In “Polykoch!”, I looked at variants on the famous Koch snowflake, which is created by erecting new triangles on the sides of an equilaternal triangle, like this:

Koch snowflake #1

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Koch snowflake #2

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Koch snowflake #3

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Koch snowflake #4

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Koch snowflake #5

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Koch snowflake #6

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Koch snowflake #7

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Koch snowflake (animated)

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One variant is simple: the new triangles move inward rather than outward:

Inverted Koch snowflake #1

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Inverted Koch snowflake #2

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Inverted Koch snowflake #3

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Inverted Koch snowflake #4

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Inverted Koch snowflake #5

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Inverted Koch snowflake #6

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Inverted Koch snowflake #7

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Inverted Koch snowflake (animated)

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Or you can alternate between moving the new triangles inward and outward. When they always move outward and have sides 1/5 the length of the sides of the original triangle, the snowflake looks like this:

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When they move inward, then always outward, the snowflake looks like this:

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And so on:

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Now here’s a Koch square with its new squares moving inward:

Inverted Koch square #1

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Inverted Koch square #2

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Inverted Koch square #3

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Inverted Koch square #4

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Inverted Koch square #5

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Inverted Koch square #6

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Inverted Koch square (animated)

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And here’s a pentagon with squares moving inwards on its sides:

Pentagon with squares #1

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Pentagon with squares #2

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Pentagon with squares #3

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Pentagon with squares #4

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Pentagon with squares #5

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Pentagon with squares #6

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Pentagon with squares (animated)

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And finally, an octagon with hexagons on its sides. First the hexagons move outward, then inward, then outward, then inward, then outward:

Octagon with hexagons #1

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Octagon with hexagons #2

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Octagon with hexagons #3

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Octagon with hexagons #4

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Octagon with hexagons #5

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Octagon with hexagons (animated)

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