Title: Semi-dimensional shapes Post by: Tglad on November 05, 2016, 12:35:14 PM I realised the other day that there is a quite systematic way to make fractals with exactly half dimensions. Here is a 1.5D curve in 3D space with no intersection:
(https://3.bp.blogspot.com/-7KcSZdGE5XM/WB2_S2OojdI/AAAAAAAAA5Y/YvSE0TQvMB8bTrsTJnQxjQMsNnKehfaYACLcB/s200/curve1pt51.png) (https://3.bp.blogspot.com/-0KQTVSDYrZM/WB2_Sw1NshI/AAAAAAAAA5U/E3u6Digl-7UKO-FbrOIJbsYWjFTOv-4mQCLcB/s200/curve1pt52.png) (https://1.bp.blogspot.com/-hvWdDKHyaTM/WB2_K2ZqrxI/AAAAAAAAA5Q/HTgZlax2jBMw8RUXuxob87EPIWwuCYCPACLcB/s320/curverender1pt5png.png) you grid the space and divide each length by a quarter, then build your replacement shape from 4^(X.5) pieces (which is integer for integer X). Here is a 2.5D surface in 3D with no intersection: (https://3.bp.blogspot.com/-DIUNaXc9bpM/WBxrKHQ9bGI/AAAAAAAAA4s/VR86XcUXBJsi4i7_Qj2-kIkJ2H3xCtabACLcB/s200/squareits1.png) (https://1.bp.blogspot.com/-eRjmRe2rCJw/WBxrOdSB2HI/AAAAAAAAA4w/9D6rDpTGFf4xQRuBAepzbEko8oM4apTEQCLcB/s200/squareits2.png) (https://3.bp.blogspot.com/-BsdeKmfYV74/WBxrVSajocI/AAAAAAAAA40/A8y7sXbU53Qe4Ec4RK2owrZZH_7bdnGxgCLcB/s1600/squarerender.png) and its 2D equivalent is a 1.5D curve in 2D space: (https://3.bp.blogspot.com/-fiy2urMPsFU/WBhiW-GNaCI/AAAAAAAAA3Y/2fEKN4Gl3eIXdAb55uC0qpA3M_yr8AhrQCLcB/s400/square1.png) http://tglad.blogspot.com.au/2016/11/semi-dimensional-shapes-and-other-curves.html (http://tglad.blogspot.com.au/2016/11/semi-dimensional-shapes-and-other-curves.html) I tried to make a 2.5D curve in 3D space without intersection... but it is pretty tricky :tease: Title: Re: Semi-dimensional shapes Post by: ciric50 on November 06, 2016, 05:07:39 PM Can you explain to me how that shape is exactly half-dimensional? It's not obvious to me.
Title: Re: Semi-dimensional shapes Post by: Tglad on November 07, 2016, 03:27:57 AM Hi ciric, for the 2.5D surface example, you start with a flat square (iteration 0) then grid it up into a 4x4x4 grid. You then want to replace the one big square with 32 of the smaller (1/4 scale on each axis) squares. so from: (https://3.bp.blogspot.com/-DIUNaXc9bpM/WBxrKHQ9bGI/AAAAAAAAA4s/VR86XcUXBJsi4i7_Qj2-kIkJ2H3xCtabACLcB/s200/squareits1.png) to: (https://1.bp.blogspot.com/-eRjmRe2rCJw/WBxrOdSB2HI/AAAAAAAAA4w/9D6rDpTGFf4xQRuBAepzbEko8oM4apTEQCLcB/s200/squareits2.png) (the darker shade here show upside-down squares). ..and then you re-apply this replacement for each of the smaller squares, and so on, forever. The resulting shape is exactly 2.5D because 4^2.5 = 32. For the curves you are replacing 1 line with 8 lines of one quarter the size. This is exactly 1.5D because 4^1.5 = 8. I hope that makes sense. |