Periodic design with 3-fold rotational symmetry from 3-dimensional space

Three dimensional space gives a three-fold rotational symmetry in the drawing plane. The designs are periodic. Note that if you put a cube on one of its points and look along its space-diagonal from above, then you see an object with three-fold rotational symmetry. This shows that there is an important similarity to isometric projection. The three unit vectors in the drawing plane are

(1, 0), (-½, ½√3) and (-½, -½√3).

They are isometric projections of the three coordinate axis and form a triangular lattice. You do not need to use all three vectors. Often, one replaces the third unit vector by

(-½, -½√3) = – (1,0) – (-½, ½√3).

But for creating designs with three-fold rotational symmetry it might be better to use all three vectors. Then, the symmetry becomes obvious. Similar to the previous post, the simplest mapping functions with three-fold rotational symmetry are

X(x,y) = cos(x) +cos(-½ x+½√3 y) +cos(-½ x -½√3 y) and

Y(x,y) = sin(x) +sin(-½ x+½√3 y) +sin(-½ x -½√3 y).

This results in images like that:

As input image I used a photo of a rosechafer on sedum spectabilis. The green and rose shapes result from the flower. The shiny yellow and blue shape is part of the body of the insect. At the ends you can discover its head.

A minstrel bug and different mapping functions give

 

This entry was posted in Anamorphosis, Kaleidoscopes, Tilings and tagged , , . Bookmark the permalink.

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