Rep-tiles are figures which are dissected by the same figures as itself.

Examples of rep-tiles

As you can see, the rep-tile of order 4 is also a rep-tile of order 9 in the above figures: Compare the L-shaped figure at row 2 column 1 and one at row 3 and column 1, and compare two trapezoids at row 1 and row 3.

As I know, Golomb found that a few rep-tiles of order 9 which are not of order 4. But, those examples are not simple closed curves.

Golomb's fish-like rep-tile

So, my questions:

(1) Is any rep-tile of order 4 always of order 9?

(2) If a rep-tile of order 9 is a simple close curve, is it also of order 4?

  • 2
    $\begingroup$ Related questions are discussed in Viorel Nitica, Rep-tiles revisited, MASS selecta, 205-217, Amer Math Soc, MR2027179 (2004j:52020). $\endgroup$ – Gerry Myerson May 18 '18 at 7:10
  • $\begingroup$ @GerryMyerson Thank you! But, I don't have Nitica's paper and Google Books does not provide the whole paper. :-( Could you let me know the answer before I purchase the book. $\endgroup$ – P.-S. Park May 18 '18 at 10:21
  • $\begingroup$ All I know about the paper is what I read in Math Reviews, and that wasn't all that detailed. $\endgroup$ – Gerry Myerson May 18 '18 at 12:40
  • $\begingroup$ @GerryMyerson OK. I see. Thank you anyway. $\endgroup$ – P.-S. Park May 19 '18 at 8:01
  • 1
    $\begingroup$ Make a triangle from 9 triangles, remove a corner. This shape is rep-9 but not rep 4. demonstrations.wolfram.com/FactoryForFracTilesOfOrderFour -- 4-fractiles often aren't 9-fractiles. $\endgroup$ – Ed Pegg Jun 20 '18 at 19:19

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