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hat-test: option to generate four-coloured hat tilings.

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This commit is purely frivolous even by Puzzles standards, in that
it's totally unrelated to any actual puzzle. But I know at least one
person has already used the 'hat-test' tool in this code base to
generate a patch of hat tiling for decorative purposes, so it's useful
in its own right. Also, now that I've worked out _how_ to do this,
it's a shame not to keep the code.

Of course, any tiling of the plane _can_ be four-coloured, just by the
Four Colour Theorem. But for a tiling with structure it's nicer if the
colouring is related to the structure in some way. And there's a
reasonably nice explicit construction that does just that: the paper
introducing the tiling observes that if each reflected hat is fused
with a particular one of its neighbours, the resulting tiling is
graph-theoretically equivalent to a tiling of the plane by hexagons.
And _that_ tiling can be three-coloured, in a unique way up to colour
choices. This induces a four-colouring of the hat tiling in which the
reflected hats have a colour to themselves, and everything else is
coloured the same as its corresponding hexagon in the three-colouring.

Actually implementing this turns out not to be too difficult using my
coordinate system. I hand-wrote tables giving a patch of colouring for
each of the four kitemaps; then, whenever two kitemaps meet, you can
determine how the colours map to each other by looking at the
overlapping tiles. So I can have hat-test work out the colour of each
tile as it goes.

So hat-test now supports a '--fourcolour' option to apply this
colouring to the output tiling.
This commit is contained in:
Simon Tatham
2023-03-31 18:35:43 +01:00
parent 52d801a06a
commit 1af1204b9c
1 changed files with 231 additions and 7 deletions
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238
hat.c
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@ -1204,6 +1204,195 @@ static bool hc_expect(const char *file, int line, HatCoords *hc,
fails++; \
} while (0)
/*
* For four-colouring the tiling: these tables give a colouring of
* each kitemap, with colour 3 assigned to the reflected tiles in the
* middle of the H, and 0,1,2 chosen arbitrarily.
*/
static const int fourcolours_H[] = {
/* 0 */ 0, 2, 1, 3,
/* 1 */ 1, 0, 2, 3,
/* 2 */ 0, 2, 1, 3,
/* 3 */ 1, -1, -1, -1,
/* 4 */ 1, 2, -1, -1,
/* 5 */ 1, 2, -1, -1,
/* 6 */ 2, 1, -1, -1,
/* 7 */ 0, 1, -1, -1,
/* 8 */ 2, 0, -1, -1,
/* 9 */ 2, 0, -1, -1,
/* 10 */ 0, 1, -1, -1,
/* 11 */ 0, 1, -1, -1,
/* 12 */ 2, 0, -1, -1,
};
static const int fourcolours_T[] = {
/* 0 */ 1, 2, 0, 3,
/* 1 */ 2, 1, -1, -1,
/* 2 */ 0, 1, -1, -1,
/* 3 */ 0, 2, -1, -1,
/* 4 */ 2, 0, -1, -1,
/* 5 */ 0, 1, -1, -1,
/* 6 */ 1, 2, -1, -1,
};
static const int fourcolours_P[] = {
/* 0 */ 2, 1, 0, 3,
/* 1 */ 1, 2, 0, 3,
/* 2 */ 2, 1, -1, -1,
/* 3 */ 0, 2, -1, -1,
/* 4 */ 0, 1, -1, -1,
/* 5 */ 1, 2, -1, -1,
/* 6 */ 2, 0, -1, -1,
/* 7 */ 0, 1, -1, -1,
/* 8 */ 1, 0, -1, -1,
/* 9 */ 2, 1, -1, -1,
/* 10 */ 0, 2, -1, -1,
};
static const int fourcolours_F[] = {
/* 0 */ 2, 0, 1, 3,
/* 1 */ 0, 2, 1, 3,
/* 2 */ 1, 2, -1, -1,
/* 3 */ 1, 0, -1, -1,
/* 4 */ 0, 2, -1, -1,
/* 5 */ 2, 1, -1, -1,
/* 6 */ 2, 0, -1, -1,
/* 7 */ 0, 1, -1, -1,
/* 8 */ 0, 1, -1, -1,
/* 9 */ 2, 0, -1, -1,
/* 10 */ 1, 2, -1, -1,
};
static const int *const fourcolours[] = {
fourcolours_H, fourcolours_T, fourcolours_P, fourcolours_F,
};
/*
* Structure that describes how the colours in the above maps are
* translated to output colours. This will vary with each kitemap our
* coordinates pass through, in order to maintain consistency.
*/
typedef struct FourColourMap {
unsigned char map[4];
} FourColourMap;
/*
* Make an initial FourColourMap by choosing the initial permutation
* of the three 'normal' hat colours randomly.
*/
static inline FourColourMap fourcolourmap_initial(random_state *rs)
{
FourColourMap f;
unsigned i;
/* Start with the identity mapping */
for (i = 0; i < 4; i++)
f.map[i] = i;
/* Randomly permute colours 0,1,2, leaving 3 as the distinguished
* colour for reflected hats */
shuffle(f.map, 3, sizeof(f.map[0]), rs);
return f;
}
static inline FourColourMap fourcolourmap_update(
FourColourMap prevm, HatCoords *prevc, HatCoords *currc, KiteStep step,
HatCoordContext *ctx)
{
size_t i, m1, m2;
const int *f1, *f2;
unsigned sum;
int missing;
FourColourMap newm;
HatCoords *prev2c;
/*
* If prevc and currc are in the same kitemap anyway, that's the
* easy case: the colour map for the new kitemap is the same as
* for the old one, because they're the same kitemap.
*/
ensure_coords(ctx, prevc, currc->nc);
ensure_coords(ctx, currc, prevc->nc);
for (i = 3; i < prevc->nc; i++)
if (currc->c[i].index != prevc->c[i].index)
goto mismatch;
return prevm;
mismatch:
/*
* The step_coords algorithm guarantees that the _new_ coordinate
* currc is expected to be in a kitemap containing both this kite
* and the previous one (because it first transformed the previous
* coordinate until it _could_ take a step within the same
* kitemap, and then did).
*
* So if we reverse the last step we took, we should get a second
* HatCoords describing the same kite as prevc but showing its
* position in the _new_ kitemap. This lets us figure out a pair
* of corresponding metatile indices within the old and new
* kitemaps (by looking at which metatile prevc and prev2c claim
* to be in).
*
* That metatile will also always be a P or an F (because all
* metatiles overlapping the next kitemap are of those types),
* which means it will have two hats in it. And those hats will be
* adjacent, so differently coloured. Hence, we have enough
* information to decide how two of the new kitemap's three normal
* colours map to the colours we were using in the old kitemap -
* and then the third is determined by process of elimination.
*/
prev2c = step_coords(
ctx, currc, (step == KS_LEFT ? KS_RIGHT :
step == KS_RIGHT ? KS_LEFT :
step == KS_F_LEFT ? KS_F_RIGHT : KS_F_LEFT));
/* Metatile indices within the old and new kitemaps */
m1 = prevc->c[2].index;
m2 = prev2c->c[2].index;
/* The colourings of those metatiles' hats in our fixed fourcolours[] */
f1 = fourcolours[prevc->c[3].type] + 4*m1;
f2 = fourcolours[prev2c->c[3].type] + 4*m2;
/*
* Start making our new output map, filling in all three normal
* colours to 255 = "don't know yet".
*/
newm.map[3] = 3;
newm.map[0] = newm.map[1] = newm.map[2] = 255;
/*
* Iterate over the tile colourings in fourcolours[] for these
* metatiles, matching up our mappings.
*/
for (i = 0; i < 4; i++) {
/* They should be the same metatile, so have same number of hats! */
assert((f1[i] == -1) == (f2[i] == -1));
if (f1[i] != 255)
newm.map[f2[i]] = prevm.map[f1[i]];
}
/*
* We expect to have filled in exactly two of the three normal
* colours. Find the missing index, and fill in its colour by
* arithmetic (using the fact that the three colours add up to 3).
*/
sum = 0;
missing = -1;
for (i = 0; i < 3; i++) {
if (newm.map[i] == 255) {
assert(missing == -1); /* shouldn't have two missing colours */
missing = i;
} else {
sum += newm.map[i];
}
}
assert(missing != -1);
assert(0 < sum && sum <= 3);
newm.map[missing] = 3 - sum;
return newm;
}
static bool unit_tests(void)
{
int fails = 0;
@ -1295,10 +1484,14 @@ static inline void psbbox_add(psbbox *bbox, pspoint p)
}
typedef enum OutFmt { OF_POSTSCRIPT, OF_PYTHON } OutFmt;
typedef enum ColourMode { CM_SEMANTIC, CM_FOURCOLOUR } ColourMode;
typedef struct drawctx {
OutFmt outfmt;
ColourMode colourmode;
psbbox *bbox;
KiteEnum *kiteenum;
FourColourMap fourcolourmap[KE_NKEEP];
} drawctx;
static void bbox_add_hat(void *vctx, Kite kite0, HatCoords *hc, int *coords)
@ -1380,13 +1573,36 @@ static void draw_hat(void *vctx, Kite kite0, HatCoords *hc, int *coords)
printf(" %f %f %s", p.x, p.y, i ? "lineto" : "moveto");
}
printf(" closepath gsave");
if (hc->c[2].type == TT_H) {
colour = (hc->c[1].index == 3 ? "0 0.5 0.8 setrgbcolor" :
"0.6 0.8 1 setrgbcolor");
} else if (hc->c[2].type == TT_F) {
colour = "0.7 setgray";
} else {
colour = "1 setgray";
switch (ctx->colourmode) {
case CM_SEMANTIC:
if (hc->c[2].type == TT_H) {
colour = (hc->c[1].index == 3 ? "0 0.5 0.8 setrgbcolor" :
"0.6 0.8 1 setrgbcolor");
} else if (hc->c[2].type == TT_F) {
colour = "0.7 setgray";
} else {
colour = "1 setgray";
}
break;
default /* case CM_FOURCOLOUR */: {
/*
* Determine the colour of this tile by translating the
* fixed colour from fourcolours[] through our current
* FourColourMap.
*/
FourColourMap f = ctx->fourcolourmap[ctx->kiteenum->curr_index];
const int *m = fourcolours[hc->c[3].type];
static const char *const colours[] = {
"1 0.7 0.7 setrgbcolor",
"1 1 0.7 setrgbcolor",
"0.7 1 0.7 setrgbcolor",
"0.6 0.6 1 setrgbcolor",
};
colour = colours[f.map[m[hc->c[2].index * 4 + hc->c[1].index]]];
break;
}
}
printf(" %s fill grestore", colour);
printf(" stroke\n");
@ -1431,6 +1647,8 @@ int main(int argc, char **argv)
drawctx dctx[1];
dctx->outfmt = OF_POSTSCRIPT;
dctx->colourmode = CM_SEMANTIC;
dctx->kiteenum = s;
while (--argc > 0) {
const char *arg = *++argv;
@ -1444,6 +1662,8 @@ int main(int argc, char **argv)
return unit_tests() ? 0 : 1;
} else if (!strcmp(arg, "--python")) {
dctx->outfmt = OF_PYTHON;
} else if (!strcmp(arg, "--fourcolour")) {
dctx->colourmode = CM_FOURCOLOUR;
} else if (!strncmp(arg, "--seed=", 7)) {
random_seed = arg+7;
} else if (arg[0] == '-') {
@ -1493,12 +1713,16 @@ int main(int argc, char **argv)
first_kite(s, w, h);
coords[s->curr_index] = initial_coords(ctx);
dctx->fourcolourmap[s->curr_index] = fourcolourmap_initial(rs);
maybe_report_hat(w, h, *s->curr, coords[s->curr_index],
draw_hat, dctx);
while (next_kite(s)) {
hc_free(coords[s->curr_index]);
coords[s->curr_index] = step_coords(
ctx, coords[s->last_index], s->last_step);
dctx->fourcolourmap[s->curr_index] = fourcolourmap_update(
dctx->fourcolourmap[s->last_index], coords[s->last_index],
coords[s->curr_index], s->last_step, ctx);
maybe_report_hat(w, h, *s->curr, coords[s->curr_index],
draw_hat, dctx);
}