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So I need a new knitting pattern, but I figured this might be a nice puzzle.

I'm looking for a pixelated 2D-shape that can be used in a room-filling pattern (rotation and flipping allowed). The pattern should fulfil the following requirements:

  • Not all pieces should have the same orientation.
  • Horizontally, the pattern must repeat itself every 8th pixel (colouring excluded).
  • No adjacent pieces can have the same colour.
  • Each piece can not have more than one colour.
  • Each horizontal line must consist of two colours.

This is an open-ended puzzle, and the more intricate pattern the better.

( Here's an example of such a pattern that repeats itself every 6th pixel:

enter image description here

© Sandra Jäger, Okt. 2010. Source)

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  • $\begingroup$ What’s ”room filling”? Haven’t heard that term before. Something related to ”plane tiling” or ”space filling”, probably? $\endgroup$ – Bass Dec 30 '17 at 21:08
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The jigsaw pattern with period 6 can be made into one with period 8 by using width 2:

Tile has six rows of widths 2, 4, 2, 2, 4, 2, centred

(Using width 1 you would end up having to split 9 pixels between four tiles meeting at a corner. It can be done, but obviously the tiles lose their symmetry and it's not particularly aesthetically pleasing. Period 10 allows quite an interesting tile but is out of spec).

The jigsaw pieces can be divided into T tetrominos:

4 by 4 squares contain four tetrominos and have rotational symmetry of order 4; the squares are then rotated and coloured appropriately to tile together][1

One way of viewing the jigsaw piece is a simple chequer pattern where the white pieces push out left and right and the red pieces push out up and down. If each piece instead pushes out a pentomino in one direction they can make a key-like tile. There are (at least) two different ways to do this, which have different symmetry groups:

All white "keys" point right, and all red "keys" point up][1

or

An eight by eight square has four "keys" and rotational symmetry][1

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  • $\begingroup$ The last pattern is exactly the kind of "complex" pattern I was looking for, so this gets to be the accepted answer. Although it wasn't an easy choice. $\endgroup$ – blupp Jan 7 '18 at 12:33
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Here are a few patterns I like.

The first one is essentially the same as the one you linked to. It has the same symmetry, and you can colour each row of horizontal shapes a different colour if you like.

enter image description here

By splitting each shape of the above pattern into four smaller shapes, you get the pattern below. I think this is the only pattern possible where the shape occurs in all 8 orientations (i.e. 4 rotations + mirror images) in the same horizontal strip.

enter image description here

Next I constructed a nice spiral pattern, where the centre of each spiral is based on the pattern above. The pieces occur in only two orientations (turned 180 degrees). Note that you can give each row of spirals a different pair of colours.

enter image description here

You can take the above pattern with spirals, and mirror every other row. The pieces then occur in 4 orientations.

enter image description here


Edited to add one more pattern.

This pattern is based on an 8x8 square with 90 degree symmetry. Alternate strips are mirrored so that all 8 orientations occur, though it is not necessary. Again, horizontal strips can be given different pairs of colours.

enter image description here


Edit2, yet another pattern.

This pattern uses the same 8x8 square as the previous pattern, but arranged so that the parts join to make a shape that is twice as wide and twice as high. It does however violate the requirement that the shape should occur in different orientations.

enter image description here

By mirroring every other horizontal strip, you get an asymmetric shape that occurs in two orientations.

enter image description here

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    $\begingroup$ The second one looks like it could vanish any minute, giving the most insane high score ever. $\endgroup$ – Bass Jan 1 '18 at 10:56
  • $\begingroup$ Wow, I really like your edits. Much Escher. $\endgroup$ – blupp Jan 7 '18 at 12:19
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Well, I have absolutely no idea what you wanted, but you got me playing around with tilings, so here's an interesting one:

enter image description here

I'd like to name this The Holy Pattern, because it's literally just a bunch of holes connected to each other, leaving holes in between. Out of the 17 possible wallpapers, this is the pattern "p4".

You can make this pattern with various different hole sizes and different sized gaps in between. That's not the clever bit.

Each of those patterns nicely tiles a plane. That's not the clever bit either.

The clever bit is how you can construct the pattern. Holes in various colours, then placed at the proper slots? Or just pick the smallest rectangle that repeats? Well, you could do either. However, it's possible to construct the pattern from a single piece containing a single hole. I'll use a different sized pattern with different eye-assaulting colours to demonstrate.

Start with a single hole not surrounded by a single colour:

enter image description here

Mark the centres of the adjacent holes. Using one of those points as a pivot, make four rotated copies of the initial piece. Notice how this creates additional pivot points.

enter image description here

Then you can either keep pivoting, OR you can notice that the resulting piece is rotationally symmetric (well, we constructed it by rotating stuff around the centre, so, yeah..), and start to pivot using that piece. This is so much nicer, since we can just copy and paste:

enter image description here

And so on, ad infinitum.

It's probably a good idea to add some thickness to the walls, and you can even add colour to the some of the holes by choosing a different shaped starting piece:

enter image description here

Repeating the whole process with this piece will result in a pattern that's hopefully at least a little less.. jarring than the ones above.

enter image description here

If this wasn't at all what you were looking for, my apologies. At least I had fun :-)

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  • $\begingroup$ Keep in mind that there is a rule of "Each horizontal line must consist of two colours." I guess it's for knitting purpose. $\endgroup$ – athin Dec 31 '17 at 2:22
  • $\begingroup$ Yeah, I know next to nothing about knitting. The starting piece does satisfy that condition, but the resulting pattern won’t, if you use more than two colours. I guess you could still make a pillow cover or something like that where you can connect each colour by teleporting (pretty sure that’s the correct term) the yarn on the hidden side. Being able to prove the Pythagorean theorem should, in my opinion, more than offset that hardship :-) $\endgroup$ – Bass Dec 31 '17 at 10:13
  • $\begingroup$ Unfortunately it repeats horizontally every tenth pixel, instead of every eighth pixel as required. $\endgroup$ – Jaap Scherphuis Dec 31 '17 at 20:17
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    $\begingroup$ well, yeah, there’s that. The generating piece fits within 8 horizontal pixels though, which might be enough for OP’s purposes. Or it might not. There’s really no way of knowing until OP returns. I completely agree that as a puzzle answer, my post is utterly useless. As a knitting pattern, quite likely, ditto. But as an interesting pattern that I would never have discovered if not for this question, I’m kind of obligated to share, regardless. $\endgroup$ – Bass Dec 31 '17 at 21:15
  • $\begingroup$ Yep, those patterns doesn't fulfill the requirements, so I can't use them for my current knitting project. But I really like them, and your explanation, so thank you for posting. :) $\endgroup$ – blupp Jan 7 '18 at 12:10

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