Steven Dutch, Natural and Applied Sciences, University
of Wisconsin  Green Bay
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Reptiles, or Replicating Tiles, are tiles that can be joined together to make larger replicas of themselves. The term "reptile" was coined by mathematician Solomon Golomb, a prolific contributor to geometrical recreations of all sorts.
Reptiles that require n tiles to build a larger version of themselves are said to be repn. For example, we can combine four squares to make a bigger square, so a square is rep4. Since we can combine any of those replicas into a secondgeneration copy, a repn tile is also repn^{2}, repn^{3}, and so on. Often tiles have several repnumbers. If a tile is repn and repm, it is also repmn, since we can build replicas with n tiles, then combine m of those into a yet larger version. Obviously, everything is also (trivially) rep1.
All triangles and parallelograms are repk^{2}, where k is some integer. Repk parallelograms for any value of k are easy to construct; they have edges of 1 and 1/sqrt(k). They need not be rectangles. 


The 306090 triangle is both rep25 (5^{2}) and rep27 (3^{3}, by combining three generations of rep3 triangles). 
Four squares can be arranged around a central square to form a Greek Cross. Greek Crosses make a lovely plane tesselation, but they are not reptiles. We can try modifying a Greek Cross tesselation by replacing each cross with the compound of five crosses. It's closer, but still not an exact reptile. We can repeat the process, each time getting figures that are more and more crinkled and closer to a true reptiling. But at every step, we see that the perimeter of the tiling always has twice as many crinkles as each tile. 
If we continue the process to infinity, however, then the tiles would have infinitely crinkly edges and would tesselate to make a true rep5 tiling. The compound of five figures would be a scaledup exact replica of each tile. Amazingly, the perimeter of each tile is infinite, even though the area of the tiles is obviously finite. A weird figure of this sort, whose perimeter behaves as if it's not merely onedimensional, but not quite twodimensional either, is called a fractal. The fact that the compound of five tiles is a bigger, but still exact, replica of the individual tiles, is called selfsimilarity. Since tiling deals with similar figures, and many fractals are selfsimilar, it should be no surprise that many fractals are reptiles.
Obviously, since neither computers nor human draftsmen can generate curves that are infinitely long and infinitely crinkly, nobody has ever drawn a true fractal. Since no physical system that we know of is infinite, nobody has ever seen a real fractal either. True fractals are ideal concepts that are only approximately achieved in physical reality  Plato would have been enthralled by them. Most fractals are approximately represented as in the last figure above, with detail close to the resolution limit of the picture but not quite, so that the fine structure is still visible.
At left is another example. Hexagons cannot be arranged to form a reptile, but the cluster of six around a central seventh hexagon comes close. If we replace each hexagon with a compound of seven, we get closer yet, and after an infinite number of cycles we achieve a true rep7 fractal. 
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Created July 9, 1999, Last Update July 9, 1999
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