The geometry of quadrangular convex pyramids
| Autor: | Kochetkov, Yury |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 5 pages, 1 figure |
| Databáze: | arXiv |
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