## New answers tagged polygons

6

It is not possible with 7 (i.e., with a triangle $T$ and a quadrilateral $Q$). I write a rough proof.
First, any quadrilateral $Q$ lying in a plane $\pi$ can be partitioned in two triangles $Q_1$ and $Q_2$, whose common edge is a diagonal $d$ of $Q$. Now the intersection of the triangle $T$ with $\pi$ consists of two points (otherwise they are coplanar and ...

4

Here is another example with 8 vertices: a short fat Star Trek symbol and a square in orthogonal planes.
Since the distance between the base points of the red figure is greater than its height, one cannot rotate the square to take it out.
Addendum: As Saúl Rodríguez Martín mentions, this example may not work. If we assume, however, that the links are ...

14

As Sam Hopkins commented, 8 vertices are enough. Let $Q$ be the pentagon from the picture and let $\pi$ be the plane containing it. Now we can define the triangle $P$ as a triangle of less diameter than the black segment and intersecting $\pi$ at two points: one point $a_0$ in the open blue region $B$ and one point $b_0$ in the open green region $G$. $P$ and ...

3

I think that 8 might be possible, by interlocking two Star Trek symbols as shown below.
Adendum: This candidate may not work, as quarague points out, but I leave it as a potential "how not to" example. There are other ways to cross the two figures, while they remain unlinked, and one of these variations could be more promising. Also I think it is ...

1

Here is a proof using analytic tools, complex numbers, and formulas for the involved points in terms of them.
Points in the plane will be denoted by capital letters like $A,B,C,D;N;P,Q,R,S$ and the corresponding affixes will
be their lower cousins, respectively $a,b,c,d;n;p,q,r,s\in\Bbb C$, and decorations (sub- or upper indices) will be kept.
First let us ...

Top 50 recent answers are included

#### Related Tags

polygons × 108mg.metric-geometry × 65

plane-geometry × 27

euclidean-geometry × 26

discrete-geometry × 20

computational-geometry × 20

convex-geometry × 14

elementary-proofs × 10

algorithms × 8

convex-polytopes × 7

reference-request × 6

co.combinatorics × 5

graph-theory × 5

computational-complexity × 4

gt.geometric-topology × 3

billiards × 3

ag.algebraic-geometry × 2

complex-geometry × 2

inequalities × 2

sequences-and-series × 2

terminology × 2

convex-optimization × 2

curves-and-surfaces × 2

polyhedra × 2

fixed-point-theorems × 2