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
Koch snowflake #2
Koch snowflake #3
Koch snowflake #4
Koch snowflake #5
Koch snowflake #6
Koch snowflake #7
Koch snowflake (animated)
One variant is simple: the new triangles move inward rather than outward:
Inverted Koch snowflake #1
Inverted Koch snowflake #2
Inverted Koch snowflake #3
Inverted Koch snowflake #4
Inverted Koch snowflake #5
Inverted Koch snowflake #6
Inverted Koch snowflake #7
Inverted Koch snowflake (animated)
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:
When they move inward, then always outward, the snowflake looks like this:
And so on:
Now here’s a Koch square with its new squares moving inward:
Inverted Koch square #1
Inverted Koch square #2
Inverted Koch square #3
Inverted Koch square #4
Inverted Koch square #5
Inverted Koch square #6
Inverted Koch square (animated)
And here’s a pentagon with squares moving inwards on its sides:
Pentagon with squares #1
Pentagon with squares #2
Pentagon with squares #3
Pentagon with squares #4
Pentagon with squares #5
Pentagon with squares #6
Pentagon with squares (animated)
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
Octagon with hexagons #2
Octagon with hexagons #3
Octagon with hexagons #4
Octagon with hexagons #5
Octagon with hexagons (animated)