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The below quadrilateral can be divided into four congruent quadrilaterals which are similar to the parent quadrilateral. Each of which can be again divided in a similar way, and so on infinitely.

Parent quadrilateral
Parent quadrilateral

Child quadrilaterals
Child quadrilaterals

How are these types of polygons classified? Are there more such polygons?

Some other examples are:

  • The A series paper size
  • Dividing a square into four and so on.
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  • $\begingroup$ Penrose tilings have this property, though they contain more than 1 kind of polygon. $\endgroup$
    – Wouter
    Aug 2, 2016 at 10:38
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    $\begingroup$ This sort of stuff is called rep-tile (not the animal), Simple ones like square, rectangle, parallelogram, rhombus, triangle and the more complicated ones like sphinx and Gosper island. $\endgroup$ Aug 3, 2016 at 4:39

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