Image:Perrontree2.jpg

[edit] Summary

This is an illustration of the method of constructing different sizes of Kakeya sets using the 'sprouting' method, in which the final shapes are what are called Perron trees. One starts with a triangle, and then partitions it into 2^n subtriangles (for some integer n) and then shifts each consecutive pair of triangles over each other so they over lap, then shift the consecutive remaining pieces, and so on until you get a 'tree' shape. For large enough n, one can shift these triangles in such a way to get a tree as small as you like. The image is based off of the construction described in Falconer's book, Geometry of Sets and Fractals. I made this image myself on Flash MX.

[edit] Licensing

I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law. Afrikaans | Alemannisch | Aragonés | العربية | Asturianu | Български | Català | Česky | Cymraeg | Dansk | Deutsch | Eʋegbe | Ελληνικά | English | Español | Esperanto | Euskara | Estremeñu | فارسی | Français | Galego | 한국어 | हिन्दी | Hrvatski | Ido | Bahasa Indonesia | Íslenska | Italiano | עברית | Kurdî / كوردی | Latina | Lietuvių | Latviešu | Magyar | Македонски | Bahasa Melayu | Nederlands | ‪Norsk (bokmål)‬ | ‪Norsk (nynorsk)‬ | 日本語 | Polski | Português | Ripoarisch | Română | Русский | Shqip | Slovenčina | Slovenščina | Српски / Srpski | Svenska | ไทย | Tagalog | Türkçe | Українська | Tiếng Việt | Walon | ‪中文(简体)‬ | ‪中文(繁體)‬ | zh-yue-hant | +/-

• Kakeya set