Sierpinski carpet and gasket
Both the Carpet and Gasket were first described by Waclaw Sierpinski in 1915.
Sierpinski Carpet
This is produced in a very similar way to the Cantor set, just in 2 dimensions, so we start off with a square.
Which is then divided into 9 equal squares.
And the middle one is removed.
The process is then repeated indefinately, and the resulting set of points is the Sierpinski carpet.
As can be seen on the right.
Again similarly to the Cantor set, it can be proved that this has no area, but again points remain.
This can then be extended into 3 dimensions, and then the Menger sponge is created
As can be expected, this starts off as a cube, which is then divided into 27 equal cubes and the middle one is removed as well as the ones at the middle of every face (so 7 cubes are removed), this is then appled to each remaining cube
And this like its 2 predeccessors has appropriately no volume. And again has points which definately remain.
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This gasket is usually shown as starting from an equilateral triangle,
Which is then shrunk by 1/2, and 2 copies are made and positioned such that the three shapes form a new triangle.
This is then repeated ad infinitum, and the limit of this is called the Sierpinski Triangle/gasket.
Bear in mind, that even though it is commonly seen starting with a triangle, this is merely for conveniance, as at the limit each starting shape is now effectively shrunk to a point. So any starting shape may be used. Like the diagram below which shows the sequance starting with a square.
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