Dissecting the Diamond

Pre-previously on O.o.t.Ü.-F., I dilated the delta. Now I want to dissect the diamond. In geometry, a shape is dissected when it is completely divided into smaller shapes of some kind. If the smaller shapes are identical (except for size) to the original, the original shape is called a rep-tile (because it can be tiled with repeating versions of itself). If the smaller identical shapes are equal in size to each other, the rep-tile is regular; if the smaller shapes are not equal, the rep-tile is irregular. This diamond is an irregular rep-tile or irrep-tile:

Dissectable diamond

Dissected diamond


As you can see, the diamond can be dissected into five smaller versions of itself, two larger ones and three smaller ones. This makes it a rep-5 irrep-tile. And the smaller versions, or sub-diamonds, can themselves be dissected ad infinitum, like this:

Dissected diamond stage #1


Dissected diamond #2


Dissected diamond #3


Dissected diamond #4


Dissected diamond #5


Dissected diamond #6


Dissected diamond #7


Dissected diamond #8


Dissected diamond #9


Dissected diamond (animated)


The full dissected diamond is a fractal, or shape that is similar to itself at varying scales. However, the fractality of the diamond becomes most obvious when you dissect-and-discard. That is, first you dissect the diamond, then you discard one (or more) of the sub-diamonds, like this:

Diamond fractal (retaining sub-diamonds 1,2,3,4) stage #1


1234-Diamond #2


1234-Diamond #3


1234-Diamond #4


1234-Diamond #5


1234-Diamond #6


1234-Diamond #7


1234-Diamond #8


1234-Diamond #9


1234-Diamond (animated)


Here are some more fractals created by dissecting and discarding one sub-diamond:

Diamond fractal (retaining sub-diamonds 1,2,4,5)


1245-Diamond (anim)


2345-Diamond


2345-Diamond (anim)


The 2345-diamond fractal has variants created by mirroring one or more sub-diamonds, so that the orientation of the sub-dissections changes. Here is one of the variants:

2345-Diamond (variation)


2345-Diamond (variant) (anim)


And here is a fractal created by dissecting and discarding two sub-diamonds:

Diamond fractal (retaining sub-diamonds 1,2,3)


123-Diamond (anim)


Again, the fractal has variants created by mirroring one or more of the sub-diamonds:

123-Diamond (variant #1)


123-Diamond (variant #2)


123-Diamond (variant #3)


123-Diamond (variant #4)


Some more fractals created by dissecting and discarding two sub-diamonds:

125-Diamond


125-Diamond (anim)


134-Diamond


134-Diamond (anim)


235-Diamond


235-Diamond (anim)


135-Diamond


135-Diamond (anim)


A variant of the 135-Diamond fractal looks like one side of a Koch snowflake:

135-Diamond (variant #1) — like Koch snowflake


135-Diamond (variant #2)


Finally, here are some colour variants of the full dissected diamond:






Full diamond colour variants (anim)


Elsewhere other-engageable:

Dilating the Delta

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