Tiling challenge

[ParadigmShifter]

Hi I'm looking for some attribute based tiling that's more interesting than a checkerboard or a square pattern.

Here is something I made earlier, as they say on Blue Peter



I used a spreadsheet, just set widths and heights of cells to be the same and use spectrumish colours. (which I think are RGB values 128 for dark and 192 for bright?)

Tiled area is 18 wide by 24 high.

I'm sure people can do better than me of course

Bonus points for only using 4 colours (no neighbouring region shares same colour) in the tiled area which is possible cos the 4 colour theorem proved via computers in the 1970s.

EDIT: Nicked the 2nd tiling of course, from this site

https://besttile.com/patterns

[equinox]

This does not seem like a "challenge", rather "I want to produce certain patterns obeying certain rules, and I haven't worked out the rules yet"...

How big must the repeat be (4x4? 5x5? ...) and what are the real rules? We can't use the same colour within x squares?

Emergency rope: the old thread https://spectrumcomputing.co.uk/forums/viewtopic.php?t=11092 ...

[ParadigmShifter]

Any repeating pattern is fine, it just has to be attribs only.

Clearly the rules are already well known

https://en.wikipedia.org/wiki/Wallpaper_group

EDIT: Your link just takes me to an empty page though

[Stefan]

EDIT: Your link just takes me to an empty page though

It was trying to get to viewtopic.php?t=11092

[ParadigmShifter]

Ok I have read that.

I'm just looking for more interesting background patterns for my columns game engine



EDIT: I'd also like a random number generator with period 18x24 = 432 of course so I can do a nice dissolve effect but I'm not sure if I will be able to find one which hits every attribute square once... am looking though.

Of course I can just use a PRNG with a longer period and skip tiles I have already dissolved. I'd use that for a transition effect between more than 1 background pattern (when level increases etc.).

I'm also planning to do non tiled backgrounds that just animate (so fire effect and so on) like this

https://www.hanshq.net/fire.html

not sure how that will look with such a low resolution and limited colours though

Might do a Conway's game of life or a snake or a pong background that animates as well

[Stefan]

How about a live attribute version of a portion of the tiles? This portion could also move and sway around 'magnifying' different areas inducing sea sickness?

[ParadigmShifter]

Yeah I am planning to animate as per my edit to previous post. Simple colour cycling at first then get weirder as you go on.

I'll use a lookup table to map indices to colours and if they only use 4 colours I can compress each into 2 bits (worst case is 16 colours - well 15) which packs into 4 bits.

EDIT: As mentioned in post #1 it's always possible to colour regions with neighbours having a different colour with only 4 colours although I think limiting the tiling to just 4 colours might be a bit restrictive. https://en.wikipedia.org/wiki/Four_color_theorem EDIT: I'm not going to set that as an exercise as a proof like some teacher did in the early 20th century, they needed a large computer to check the proof in the 70s

Proof 5 colours are sufficient isn't too hard though... Numberphile did a video on it



Probably want at least 40 variations, there's only 5 tile types so it should not be hard to play for a long time so I'll need at least 40 backgrounds before they repeat I think.

EDIT: Current plan for animating is have a list of at most 64 attribs which change per frame and there will be a call which is done to generate the list of tiles to change at the end of each frame.

EDIT2: Non rectangular regions I am fine with as well by the way... just want some interesting backgrounds. L shapes are obviously good for tiling.

[ParadigmShifter]

Found the theorem that allows you to have a sequence that doesn't repeat and covers every number in the range, it's the Hull-Dobell theorem

If you have an LCG generator i.e.

x(n+1) = (a * x(n) + c) mod m

then this will have period m if and only if:

1. m and c are coprime
2. a-1 is divisible by all prime factors of m
3. a-1 is divisible by 4 if m is divisible by 4

So I can get a period 432 sequence by picking say

m = 432 = 2*2*2*2 * 3*3*3 so prime factors are 2 and 3, and it's divisible by 4
c = something coprime to 432 i.e. one of {415, 419, 421, 425, 427, 431, 433, 437, 439, 443} EDIT: There's more than that though, that's just what WolframAlpha gave as the first result, can keep hitting the "more" button here https://www.wolframalpha.com/input?i=in ... ime+to+432 only numbers c < 432 give different results obviously because of the modulus
a = 205 (204 is divisible by 2 and 3 i.e. a multiple of 6), and 204 is divisible by 4 since m is also divisible by 4, so a-1 has to be divisible by 12. So anything of the form a = 1 + 12n will do.

Some a and c are worse than others though (or better?) in that you see stripes, which might look ok for a swipe/dissolve effect.

I might still use 512 as the period and ignore anything >= 432 though since that is easy to do with bitwise AND.

BASIC code to test