The spiral hepta-hextals are 4 of many hepta-hextals. These particular fractals, unlike many others, are not known to be members of a rich set of hextals with a common construction.
All 4 spiral hepta-hextals are c3-symmetric, and tile the plane with a single copy in the unit cell. If the vector connecting the center of the 1st (red) and 2nd (blue) elements is (1,0) the tiling vectors are (2,2sin(60)) and (2.5,-sin(60)).
| Image | Tiling | Name |
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cis-0-spiral hepta-hextal |
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trans-0-spiral hepta-hextal |
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cis-60-spiral hepta-hextal |
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trans-60-spiral hepta-hextal |
Construction: These fractals are hextals, and hence the characteristic hepta-hextal rotation of arctan(sin(60)/2.5) is involved. The fractals are based on the spiral 7 element polyhex, with a central element, alternating elements for the surrounding ring, and 3 symmetrically placed elements from the next ring. If the coordinates of the centre of the central element (red) are (0,0) and of the blue element are (1,0) then the coordinates of the magenta element are (1+cos(60),sin(60); the remaining coordinates can be determined by rotation are are (-cos(60),sin(60)) (gold), (-cos(60),-sin(60)) (green), (0,2*sin(60)) (yellow) and (-1-cos(60),-sin(60)) (cyan). The elements of the cis forms are rotation by arctan(sin(60)/2.5) or 60+arctan(sin(60)/2.5), and the elements of the trans forms by -arctan(sin(60)/2.5) or 60-arctan(sin(60)/2.5). These rotations are reversed for the mirror image fractals.
Related fractals - hextals, grouped element derivatives of the spiral hepta-hextals
Sources: Independent invention. R. William Gosper has independently discovered the 0-spiral hepta-hextals.
data file spiralheptahextal.xil; generator spiralheptahextal.pl; images generated at 6x5 per screen at 1280x1024
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