Bent for the Pent

A triangle can be tiled with triangles and a square with squares, but a pentagon can’t be tiled with pentagons. At least, not in the same way, using smaller copies of the same shape. The closest you can get is this:

Pentaflake #1

If you further subdivide the pentagon, you create what is known as a pentaflake:

Pentaflake #2

Pentaflake #3

Pentaflake #4

Pentaflake (animated)

Pentaflake (static)

But if you bend the rules and use irregular smaller pentagons, you can tile a pentagon like this, creating what I called a pentatile:

Pentatile stage 1

Further subdivisions create an interesting final pattern:

Pentatile #2

Pentatile #3

Pentatile #4

Pentatile #5

Pentatile #6

Pentatile (animated)

Pentatile (static)

By varying the size of the central pentagon, you can create other patterns:

Pentatile #1 (animated)

Pentatile #2 (animated)

Pentatile #2

Pentatile with no central pentagon

And here are various pentatiles in an animated gif:

And here are some variations on the pentaflake:

Elsewhere other-posted:

Bent for the Rent (1976) — the title of the incendiary intervention above is of course a reference to the “first and last glitter-rock album” by England’s loudest band, Spinal In Terms Of Tap
Phrallic Frolics — more on pentaflakes

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