta...

You might consider first the distribution in two special cases: when the (open) segment is first placed in a (closed) ball of diameter equal to the length of the ball, and when the ball has two (diame …
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