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Interesting Esoterica

The Flapping Birds in the Pentagram Zoo

Article by Richard Evan Schwartz
  • Published in 2024
  • Added on
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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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key
TheFlappingBirdsinthePentagramZoo
type
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}
}