A plane continuum $X$ is said to admit an inscribed polygon, $P$, if every embedding of $X$ into $\mathbb{R}^2$ (the Euclidean plane) contains the vertices of a polygon similar to $P$. In this talk we adapt this definition to the hyperbolic geometry setting: A plane continuum quasi-inscribes a polygon $Q$ in the hyperbolic plane $\mathbb{H}$, if given any embedding $\gamma:X\hookrightarrow\mathbb{H}$, we have that for all $\varepsilon>0$, $\gamma(X)$ admits a polygon whose inner angular sum is $\varepsilon$-close to the sum in $Q$; and both polygons share geometric structure. In particular, we show that there is a wide class of continua that quasi-inscribe rectangles. We will also include some results, obtained in the Euclidean setting, regarding the inscription of squares and rectangles in continua, in this case we will focus on continua that are ray compactifications.

1. Morales-Fuentes Ulises; Villanueva-Segovia Cristina, “Circularly chainable continua that annularly inscribe squares”, Topology and its Applications, vol. 368, (2025).

2. Morales-Fuentes Ulises; Villanueva-Segovia Cristina, “Rectangles Inscribed in Locally Connected Plane Continua”, Topology Proceedings, 58, (2021).

3. Díaz Juan Pablo; Hinojosa Gabriela; Morales-Fuentes Ulises and Valdez Rogelio. “Classification of trees that quasi-inscribe rectangles in the hyperbolic plane”, Advances in Geometry, vol. 25, no. 4, (2025).