The geometry of quadrangular convex pyramids

TL;DR

This paper proves the strong rigidity of convex quadrangular pyramids, showing they cannot be continuously deformed while preserving edge lengths, and explores realizations of such pyramids with given edge lengths.

Contribution

It establishes the strong rigidity of convex quadrangular pyramids and bounds the number of realizations for given edge lengths, providing an explicit example.

Findings

Convex quadrangular pyramids are strongly rigid.

The number of realizations with fixed edge lengths is at most four.

An example with three non-congruent realizations is provided.

Abstract

A convex quadrangular pyramid $ABCDE$, where $ABCD$ is the base and $E$ -- the apex, is called \emph{strongly flexible}, if it belongs to a continuous family of pairwise non-congruent quadrangular pyramids that have the same lengths of corresponding edges. $ABCDE$ is called \emph{strongly rigid}, if such family does not exist. We prove the strong rigidity of convex quadrangular pyramids and prove that strong rigidity fails in the self-intersecting case. Let $L=\{l_1,\ldots,l_8\}$ be a set of positive numbers, then a \emph{realization} of $L$ is a convex quadrangular pyramid $ABCDE$ such, that $|AB|=l_1$, $|BC|=l_2$, $|CD|=l_3$, $|DA|=l_4$, $|EA|=l_5$, $|EB|=l_6$, $|EC|=l_7$, $|ED|=l_8$. We prove that the number of pairwise non-congruent realizations is $\leqslant 4$ and give an example of a set $L$ with three pairwise non-congruent realizations.