The Flapping Birds in the Pentagram Zoo
- Published in 2024
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We study the (k+1,k) diagonal map for k=2,3,4,.... We call this map Δk. The map Δ1 is the pentagram map and Δk is a generalization. Δk does not preserve convexity, but we prove that Δk preserves a subset Bk of certain star-shaped polygons which we call k-birds. The action of Δk on Bk seems similar to the action of Δ1 on the space of convex polygons. We show that some classic geometric results about Δ1 generalize to this setting.
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- TheFlappingBirdsinthePentagramZoo
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- article
- date_added
- 2024-03-14
- date_published
- 2024-03-26
BibTeX entry
@article{TheFlappingBirdsinthePentagramZoo, key = {TheFlappingBirdsinthePentagramZoo}, type = {article}, title = {The Flapping Birds in the Pentagram Zoo}, author = {Richard Evan Schwartz}, abstract = {We study the {\$}(k+1,k){\$} diagonal map for {\$}k=2,3,4,...{\$}. We call this map {\$}\Delta{\_}k{\$}. The map {\$}\Delta{\_}1{\$} is the pentagram map and {\$}\Delta{\_}k{\$} is a generalization. {\$}\Delta{\_}k{\$} does not preserve convexity, but we prove that {\$}\Delta{\_}k{\$} preserves a subset {\$}B{\_}k{\$} of certain star-shaped polygons which we call {\$}k{\$}-birds. The action of {\$}\Delta{\_}k{\$} on {\$}B{\_}k{\$} seems similar to the action of {\$}\Delta{\_}1{\$} on the space of convex polygons. We show that some classic geometric results about {\$}\Delta{\_}1{\$} generalize to this setting.}, comment = {}, date_added = {2024-03-14}, date_published = {2024-03-26}, urls = {http://arxiv.org/abs/2403.05735v1,http://arxiv.org/pdf/2403.05735v1}, collections = {animals}, url = {http://arxiv.org/abs/2403.05735v1 http://arxiv.org/pdf/2403.05735v1}, year = 2024, urldate = {2024-03-14}, archivePrefix = {arXiv}, eprint = {2403.05735}, primaryClass = {math.DS} }