Pixellated triangles can only approximate to Koch snowflakes/antisnowflakes, but they are all produced by following the rules I developed for them.
Trefoils: from the centre of each side, remove the largest triangular segment that leaves the remainder intact.
Hexagrams: to the centre of each side add a triangle that creates a regular hexagram.
Only G-triangles with a single central counter) will give regular hexagrams.
All three classes (G-triangles, G2-triangls, G3-triangles) can be transformed into trefoils.
Hexagrams
To each side of GTn, add T(n-1).
So to each side of GT4 (tr. 55) we add T3 (tr. 6), giving Hgr. 4 (her. 73).
Trefoils
G-triangles: To each side of GTn, subtract T(n-2).
So to each side of GT4 (tr. 55) we subtract T2 (tr. 3).
G2-triangles: To each side of (G2)Tn, subtract T(n-1).
So to each side of (G2)T3 (tr. 36) we subtract T2 (tr. 3).
G3-triangles: To each side of (G3)Tn, subtract (Tn - T1).
So to each side of (G3)T3 (triangle 45) we subtract T3-T1 (tr.6 - tr.1))
- Trefoils Hexagrams.png (245.89 KiB) Viewed 98 times
There is good evidence that all three classes of trefoil have been utilized. The first and second groups of nine words in the NIV '84 have OVs of 378 and 528. I puzzled over these for years until I tried one Koch antisnowflake iteration on each. This was the result.
- 378 528.png (157.93 KiB) Viewed 98 times