# Rep-tiles of order 4 and order 9

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

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.

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?

• Related questions are discussed in Viorel Nitica, Rep-tiles revisited, MASS selecta, 205-217, Amer Math Soc, MR2027179 (2004j:52020). May 18, 2018 at 7:10
• @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. May 18, 2018 at 10:21
• All I know about the paper is what I read in Math Reviews, and that wasn't all that detailed. May 18, 2018 at 12:40
• @GerryMyerson OK. I see. Thank you anyway. May 19, 2018 at 8:01
• 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. Jun 20, 2018 at 19:19