This is how you form the famous Koch snowflake, in which at each stage you erect a new triangle on the middle of each line whose sides are 1/3 the length of the line:

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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Here’s a variant of the Koch snowflake, with new mid-triangles whose sides are 1/2 the length of the lines:

Koch snowflake (1/2 side) #1

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Koch snowflake (1/2 side) #2

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Stage #3

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Stage #4

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Stage #5

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Stage #6

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Stage #7

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Stage #8

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Koch snowflake (1/2 side) (animated)

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But why stop at triangles? This is a Koch square, in which at each stage you erect a new 1/3 square on the middle of each line:

Koch square #1

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Koch square #2

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Koch square #3

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Koch square #4

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Koch square #5

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Koch square #6

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Koch square (animated)

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And a Koch pentagon, in which at each stage you erect a pentagon on the middle of each line whose sides are 1 – (1/φ^2 * 2) = 0·236067977… the length of the line (I used 55/144 as an approximation of 1/φ^2):

Koch pentagon (side 55/144) #1

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Koch pentagon #2

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Koch pentagon #3

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Koch pentagon #4

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Koch pentagon #5

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Koch pentagon #6

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Koch pentagon (animated)

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In this close-up, you can see how precisely the sprouting pentagons kiss at each stage:

Koch pentagon (close-up) #1

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Koch pentagon (close-up) #2

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Koch pentagon (close-up) #3

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Koch pentagon (close-up) #4

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Koch pentagon (close-up) #5

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Koch pentagon (close-up) #6

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Koch pentagon (close-up) (animated)

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