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Rep-4 = Rep-9?
editI just reverted the addition of the sentence "Every known rep-9 tile is also rep-4, and every known rep-4 tile is also rep-9 [1]". I don't think it's true — see e.g. the n=4 line of [2]. And the given source doesn't support it — it only says that it's true of the ones shown in that source. But maybe it's true for non-fractal reptiles? Does anyone know of additional sources that can shed light on this? —David Eppstein (talk) 07:40, 22 March 2012 (UTC)
Replacing image
editCould someone who edits this page tell me how to replace an image added to the page with a new version? I'm assuming there's a way to use the original name, info, etc with a new version, but the help pages don't help v. quickly and I don't have time to learn for myself what is possible on Wikipedia. Thanks. MagistraMundi (talk) 09:57, 29 November 2012 (UTC)
- You could go to the original place the image was uploaded (probably on commons.wikimedia.org) and click on the "Upload a new version of this file" link. But you should only do it this way if your changes to the appearance of the image are minor. —David Eppstein (talk) 13:28, 29 November 2012 (UTC)
- Ah, thanks very much. The problem was that I wasn't logged onto Commons, so that new-upload-link wasn't showing. MagistraMundi (talk) 09:18, 1 December 2012 (UTC)
Right Triangles
editNot sure if this qualifies, but if you start with any right triangle (other than a 45-45-90), then divide that triangle into two by drawing a perpendicular from the hypotenuse to the opposite vertex, you get two congruent right triangles which are smaller copies of the original. Repeat this process on the two smaller triangles, and then on the resulting four triangles, etc. as often as you like, and you'll end up with a non-periodic self-tiling of smaller copies of the original triangle. (I only exclude 45-45-90 because this would result in a regular periodic tiling, which is a special case.) Lurlock (talk) 14:32, 12 November 2015 (UTC)
No, the two triangles are similar, but they are not congruent. Gnomon (talk) 09:11, 3 July 2020 (UTC)
N2 Tiling?
editI believe that most of these shapes can tile themselves N2 times for any N. I'm not sure how much research has been done on this, but to take an example:
┌─────┬─────┐ ┌─────┬───┬─────┬─┐ ┌─┬─────┬───┬─────┬───┬───┬───┐ ┌─────┬─────┬───┬───┬───┬───┬───┬───┬─────┐
│ . ┌─┴─┐ . │ ├─┐ . │ . │ . ┌─┘ │ │ └─┐ . │ . ├─┐ . │ . │ . │ . │ │ . ┌─┴─┐ . │ . │ . │ . │ . │ . │ . └─┐ . │
├───┤ . ├───┘ │ └─┬─┤ ┌─┴───┤ . │ │ . ├───┤ ┌─┘ ├───┴─┐ │ ┌─┤ ┌─┤ ├───┤ . └─┬─┤ ┌─┤ ┌─┤ ┌─┤ ┌─┤ ┌─┴─┬───┼───┤
│ . └─┐ │ │ . │ └─┤ . ┌─┴───┘ ├───┤ . └─┤ . │ . ┌─┴─┼─┘ ├─┘ │ │ . └─┬───┤ └─┤ └─┤ └─┤ └─┤ └─┤ . │ . │ . │
└─────┴─┘ ├───┤ . ├───┤ │ . ├───┬─┴─┬─┴───┤ . │ . │ . │ ├───┬─┘ . │ . │ . │ . │ . │ . ├─┐ ├─┐ ├─┐ │
│ . └─┬─┘ . │ │ ┌─┘ . │ . │ . ┌─┤ ┌─┴───┴───┘ │ . └─┬───┴─┬─┴─┬─┴───┼───┼───┤ └─┤ └─┤ └─┤
└─────┴─────┘ ├─┴───┬─┴─┐ ├───┤ └─┤ ├───┬─┴─┐ . │ . └─┐ . │ . │ . │ . │ . │ . │
├─┐ . │ . ├─┤ . │ . │ │ . │ . └─┬─┴─┬───┼───┤ ┌─┤ ┌─┴───┴───┴───┘
│ └─┬─┴─┐ │ └─┐ ├───┤ │ ┌─┴───┬─┘ . │ . │ . └─┤ └─┤
│ . │ . └─┤ . ├─┘ . │ ├─┤ . ┌─┼───┬─┤ ┌─┴─┬───┤ . │
└───┴─────┴───┴─────┘ │ └─┬─┘ │ . │ └─┤ . │ . ├───┤
│ . │ . │ ┌─┤ . ├─┐ │ ┌─┘ . │
├───┴─┬─┴─┤ └─┬─┘ ├─┴─┼─────┤
│ . ┌─┘ . │ . │ . │ . └─┐ . │
└───┴─────┴───┴───┴─────┴───┘
This shows a shape tiling itself 22, 32, 52 and 72 times. It can also trivially tile itself once, and you can get 42 by doing the 4-tiling twice, or 62 by doing 4 and then 9 or vice versa, or 82 by doing 4-tiling three times, or 92 by doing 9-tiling twice, etc. (You really only have to prove it for the prime numbers, since any square of a composite number can be achieved by using multiples of the prime squares.) I've also done the same for the Sphinx Tile, up to 52 and 72-tiling. (Still working on 112.) These are harder to do with ASCII art, but I could throw together a picture. Not sure how one would go about proving that it's possible for any N2, or for how many of these shapes it is the case, but I have strong suspicions that it is. My math isn't rigorous enough for a proof, and even if it was this might fall under "original research", but I still feel it's worth mentioning... Lurlock (talk) 14:32, 12 November 2015 (UTC)
- I just posted a proof that the P pentomino is rep-n for all square n to http://11011110.livejournal.com/319787.html. I'm not sure about the sphinx, but it seems likely. —David Eppstein (talk) 08:39, 11 November 2015 (UTC)
- Nice. I have managed an 112 tiling for the P pentomino, and I've gotten 52, 72, and 112 for several others - both the regular trapezoid (half-regular hexagon) and the trapezoid that's one square plus a diagonally cut half-square, as well as the L tetromino and L tromino. I think it works for the two triangles one pretty easily as well - the one shaped like this:
___ | /|\ | / | \ |/ |__\
- I still haven't managed 112 tiling on the Sphinx, but I'm pretty sure it can be done. I just don't have any organized approach to doing it. I should try cutting some shapes out of cardboard, they'd be easier to move around than just pen-and-pencil on paper. I can e-mail you my 52 and 72 Sphinx solutions when I get home. I can prove that it doesn't work for some shapes, such as the T tetromino or the J hexomino. Lurlock (talk) 14:32, 12 November 2015 (UTC)
- Actually, this is easier:
/\
/\ \
/ /__\
/__\ /\
/\ /\ /\ \ \
/_ \ / / / _\/__\
/\ \_\ ______________/__\/_/\ __/\
/_ \ /\ /\ __/\ __/ /\ /____\/____\
__________/\ \_\/_ \ / _\/____\/\ / / /\__ /\__ /\
/\ /\ /____\ /\ \_\ /_/ /\__ / _\/__\/____\/____\/\ \
/ / / / /\__ /\/_ \ /\ /\ /____\/_/\__ /\__ /\__ / /__\
/__\/__\/____\/____\_\/_ \ / _\/ /\ __/____\/____\/____\/__\ /\
/\__ /\__ /\__ /\__ /\_\ /_/\ / _\/\ __/\ __/\ __/\ __/\ \ \
/____\/____\/____\/____\/____\ /____\/_/____\/____\/____\/____\/____\/__\
(Edit: Remembered my login)
- Success!
/\
/_ \
/\ \_\
/_ \ /\
/\ \_\/_ \
/_ \ /\ \_\
/\ \_\/_ \ /\
/\ \ /\ \_\/_ \
/ /__\/_ \ /\ \_\
/__\__ /\_\/_ \ /\
______________________/\ __/\/____\ \_\/_ \
/\ /\ /\ __/ /\ /____\/____\ __/\ /\ \_\
/ / / / / _\/\ / / /\__ /\__ /\/____\/_ \ /\
/__\/__\/_/____\/__\/____\/____\/_ \ __/\ \_\/ _\
/\__ /\ /\ __/\ __/\ __/\ __/\_\/ /_ \ /_/ /\
/____\/_ \ \ \/____\/ / _\/____\/____\ /\ \_\/\ / _\
/\ /\ /\_\/__\ /\ /_/\ __/\ \__ /\/_ \ / _\/_/ /\
/ _\ \ \ \ \ __/\ \ \/____\/____\ /\/____\_\/_/ /\ /_ \
/_/ /\/__\/__\/____\/__\ __/\ __/\/____\ __/\ / _\/ /\_\
/\ /____\ /\ /\ /\ /\/____\/____\ /\ \/____\/_/\ /____\
/ _\/\ __/\ \ \ \ \ \ \ \ \ __/\ __/\ \ \ /\__ / _\/\ __/\
/_/____\/____\/__\/__\/__\/__\/____\/____\/__\/____\/_/____\/____\
- Took a while, but I finally managed an 112 Sphinx tiling. That's about as far as I care to go by hand, but I feel it's most likely at this point that if it can be done for any n2 up to n = 11 (and of course 12 by doing two 22 and one 32 tilings), then it can be done for any n2. That does seem kindy of wishy-washy for a proof, but I'm not sure if there's any good systematic way of proving this one. I only know that increasing n generally only gives you more freedom of movement. For example, there's only ONE valid 22 or 32 tiling for most of these (some of the simpler ones have multiple 32 solutions, but none that I am aware of can do 22 in more than one way), but there are definitely more that one way to do the higher level ones, they just become harder to find due to the sheer number of variables. Incidentally, I can rule out a few more of them as possible pan-rep-tiles. The two-squares touching on a corner won't work, nor will the four-squares in a zig-zag corner-touching arrangement. Neither of the "fish" tilings will work. That doesn't mean that all corner-touching configurations are out, per se, as I'm pretty sure the two-triangle one I mentioned above works for all n2. Lurlock (talk) 15:44, 13 November 2015 (UTC)
- There's a paper in JCDCG this year that proves that it's always possible for the sphinx. Unfortunately it's not online. —David Eppstein (talk) 17:04, 13 November 2015 (UTC)
- The 1-2-2-3 trapezoid is an unusual variant of this. It appears that it's possible to tile this with n2 for any odd n, but not for evens. Proving it is pretty simple in this case. Start with 1 tile, and you can surround it in a layer of duplicates - this is the 32 seen in the article. You can then surround that with another layer and you'll get 52. Continue the process, and you can get 72, 92, 112, 132, ad infintum. As an example, here's a 92 tiling.
__________________
/9 \9 /9 \9 /9 \
/____\__/____\__/____\
/9 /7 /7 \7 /7 \9 \
/\ /\ /____\__/____\ /\
/9 \/7 \/5 /5 \5 \7 \/9 \
/___/___/\ /____\ /\ /\___\
/9 /7 /5 \/3 \3 \/5 \/7 \9 \
/\ /\ /___/____\ /\___\___\ /\
/9 \/7 \/5 /3 /1 \/3 \5 \7 \/9 \
/___/___/\ /\ /____\___\ /\ /\___\
/9 /7 /5 \/3 \/3 \3 /3 \/5 \/7 \9 \
/\ /\ /___/___/____\__/____\___\___\ /\
/9 \/7 \/5 \5 /5 \5 /5 \5 /5 \7 \/9 \
/___/___/____\__/____\__/____\__/____\ /\___\
/9 /7 \7 /7 \7 /7 \7 /7 \7 /7 \/7 \9 \
/\ /____\__/____\__/____\__/____\__/____\___\ /\
/9 \/9 \9 /9 \9 /9 \9 /9 \9 /9 \9 /9 \/9 \
/___/____\__/____\__/____\__/____\__/____\__/____\___\
- You can clearly see the 72, 52, 32, and 12 nested within it. In a lot of ways, it's similar to simply tiling around a square, only you take one off the top and squeeze it onto the bottom. Unlike a square, though, where you can just add onto 2 of the 4 sides to get the even square numbers, this robbing from the top to pay the bottom makes this formation impossible. (You pretty much always end up with half a tile on top and half on the bottom and no way to join them.) Lurlock (talk) 18:56, 13 November 2015 (UTC)
It turns out to be known since at least 2003 that many rep-tiles are rep-n^2 for all n. I added a reference to the article. —David Eppstein (talk) 16:53, 16 November 2015 (UTC)
- Nice. Did you mean the L-tetromino? You listed the L-pentomino, but so far I've been unable to make that work for any N. Incidentally, I have a generalized proof that it works for all non-uniform scalings of the L-tromino, as you can just multiply as much as you want in any dimension and they'll still be all the same shape. For example, simply adding a square in one dimension gets you the P-hexomino, P-nonomino, P-dodecomino, etc.:
┌─────┐ ┌─────────┐ ┌─────────────┐ ┌─────────────────┐ │ ┌──┤ │ ┌────┤ │ ┌──────┤ │ ┌────────┤ ├──┤ └──┬──┐ ├────┤ └────┬────┐ ├──────┤ └──────┬──────┐ ├────────┤ └────────┬────────┐ │ └──┬──┘ │ │ └────┬────┘ │ │ └──────┬──────┘ │ │ └────────┬────────┘ │ └─────┴─────┘ └─────────┴─────────┘ └─────────────┴─────────────┘ └─────────────────┴─────────────────┘
- This one may be unique in that respect due to its diagonal axis of symmetry. Lurlock (talk) 19:30, 16 November 2015 (UTC)
- Yes, I meant tetromino; now fixed. —David Eppstein (talk) 20:04, 16 November 2015 (UTC)
Do you mean rectangles?
editIn section Infinite tiling
Among regular polygons, only the triangle and square can be dissected into smaller equally sized copies of themselves.
Do you mean rectangles? Jumpow (talk) 14:11, 1 January 2016 (UTC)
- No, a rectangle isn't a regular polygon. Regular means equal sides and equal angles. Gnomon (talk) 09:25, 3 July 2020 (UTC)
In section Rep-tiles as fractals
...removing one or more copies...
In illustratin Geometrical dissection of an L-triomino I see no removed copies, so it must be
...removing zero or more copies... Jumpow (talk) 14:59, 1 January 2016 (UTC)
In section Examples
A rectangle (or parallelogram) is rep-n if its aspect ratio is √n:1.
What about rectangle √2:√3? It is rep-6 and has aspect ratio √3/√2... Jumpow (talk) 17:31, 1 January 2016 (UTC)
- It says "if", not "if and only if". —David Eppstein (talk) 17:54, 1 January 2016 (UTC)
- I understand it, but sentence is confusing. It must be extended or clarified, like
... its aspect ratio is √n:1 (for example) ...
or
... its aspect ratio may be √n:1 ... Jumpow (talk) 00:00, 2 January 2016 (UTC)
NB:I am programmer, and one of main principles of programming - do not force user to think (GUI must be clear) Jumpow (talk) 00:00, 2 January 2016 (UTC)
- Perhaps you didn't notice that the heading of the section this appears in is "Examples"? —David Eppstein (talk) 02:11, 2 January 2016 (UTC)
Pentagonal Rep-tiles
editI don't think that this statement is correct: "However, the sphinx remains the only known pentagonal rep-tile whose sub-copies are equal in size." Andrew Clarke's The Poly Pages has a page [3] that shows proper rep-tilings (components all the same size) for extended Sphinx polyiamonds. If no one objects, I will update the article accordingly. Sicherman (talk) 23:28, 5 April 2017 (UTC)
- Yes, that does seem a dubious statement. Please do fix it. —David Eppstein (talk) 23:33, 5 April 2017 (UTC)
- Done. I preserved the statement, adding the extended sphinxes. I know of no other pentagons that can be regularly rep-tiled. Sicherman (talk) 03:48, 7 April 2017 (UTC)
Regular polygon irrep-infinity tiling
editThe hexagon is not the only regular polygon that can be tiled with infinitely many similar copies. It works also for the regular pentagon.
Given regular Pentagon ABCDE, draw diagonals AC and AD. The large central triangle ACD, a golden triangle, can have a regular pentagon inscribed in it by fitting two pentagonal sides onto AC and AD, and the remaining vertex onto CD. The remaining pieces of the triangle are then golden triangles.
What about the flanking triangles ABC and ADE? In triangle ABC fit two sides of a regular pentagon to sides AB and BC, making B a vertex of the pentagon. We then make this pentagon the right size to match a third side with AC so that triangle ACD ends up divided into a regular pentagon and two golden triangles; the latter can be processed as described above for triangle ACD. Triangle ADE is treated similarly.
The resulting infinite pentagon tiling lacks the original polygonal symmetry, unlike its hexagonal counterpart, but it still works as a tiling.
I am hoping someone can put this in with pictures based on my description, as I am not good drawing pictures on a mobile device. Olthe3rd1 (talk) 02:13, 4 November 2018 (UTC)