Image discovered while doodling

[yomikoma]

Comments

[rpipuzzleguy.livejournal.com]

My barcode scanner couldn't do anything with this!

[rikchik.livejournal.com]

:D

[rymrytr.livejournal.com]

In part, and from a face-on view, I see a

Race Car Frame

white face - black eyes and mouth

Black face, black eyebrows, white eyes, nose and mouth

A 2 and a 5 or 25

A Cross

A stylized black cat with a straight, black tail

And a Daisy with 4 petals, one of which points toward the sky...

[rikchik.livejournal.com]

I can't argue with any of that.

[projectyl.livejournal.com]

The image is exactly half-black and half-white. Every column has two squares of one color and six of the other. Two of the rows are monochromatic, and the rest are exactly half-black and half-white.

None of which, I'm guessing, is what you're going for with this, but maybe it'll get someone else's thought processes going.

[projectyl.livejournal.com]

Huh... XORing it with a chessboard makes a pretty "OU" in the top half, but then the lower half is something I can't really parse.

[dougo.livejournal.com]

Hm, the top left quadrant is 101A (rows from top to bottom, and also columns from left to right). The bottom left quadrant is that ORed with 0505; the top right quadrant is that ORed with A0A0 (which is 0505 rotated 180°); and the bottom right quadrant is all three ORed.

[rikchik.livejournal.com]

I feel like you and cnoocy almost have it.

[rikchik.livejournal.com]

Those are all necessary outcomes of the process used to create this image.

[dougo.livejournal.com]

1B
00
1B
AA
1B
55
1B
FF

[rikchik.livejournal.com]

How do the columns come out?

[cnoocy]

11
05
11
AF
BB
05
BB
AF

Similar sort of symmetry, but it doesn't tell me how this came about.

[rikchik.livejournal.com]

Does the combination of rows and columns suggest any groupings?

[cnoocy]

Huh. Now I see the quadrants of those two lists, but I still don't see the root of the issue.

[cnoocy]

Ah! Writing it out into quadrants helped. It's successive applications of a 4-point square in two directions. Was the original source of the algorithm multi-dimensional?

[rikchik.livejournal.com]

In a sense. I'm not sure how guessable my original logic is.

[dougo.livejournal.com]

But the upper left quadrant doesn't have a 4-point square (though it does have 4 points). What is the significance of that pattern?

Aha, is it all 16 possible 2x2 binary grids? I'm not sure why they're in that order though. (I want to draw the order that seems natural to me, but I can't get Gimp to do pixel editing... What drawing program did you use for that image? Or did you generate it?)

[rikchik.livejournal.com]

Ding! It is exactly all 2x2 binary grids. I was pondering 2-variable logical truth tables and decided to draw them out. I wanted to make sure I got them all, so I arbitrarily chose to put A&!B in the even columns, A&B in the last two columns, B&!A in the even rows, and A NOR B in the last two rows.

Once I had drawn it out in pencil I created an 8x8 image in MSPaint and used netpbm tools to blow it up by a factor of 12.

I'd be interested in seeing your preferred order. I sketched it out switching B&!A for NOR, and it's similarly structural.

[dougo.livejournal.com]

Hm, my first idea was this:

0 0  0 1  1 0  1 1
0 0  0 0  0 0  0 0

0 0  0 1  1 0  1 1
0 1  0 1  0 1  0 1

0 0  0 1  1 0  1 1
1 0  1 0  1 0  1 0

0 0  0 1  1 0  1 1
1 1  1 1  1 1  1 1

That's basically 0 to 15 in order, going down in columns from left to right. (It has the same 3x3 white square in the upper left as your pattern, is why I thought of it.) But it would make more sense to write the squares in the same order as you read the bits in each square:

0 0  0 0  0 0  0 0
0 0  0 1  1 0  1 1

0 1  0 1  0 1  0 1
0 0  0 1  1 0  1 1

1 0  1 0  1 0  1 0
0 0  0 1  1 0  1 1

1 1  1 1  1 1  1 1
0 0  0 1  1 0  1 1

Or you could use a sort of Gray code order, where each 2x2 grid differs from its neighbors (including wrapping around) by one pixel only:

0 0  0 0  0 0  0 0
0 0  0 1  1 1  1 0

0 1  0 1  0 1  0 1
0 0  0 1  1 1  1 0

1 1  1 1  1 1  1 1
0 0  0 1  1 1  1 0

1 0  1 0  1 0  1 0
0 0  0 1  1 1  1 0

Might make good wallpaper:

[dougo.livejournal.com]

Actually I think this is a better Gray code (do you see the pattern?):

0 0  0 1  0 1  0 0
0 0  0 0  1 0  1 0

0 0  0 1  0 1  0 0
0 1  0 1  1 1  1 1

1 0  1 1  1 1  1 0
0 1  0 1  1 1  1 1

1 0  1 1  1 1  1 0
0 0  0 0  1 0  1 0

Nicely fractal wallpaper too:

[rikchik.livejournal.com]

Yes - center 2 and last 2 columns/rows. Interesting that it so closely resembles Houndstooth. (I remember being amazed at realizing that the white and black areas on a houndstooth panel of a ball were the same shape. I must have been 3 or 4.)

[dougo.livejournal.com]

I don't understand your first sentence... The way I generated it was to start with 0 in the upper left, and then use this grid:

N E
W S

so that moving in each direction turned on that bit. The rest are constrained by the Gray code.

Yes, it is interesting how close it is to Houndstooth! I never realized that was also black and white squares alternating with half-and-half. The half-and-half squares are different, though, or maybe just higher resolution?

[rikchik.livejournal.com]

I mean the upper right square is true in the 2nd and 3rd columns, and the lower left square is true in the 3rd and 4th columns (and mutatis mutandis for the rows.) My rule is isomorphic to your rule, so it's valid. :P

Houndstooth is basically a plaid alternating a thick black and a thick white section, but the fact that it is a twill rather than a plain weave causes the diagonal stripes as opposed to the simple dithering you'd see in something like a diner tablecloth. (Thanks, Wikipedia!)

[rikchik.livejournal.com]

A wonderful gallery! I kind of want to see the other ones tesselated now. My original:

[cnoocy]

I had some different patterns, which I'm not up to painting while at work.

This is the most straightforward, but they are all similar:

0 0  1 0  0 1  1 1
0 0  0 0  0 0  0 0

0 0  1 0  0 1  1 1
1 0  1 0  1 0  1 0

0 0  1 0  0 1  1 1
0 1  0 1  0 1  0 1

0 0  1 0  0 1  1 1
1 1  1 1  1 1  1 1

[rikchik.livejournal.com]

[dougo.livejournal.com]

I think you missed the 7th row?

[rikchik.livejournal.com]

Nice catch. Fixed.

[tahnan.livejournal.com]

It's a Rorschach Test for puzzlers!

[rikchik.livejournal.com]

Is that what you see? Hmm. Tell me about your mother.

[cnoocy]

It almost looks like music to me. Some sort of rhythm?

[rikchik.livejournal.com]

There is indeed some repetition going on here.

[dougo.livejournal.com]

Iambic hexameter?

[rikchik.livejournal.com]

Didn't even notice that! If I had, I would have dropped the word "here" which doesn't fit.

Ha!

[okrablossom (from livejournal.com)]

You didn't need me to post at all and say that you are clearly missing me :P

Re: Ha!

[rikchik.livejournal.com]

Tormenting my puzzler friends is how I deal with your absence :)

[obonic.livejournal.com]

Where did you discover it? :3

[rikchik.livejournal.com]

I say "discover" since I thought of the algorithm and then drew it out to see what it looked like.