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File:Abyss of regular octagons.svg

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Description
English:
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.


Français :
Bientôt…
Date 07/03/2010
Source Own work
Author Yves Baelde
Other versions

        With regular dodecagons

SVG development
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Arthur Baelde, the copyright holder of this work, hereby publishes it under the following license:
w:en:Creative Commons
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Attribution: Arthur Baelde
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Captions

On a pale disc a dark convex octagon and its star‑shaped hole share their vertices. Through the central hole we see a dark star octagon, its central zone is a duplicate of the previous octagon, in this way extended into an abyss of regular octagons.

Items portrayed in this file

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image/svg+xml

710df25307e6de4694bbed2d21d5a11047224b14

818 byte

750 pixel

750 pixel

File history

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Date/TimeThumbnailDimensionsUserComment
current12:43, 8 March 2022Thumbnail for version as of 12:43, 8 March 2022750 × 750 (818 bytes)Arthur Baelde better framing,  two lighter colors inverted  and the source code is more interesting to examine or explain 
10:24, 17 July 2012Thumbnail for version as of 10:24, 17 July 2012750 × 750 (676 bytes)BaeldeSize  750 × 750 instead of  600 × 600,  margins reduced 
21:27, 7 March 2010Thumbnail for version as of 21:27, 7 March 2010600 × 600 (677 bytes)Baelde{{Information |Description={{en|1=By tracing a regular stellated octagon,  we obtain a concentric convex octagon,  whose sides are smaller.  We can start with this convex octagon to put in the original image its reproduction.  In pure geometr

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