Exact colinearity of centroids of iterated midpoint hexagons
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911569521147904 |
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| author | Tisdell, Jack Edward |
| author_facet | Tisdell, Jack Edward |
| contents | We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact colinearity of centroids of iterated midpoint hexagons Tisdell, Jack Edward Metric Geometry 51F, 51M04 We study the iteration that replaces a planar hexagon by the hexagon formed by joining the midpoints of consecutive edges. While this iteration quickly drives any polygon toward a point and their shapes asymptotically regularize, we show a stronger and unexpected rigidity holds for hexagons: from the second iterate onward, the centroids of the filled hexagons all lie exactly on a fixed line. This exact colinearity reflects a special algebraic feature of the hexagonal case and does not hold generally for any other polygons. |
| title | Exact colinearity of centroids of iterated midpoint hexagons |
| topic | Metric Geometry 51F, 51M04 |
| url | https://arxiv.org/abs/2604.04668 |