A rep-tile is a shape that can be tiled with smaller equally sized copies of the shape. If the tiling uses copies, the shape is said to be rep-.
An irregular rep-tile (or irreptile) is a shape that can be tiled with copies of itself, not all of the same size. If the tiling uses copies, the shape is said to be irrep-. If no two tiles have the same size, the tiling is called perfect. A shape that only allows irregular self-tilings is called a puritile.
Diagrams 6, 10, 25, 33 do not show a tiling because I could not find the associated tiling. I hope a reader can fill in some of the gaps.
Irreptiles of order less than 20 were dealt with in the Demonstration Irreptiles by the same author.
The regular order of a shape is the smallest possible number of tiles in a regular self-tiling.
The irregular order of a shape is the smallest possible number of tiles in an irregular self-tiling (using more than one size of tile).
The author conjectures that there is no shape when these orders are the same.
The terms "irreptile" and "puritile" were introduced by the author in 1987 in his book A Puzzling Journey to the Reptiles and Related Animals, privately published. Since then several mathematicians have expanded on the subject in various ways.
The material of this Demonstration was mainly taken from the author's book.
Of course, this Demonstration can only give a short introduction to irreptiles. Many questions can be asked (and many are answered in the author's book), but several problem areas are still unsolved.
Content of this Demonstration
Diagrams 1 and 2 are overviews. The icons with names in parentheses have the self-tiling missing.
If you know the solution to a missing tiling, please contact the author (karlscherer.com).
If a shape is a polysquare such that two copies tile a rectangle, then the associated regular and irregular self-tilings are usually quite straightforward and obvious, if not trivial. Therefore some of these cases have been ignored in this Demonstration.