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Results tagged with mg.metric-geometry
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user 35626
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
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 …
- 2,158
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 …
CommunityBot
- 1
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
…
- 2,158
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 ( …
- 2,158
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 …
CommunityBot
- 1
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158
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 …
- 2,158