All Questions
9 questions
8
votes
4
answers
530
views
Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
- 151k
88
votes
2
answers
7k
views
17
votes
0
answers
731
views
Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 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
15
votes
1
answer
640
views
Smallest regular simplex containing the unit cube in $R^n$
What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that $e_n<\sqrt{2}(n+1-\...
- 6,101
15
votes
1
answer
530
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
3
votes
1
answer
197
views
Three-dimensional Apollonian spirals
Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let $P_{\...
- 19.7k
3
votes
0
answers
214
views
Volume of intersection of a ball and cube with arbitrary position in $n$ dimension
Let $ A(n, r, x) = B^n_r(x) \cap [0,1]^n $ denote the intersection between an $n$ ball $B^n_r(x)$ with arbitrary radius $r$ and arbitrary center $x \in \mathbb{R}^n$ that intersects a unit $n$ cube $ [...
- 365