All Questions
6 questions
13
votes
1
answer
921
views
Limiting shape for Brillouin zones
Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
- 12.3k
7
votes
1
answer
5k
views
Shrink polygon to a specific area by offsetting
I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...
- 171
6
votes
1
answer
767
views
Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
- 17.9k
6
votes
1
answer
880
views
Relation of some Euclidean geometry theorems and more conjecture generalizations
In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem, the Ptolemy's theorem and the Feuerbach-Luchterhand. Since ...
- 1,044
8
votes
2
answers
339
views
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
- 113
1
vote
1
answer
320
views
A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
- 2,661