ref: 1af1204b9c33c9c03b2e0fe66c2f07d9729cbc72
parent: 52d801a06a804244292f4a872eeaf5e84a9f70b1
author: Simon Tatham <[email protected]>
date: Fri Mar 31 14:35:43 EDT 2023
hat-test: option to generate four-coloured hat tilings. 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.
--- a/hat.c
+++ b/hat.c
@@ -1204,6 +1204,195 @@
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 @@
}
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 @@
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 @@
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 @@
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] == '-') {@@ -1493,6 +1713,7 @@
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)) {@@ -1499,6 +1720,9 @@
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);
}