When is a fish a reptile? When it looks like this:

Fish from four isosceles right triangles

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The fish-shape can be divided into eight identical sub-copies of itself. That is, it can be repeatedly tiled with copies of itself, so it’s an example of what geometry calls a rep-tile:

Fish divided into eight identical sub-copies

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Fish divided again

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Fish divided #4

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Fish divided #5

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Fish divided #6

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Fish (animated rep-tiling)

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Now suppose you divide the fish, then discard one of the sub-copies. And carry on dividing-and-discarding like that:

Fish discarding sub-copy 7 (#1)

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Fish discarding sub-copy 7 (#2)

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Fish discarding sub-copy 7 (#3)

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Fish discarding sub-copy 7 (#4)

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Fish discarding sub-copy 7 (#5)

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Fish discarding sub-copy 7 (#6)

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Fish discarding sub-copy 7 (#7)

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Fish discarding sub-copy 7 (animated)

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Here are more examples of the fish sub-dividing, then discarding sub-copies:

Fish discarding sub-copy #1

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Fish discarding sub-copy #2

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Fish discarding sub-copy #3

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Fish discarding sub-copy #4

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Fish discarding sub-copy #5

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Fish discarding sub-copy #6

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Fish discarding sub-copy #7

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Fish discarding sub-copy #8

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Fish discarding sub-copies (animated)

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Now try a square divided into four copies of the fish, then sub-divided again and again:

Fish-square #1

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Fish-square #2

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Fish-square #3

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Fish-square #4

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Fish-square #5

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Fish-square #6

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Fish-square (animated)

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The fish-square can be used to create more symmetrical patterns when the divide-and-discard rule is applied. Here’s the pattern created by dividing-and-discarded two of the sub-copies:

Fish-square divide-and-discard #1

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Fish-square divide-and-discard #2

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Fish-square divide-and-discard #3

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Fish-square divide-and-discard #4

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Fish-square divide-and-discard #5

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Fish-square divide-and-discard #6

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Fish-square divide-and-discard #7

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Fish-square divide-and-discard #8 (delayed discard)

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Fish-square divide-and-discard (animated)

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Using simple trigonometry, you can convert the square pattern into a circular pattern:

↓

Circular version

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Square to circle (animated)

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Here are more examples of divide-and-discard fish-squares:

Fish-square divide-and-discard #1

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Fish-square divide-and-discard #2

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Fish-square divide-and-discard #3

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Fish-square divide-and-discard #4

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Fish-square divide-and-discard #5

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Fish-square divide-and-discard #6

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And more examples of fish-squares being converted into circles:

Fish-square to circle #1 (animated)

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Fish-square to circle #2

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Fish-square to circle #3

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Fish-square to circle #4

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Fish-square to circle #5

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Fish-square to circle #6

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