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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 ...
Joseph O'Rourke's user avatar
88 votes
2 answers
7k views

Light reflecting off Christmas-tree balls

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Joseph O'Rourke's user avatar
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 ...
Kevin Johnson's user avatar
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 $P$ ...
Roland Bacher's user avatar
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-\...
Jan Kyncl's user avatar
  • 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 ...
Kepler's Triangle's user avatar
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 ...
Penelope Benenati's user avatar
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 $ [...
random_shape's user avatar
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_{\...
James Propp's user avatar
  • 19.7k