Abyss of regular octagons
On a pale disc, the initial dark octagon is convex, its star‑shaped hole and itself share their eight vertices of regular polygons. Their common center is the center of a homothety of ratio which shrinks repeatedly these octagons by alternating the dark color and the light grey, visible through the holes. In this way the geometric figure is extended into an abyss of similar regular octagons.
Mathematically the number of successive polygons can be infinite. Actually the SVG source code ends with </svg>, after a drawing of disc in light grey, one convex octagon and six star octagons. The six are coded each from a 'M', there are six letters 'M' in the last character string: the last value of 'd' (“d” like “drawing”, “M” like “Moveto” ). If you really want to verify the smallest star of the abyss is dark with eight triangular holes around a dark convex octagon, magnify the image through several successive keyboard shortcuts 'Ctrl + +'. You can also open the SVG file from a text editor, replace the value "‑141 ‑141 282 282" of 'viewBox' with "‑6 ‑6 12 12", save the new SVG file under a new name like Center_of_abyss_of_regular_octagons.svg and display your new image. In one way or another, you will note the smallest star‑shaped hole has not exactly its vertices at the intended locations, whereas the drawing of the very last star seems perfect. Its vertices have the successive following coordinates in the source code, with a space between two successive pairs: 0,-1.7 1.2,1.2 -1.7,0 1.2,-1.2 0,1.7 -1.2,-1.2 1.7,0 -1.2,1.2 (Ctrl + U to display the source code in a browser).
Clockwise or not around the center we imagine, a 22.5° angle rotation keeps the whole figure unchanged. A ± 180° angle rotation of this center composed with the previous homothety yields the same abyss of polygons. This new homothety of opposite is considered as a similarity of positive ratio.
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