14 problems

Let $N>3$, and consider the polar image of the Harmonic family, a Poncelet family of $N$-gons. For each polygon, let $\theta_i$ denote its relevant angles, so that the half-angle t…

Let $q_t$ and $r_t$ be the maps used to describe the two symmetry-related images of a triangle, and let a triangle be non-equilateral when it is not equilateral. Six-periodic isosc…

Let ${\mathcal S}_3$ be the space of unit-area triangles, let $f_t:{\mathcal S}_3\to{\mathcal S}_3$ be the triangle map, and let ${\mathcal A}$ be its area integral. Second-integra…

Let $f_t$ be the maps on planar polygon spaces and let $\xi$ be the vector field. The polygons evolving under these maps and under the flow of $\xi$ appear to move around invariant…

Let $n$ be odd, let ${\mathcal T}_n$ be the symplectic space of planar $n$-gons, let $f_t$ be the maps on this space, and let $\xi$ be the vector field. Let ${\mathcal A}$ denote t…

Let $\mathcal{Q}$ be a simple $n$-gon, meaning that it has no self-intersections, and consider the regions of points $M$ for which the circumcenter map produces area contraction. R…

Let $\mathcal{P}$ be a regular $n$-gon, with $n\geq 3$, and let $\mathcal{P}'=\mathcal{C}_M^n(\mathcal{P})$. Write $\alpha$ for the angle of rotation in the similarity taking…

A bicentric pedal polygon is obtained from a bicentric polygon by taking its pedal polygon with respect to a limiting point, denoted by $\ell_1$ or $\ell_2$. Cosine-sum invariance…

Let $R$ be a convex polygon with $2n+1$ sides. A quadrisection is a partition of $R$ into four regions of equal area by three concurrent cevians. Quadrisection bound conjecture. Th…

Let $n \geq 5$, let $H_{\lambda}$ be the polygon iteration on the space $\mathcal{P}_n=(\mathbb{R}P^2)^n$ of $n$-gons, and let $\lambda_0=2\cos(\pi/n)$. Affine dynamics conjecture.…

Let $n \geq 5$ and let $H_{\lambda}$ be the polygon iteration on the projective moduli space $\mathfrak{P}_n = (\mathbb{R}P^2)^n/PGL(3,\mathbb{R})$ of projective equivalence classe…

Let $n$ be odd, write $n=2m+1$, and let $bbda_0=0,bbda_1,bbda_{n-1}$ be the eigenvalues of the circulant matrix governing the linearized system, with $bbda_j=-bbda_{n-j}$ and, for…

Circle-intersection Devron conjecture. After $2n-6$ steps, one reaches another polygon for which circle intersection takes half of the vertices to a common point.

Stein's conjecture. A balanced polygon cannot be cut into an odd number of triangles of equal areas.