Aperiodic Spectre tiling

aperiodic-hat.cpp

@@ -382,6 +382,405 @@ vector<rule_recursive> rules_recursive = {
   {6, 6, 31, 31, 14},
   };
 
+vector<rule_base> spectre_rules_base = { rule_base
+{0, 0, 5, 5, -1},
+{0, 10, 7, 7, -1},
+{0, 11, 5, 8, -1},
+{0, 12, 5, 7, -1},
+{0, 13, 5, 6, -1},
+{0, 1, 3, 8, -1},
+{0, 2, 3, 7, -1},
+{0, 3, 3, 6, -1},
+{0, 4, 3, 5, -1},
+{0, 5, 1, 0, -1},
+{0, 6, 1, 13, -1},
+{0, 7, 1, 12, -1},
+{0, 8, 1, 11, -1},
+{0, 9, 7, 8, -1},
+{1, 0, 0, 5, -1},
+{1, 10, 7, 9, -1},
+{1, 11, 0, 8, -1},
+{1, 12, 0, 7, -1},
+{1, 1, 2, 6, -1},
+{1, 13, 0, 6, -1},
+{1, 2, 2, 5, -1},
+{1, 3, 2, 4, -1},
+{1, 4, 2, 13, 20},
+{1, 4, 3, 13, 14},
+{1, 4, 6, 13, 30},
+{1, 4, 7, 13, 26},
+{1, 4, 8, 13, 15},
+{1, 5, 2, 12, 20},
+{1, 5, 3, 12, 14},
+{1, 5, 6, 12, 30},
+{1, 5, 7, 12, 26},
+{1, 5, 8, 12, 15},
+{1, 6, 2, 11, 20},
+{1, 6, 3, 11, 14},
+{1, 6, 6, 11, 30},
+{1, 6, 7, 11, 26},
+{1, 6, 8, 11, 15},
+{1, 7, 2, 0, 13},
+{1, 7, 4, 0, 29},
+{1, 7, 4, 4, 14},
+{1, 7, 6, 0, 25},
+{1, 7, 8, 0, 9},
+{1, 8, 2, 13, 13},
+{1, 8, 4, 13, 29},
+{1, 8, 4, 3, 14},
+{1, 8, 6, 13, 25},
+{1, 8, 8, 13, 9},
+{1, 9, 2, 12, 13},
+{1, 9, 4, 12, 29},
+{1, 9, 4, 2, 14},
+{1, 9, 6, 12, 25},
+{1, 9, 8, 12, 9},
+{2, 0, 1, 7, 31},
+{2, 0, 2, 3, 27},
+{2, 0, 6, 3, 16},
+{2, 0, 8, 3, 22},
+{2, 10, 2, 11, 22},
+{2, 10, 2, 1, 32},
+{2, 10, 4, 11, 16},
+{2, 10, 7, 11, 31},
+{2, 11, 1, 6, 27},
+{2, 11, 2, 10, 22},
+{2, 11, 4, 10, 16},
+{2, 11, 7, 10, 31},
+{2, 1, 2, 10, 14},
+{2, 12, 1, 5, 27},
+{2, 12, 1, 9, 31},
+{2, 1, 2, 2, 27},
+{2, 12, 3, 1, 22},
+{2, 12, 5, 1, 16},
+{2, 13, 1, 4, 27},
+{2, 13, 1, 8, 31},
+{2, 13, 3, 0, 22},
+{2, 13, 5, 0, 16},
+{2, 1, 6, 10, 26},
+{2, 1, 6, 2, 16},
+{2, 1, 8, 2, 22},
+{2, 2, 2, 1, 20},
+{2, 2, 3, 1, 14},
+{2, 2, 6, 1, 30},
+{2, 2, 7, 1, 26},
+{2, 2, 8, 1, 15},
+{2, 3, 2, 0, 20},
+{2, 3, 3, 0, 14},
+{2, 3, 6, 0, 30},
+{2, 3, 7, 0, 26},
+{2, 3, 8, 0, 15},
+{2, 4, 1, 3, -1},
+{2, 5, 1, 2, -1},
+{2, 6, 1, 1, -1},
+{2, 7, 3, 4, -1},
+{2, 8, 3, 3, -1},
+{2, 9, 3, 2, -1},
+{3, 0, 2, 13, 22},
+{3, 0, 2, 3, 32},
+{3, 0, 4, 13, 16},
+{3, 0, 7, 13, 31},
+{3, 10, 4, 5, -1},
+{3, 11, 1, 6, 32},
+{3, 11, 8, 6, -1},
+{3, 1, 2, 12, 22},
+{3, 12, 1, 5, 32},
+{3, 1, 2, 2, 32},
+{3, 12, 8, 5, -1},
+{3, 13, 1, 4, 32},
+{3, 13, 8, 4, -1},
+{3, 1, 4, 12, 16},
+{3, 1, 7, 12, 31},
+{3, 2, 2, 9, -1},
+{3, 3, 2, 8, -1},
+{3, 4, 2, 7, -1},
+{3, 5, 0, 4, -1},
+{3, 6, 0, 3, -1},
+{3, 7, 0, 2, -1},
+{3, 8, 0, 1, -1},
+{3, 9, 4, 6, -1},
+{4, 0, 1, 7, 11},
+{4, 0, 4, 13, 21},
+{4, 0, 7, 13, 10},
+{4, 0, 8, 3, 28},
+{4, 10, 2, 11, 28},
+{4, 10, 6, 11, 19},
+{4, 10, 7, 11, 11},
+{4, 10, 8, 11, 23},
+{4, 11, 2, 10, 28},
+{4, 11, 6, 10, 19},
+{4, 11, 7, 10, 11},
+{4, 11, 8, 10, 23},
+{4, 12, 1, 9, 11},
+{4, 12, 3, 1, 28},
+{4, 12, 4, 1, 23},
+{4, 12, 7, 1, 19},
+{4, 13, 1, 8, 11},
+{4, 13, 3, 0, 28},
+{4, 13, 4, 0, 23},
+{4, 13, 7, 0, 19},
+{4, 1, 4, 12, 21},
+{4, 1, 7, 10, 32},
+{4, 1, 7, 12, 10},
+{4, 1, 8, 2, 28},
+{4, 2, 1, 9, 32},
+{4, 2, 8, 9, -1},
+{4, 3, 1, 8, 32},
+{4, 3, 8, 8, -1},
+{4, 4, 1, 7, 32},
+{4, 4, 8, 7, -1},
+{4, 5, 3, 10, -1},
+{4, 6, 3, 9, -1},
+{4, 7, 5, 4, -1},
+{4, 8, 5, 3, -1},
+{4, 9, 5, 2, -1},
+{5, 0, 2, 13, 28},
+{5, 0, 6, 13, 19},
+{5, 0, 7, 13, 11},
+{5, 0, 8, 13, 23},
+{5, 10, 7, 5, -1},
+{5, 11, 6, 6, -1},
+{5, 1, 2, 12, 28},
+{5, 12, 6, 5, -1},
+{5, 13, 6, 4, -1},
+{5, 1, 6, 12, 19},
+{5, 1, 7, 12, 11},
+{5, 1, 8, 12, 23},
+{5, 2, 4, 9, -1},
+{5, 3, 4, 8, -1},
+{5, 4, 4, 7, -1},
+{5, 5, 0, 0, -1},
+{5, 6, 0, 13, -1},
+{5, 7, 0, 12, -1},
+{5, 8, 0, 11, -1},
+{5, 9, 7, 6, -1},
+{6, 0, 1, 7, 17},
+{6, 0, 2, 3, 12},
+{6, 0, 6, 3, 24},
+{6, 10, 2, 1, 18},
+{6, 10, 4, 11, 24},
+{6, 10, 6, 1, 29},
+{6, 10, 7, 11, 17},
+{6, 10, 8, 1, 13},
+{6, 11, 1, 6, 12},
+{6, 11, 4, 10, 24},
+{6, 11, 7, 10, 17},
+{6, 12, 1, 5, 12},
+{6, 12, 1, 9, 17},
+{6, 1, 2, 2, 12},
+{6, 12, 5, 1, 24},
+{6, 13, 1, 4, 12},
+{6, 13, 1, 8, 17},
+{6, 13, 5, 0, 24},
+{6, 1, 6, 10, 11},
+{6, 1, 6, 2, 24},
+{6, 2, 2, 1, 28},
+{6, 2, 6, 1, 19},
+{6, 2, 7, 1, 11},
+{6, 2, 8, 1, 23},
+{6, 3, 2, 0, 28},
+{6, 3, 6, 0, 19},
+{6, 3, 7, 0, 11},
+{6, 3, 8, 0, 23},
+{6, 4, 5, 13, -1},
+{6, 5, 5, 12, -1},
+{6, 6, 5, 11, -1},
+{6, 7, 7, 4, -1},
+{6, 8, 7, 3, -1},
+{6, 9, 7, 2, -1},
+{7, 0, 2, 3, 18},
+{7, 0, 4, 13, 24},
+{7, 0, 6, 3, 29},
+{7, 0, 7, 13, 17},
+{7, 0, 8, 3, 13},
+{7, 10, 2, 11, 13},
+{7, 10, 4, 11, 29},
+{7, 10, 4, 1, 14},
+{7, 10, 6, 11, 25},
+{7, 10, 8, 11, 9},
+{7, 11, 1, 6, 18},
+{7, 11, 2, 10, 13},
+{7, 11, 4, 10, 29},
+{7, 11, 6, 10, 25},
+{7, 11, 8, 10, 9},
+{7, 12, 1, 5, 18},
+{7, 1, 2, 2, 18},
+{7, 12, 3, 1, 13},
+{7, 12, 4, 1, 9},
+{7, 12, 5, 1, 29},
+{7, 12, 7, 1, 25},
+{7, 13, 1, 4, 18},
+{7, 13, 3, 0, 13},
+{7, 13, 4, 0, 9},
+{7, 13, 5, 0, 29},
+{7, 13, 7, 0, 25},
+{7, 1, 4, 12, 24},
+{7, 1, 6, 2, 29},
+{7, 1, 7, 12, 17},
+{7, 1, 8, 2, 13},
+{7, 2, 6, 9, -1},
+{7, 3, 6, 8, -1},
+{7, 4, 6, 7, -1},
+{7, 5, 5, 10, -1},
+{7, 6, 5, 9, -1},
+{7, 7, 0, 10, -1},
+{7, 8, 0, 9, -1},
+{7, 9, 1, 10, -1},
+{8, 0, 1, 7, 10},
+{8, 0, 2, 3, 33},
+{8, 0, 6, 3, 21},
+{8, 10, 4, 11, 21},
+{8, 10, 7, 11, 10},
+{8, 10, 8, 1, 28},
+{8, 11, 1, 6, 33},
+{8, 11, 4, 10, 21},
+{8, 11, 7, 10, 10},
+{8, 12, 1, 5, 33},
+{8, 12, 1, 9, 10},
+{8, 1, 2, 2, 33},
+{8, 12, 5, 1, 21},
+{8, 13, 1, 4, 33},
+{8, 13, 1, 8, 10},
+{8, 13, 5, 0, 21},
+{8, 1, 6, 10, 31},
+{8, 1, 6, 2, 21},
+{8, 1, 8, 10, 16},
+{8, 2, 2, 1, 22},
+{8, 2, 4, 1, 16},
+{8, 2, 7, 1, 31},
+{8, 3, 2, 0, 22},
+{8, 3, 4, 0, 16},
+{8, 3, 7, 0, 31},
+{8, 4, 3, 13, -1},
+{8, 5, 3, 12, -1},
+{8, 6, 3, 11, -1},
+{8, 7, 4, 4, -1},
+{8, 8, 4, 3, -1},
+{8, 9, 4, 2, -1},
+  };
+
+vector<rule_recursive> spectre_rules_recursive = { rule_recursive
+{0, 1, 27, -1, 20},
+{0, 1, 9, 18, 10},
+{0, 2, 32, -1, 14},
+{0, 3, 11, -1, 29},
+{0, 3, 9, 32, 10},
+{0, 4, 15, 31, 33},
+{0, 4, 17, -1, 25},
+{0, 4, 9, 27, 10},
+{0, 5, 15, 17, 33},
+{0, 5, 9, 12, 10},
+{0, 6, 15, 11, 33},
+{0, 6, 30, 32, 12},
+{0, 7, 15, 10, 33},
+{0, 7, 9, 33, 10},
+{1, 0, 10, 26, 9},
+{1, 0, 20, -1, 27},
+{1, 1, 21, 17, 23},
+{1, 1, 23, 25, 21},
+{1, 2, 13, -1, 31},
+{1, 2, 21, 11, 23},
+{1, 3, 21, 31, 23},
+{1, 4, 19, 26, 24},
+{1, 4, 22, 31, 22},
+{1, 5, 12, -1, 30},
+{1, 5, 19, 11, 24},
+{1, 5, 22, 17, 22},
+{1, 6, 16, 32, 28},
+{1, 6, 21, 10, 23},
+{1, 6, 22, 11, 22},
+{1, 6, 23, 19, 21},
+{1, 7, 19, 31, 24},
+{1, 7, 22, 10, 22},
+{2, 0, 14, -1, 32},
+{2, 1, 23, 29, 21},
+{2, 1, 31, -1, 13},
+{2, 3, 18, -1, 26},
+{2, 4, 23, 16, 21},
+{2, 5, 10, -1, 9},
+{2, 5, 23, 24, 21},
+{2, 6, 12, -1, 30},
+{2, 7, 23, 21, 21},
+{3, 0, 10, 14, 9},
+{3, 0, 29, -1, 11},
+{3, 1, 23, 13, 21},
+{3, 2, 26, -1, 18},
+{3, 4, 12, -1, 30},
+{3, 4, 19, 14, 24},
+{3, 4, 23, 22, 21},
+{3, 6, 23, 28, 21},
+{3, 6, 31, -1, 13},
+{3, 7, 10, -1, 9},
+{4, 0, 10, 20, 9},
+{4, 0, 25, -1, 17},
+{4, 0, 33, 13, 15},
+{4, 1, 22, 13, 22},
+{4, 1, 24, 18, 19},
+{4, 2, 21, 28, 23},
+{4, 3, 21, 22, 23},
+{4, 3, 24, 32, 19},
+{4, 3, 30, -1, 12},
+{4, 4, 16, 14, 28},
+{4, 4, 19, 20, 24},
+{4, 4, 22, 22, 22},
+{4, 4, 24, 27, 19},
+{4, 4, 28, 32, 16},
+{4, 5, 19, 28, 24},
+{4, 5, 24, 12, 19},
+{4, 5, 28, 18, 16},
+{4, 6, 22, 28, 22},
+{4, 7, 19, 22, 24},
+{4, 7, 24, 33, 19},
+{5, 0, 10, 30, 9},
+{5, 0, 33, 25, 15},
+{5, 1, 22, 25, 22},
+{5, 1, 24, 29, 19},
+{5, 1, 30, -1, 12},
+{5, 2, 21, 19, 23},
+{5, 2, 9, -1, 10},
+{5, 4, 16, 26, 28},
+{5, 4, 19, 30, 24},
+{5, 4, 24, 16, 19},
+{5, 5, 16, 11, 28},
+{5, 5, 19, 19, 24},
+{5, 5, 24, 24, 19},
+{5, 5, 28, 29, 16},
+{5, 6, 22, 19, 22},
+{5, 7, 16, 31, 28},
+{5, 7, 24, 21, 19},
+{6, 0, 12, 14, 30},
+{6, 0, 33, 29, 15},
+{6, 1, 21, 24, 23},
+{6, 1, 22, 29, 22},
+{6, 1, 23, 9, 21},
+{6, 1, 28, 14, 16},
+{6, 2, 30, -1, 12},
+{6, 3, 13, -1, 31},
+{6, 3, 21, 16, 23},
+{6, 4, 22, 16, 22},
+{6, 5, 22, 24, 22},
+{6, 6, 21, 21, 23},
+{6, 6, 23, 23, 21},
+{6, 7, 12, -1, 30},
+{6, 7, 19, 16, 24},
+{6, 7, 22, 21, 22},
+{7, 0, 10, 15, 9},
+{7, 0, 33, 9, 15},
+{7, 1, 22, 9, 22},
+{7, 1, 24, 13, 19},
+{7, 2, 21, 23, 23},
+{7, 3, 9, -1, 10},
+{7, 4, 19, 15, 24},
+{7, 4, 24, 22, 19},
+{7, 5, 19, 23, 24},
+{7, 5, 28, 13, 16},
+{7, 6, 22, 23, 22},
+{7, 6, 24, 28, 19},
+{7, 6, 30, -1, 12},
+{7, 7, 16, 16, 28},
+{7, 7, 28, 28, 16},
+  };
+
 EX ld hat_param = 1;
 EX ld hat_param_imag = 0;
 
@@ -390,6 +789,8 @@ struct hrmap_hat : hrmap {
   // always generate the same way
   std::mt19937 hatrng;
 
+  bool is_spectre;
+
   int hatrand(int i) {    
     return hatrng() % i;
     }
@@ -412,6 +813,9 @@ struct hrmap_hat : hrmap {
   memo_matrix adj_memo[2][2][14][14];
   vector<vector<memo_matrix>> long_transformations;
 
+  vector<rule_base> get_rules_base() { return is_spectre ? spectre_rules_base : rules_base; }
+  vector<rule_recursive> get_rules_recursive() { return is_spectre ? spectre_rules_recursive : rules_recursive; }
+
   void fill_transform_levels(int lev) {
     int clev = isize(long_transformations);
     while(clev <= lev) {
@@ -439,11 +843,11 @@ struct hrmap_hat : hrmap {
           else unknown++;
           };
 
-        if(clev == 1) for(auto& b: rules_base) {
+        if(clev == 1) for(auto& b: get_rules_base()) {
           products_equal(lt[0][b.id0+1], adj2(b.id0==0, fix(b.edge0), b.id1==0, fix(b.edge1)), lt[1][b.master_connection+1], lt[0][b.id1+1]);
           }
 
-        if(clev >= 2) for(auto& b: rules_recursive) {
+        if(clev >= 2) for(auto& b: get_rules_recursive()) {
           products_equal(lt[clev][b.id0+1], lt[clev-1][b.child+1], lt[clev][b.parent+1], lt[clev][b.id1+1]);
           }
 
@@ -464,6 +868,8 @@ struct hrmap_hat : hrmap {
 
   void init() {
 
+    relations = 34;
+
     transmatrix T = Id;
     auto& hc = hatcorners[0];
     hc.clear();
@@ -535,14 +941,22 @@ struct hrmap_hat : hrmap {
       for(auto& h: hc) h = gpushxto0(ctr) * h;
       };
 
-    if(hat_param_imag)  {
+    if(is_spectre) {
+      println(hlog, "eshort = ", eshort, " elong = ", elong);
+      swap(eshort, elong);
+      swap(eshorti, elongi);
+      hat(hatcorners[1]);
+      }
+    else if(hat_param_imag) {
       eshorti *= -1;
       elongi *= -1;
       hat(hatcorners[1]);
       }
     else hatcorners[1] = hc;
-    for(auto& h: hc) h = MirrorX * h;
+    if(!is_spectre) {
-    reverse(hatcorners[1].begin(), hatcorners[1].end());
+      for(auto& h: hc) h = MirrorX * h;
+      reverse(hatcorners[1].begin(), hatcorners[1].end());
+      }
 
     if(q == 6) {
       ld phi = (1 + sqrt(5)) / 2;
@@ -577,7 +991,7 @@ struct hrmap_hat : hrmap {
     long_transformations.clear();
     }
 
-  constexpr static int relations = 34;
+  int relations;
 
   // heptagons represent clusters
   // heptagon->distance is 0 for clusters of hats, d+1 for supercluster of heptagon d
@@ -622,7 +1036,7 @@ struct hrmap_hat : hrmap {
         }
       return h1;
       }
-    if(dir <= 7 - h->zebraval) {
+    if(dir <= (is_spectre ? 8 : 7) - h->zebraval) {
       // create child
       auto h1 = init_heptagon(relations); 
       h1->distance = h->distance - 1;
@@ -635,11 +1049,13 @@ struct hrmap_hat : hrmap {
     createStep(h, 0);
     int id = h->c.spin(0)-1;
     indenter ind(2);
-    for(auto& ru: rules_recursive) {
+    for(auto& ru: get_rules_recursive()) {
-      if(ru.id0 == id && ru.child == dir) {
+      int i0 = ru.id0, i1 = ru.id1;
+      if((h->distance & 1) && is_spectre) swap(i0, i1);
+      if(i0 == id && ru.child == dir) {
         heptagon *h1 = get_step(h->move(0), ru.parent);
         if(!h1) continue;
-        heptagon *h2 = get_step(h1, ru.id1+1);
+        heptagon *h2 = get_step(h1, i1+1);
         if(!h2) continue;
         h->c.connect(dir, h2, ru.rev_child, false);
         return h2;
@@ -662,7 +1078,7 @@ struct hrmap_hat : hrmap {
   void find_cell_connection(cell *c, int d) override {
     int id = hat_id(c);
     indenter ind(2);
-    for(auto& ru: rules_base) {
+    for(auto& ru: get_rules_base()) {
       if(ru.id0 == id && ru.edge0 == fix(d)) {
         heptagon *h = get_step(c->master, ru.master_connection);
         if(!h) continue;
@@ -728,12 +1144,13 @@ struct hrmap_hat : hrmap {
   void build_cells(heptagon *h) {
     if(h->c7) return;
     auto& ha = hats[h];
-    ha.resize(8 - h->zebraval);
+    ha.resize((is_spectre ? 9 : 8) - h->zebraval);
     for(auto& hac: ha) hac = newCell(isize(hatcorners[0]), h);
     h->c7 = ha[0];
     }
 
   hrmap_hat() {
+    is_spectre = geometry == gAperiodicSpectre;
     hatrng.seed(37);
     init();
     origin = init_heptagon(relations);

classes.cpp

@@ -752,6 +752,7 @@ enum eGeometry {
   gHalfBring,
   gAperiodicHat,
   gSierpinski3, gSierpinski4, gSixFlake, gMengerSponge, gSierpinskiTet,
+  gAperiodicSpectre,
   gGUARD};
 
 enum eGeometryClass { gcHyperbolic, gcEuclid, gcSphere, gcSol, gcNIH, gcSolN, gcNil, gcProduct, gcSL2 };
@@ -967,6 +968,7 @@ EX vector<geometryinfo> ginf = {
   {"6-flake","none",    "6-flake fractal",                            "S6",       6, 3, qFRACTAL,  giEuclid2, {{10, 10}}, eVariation::pure},
   {"{4,3,4}","none",    "Menger sponge",                              "S8",       6, 4, qFRACTAL,  giEuclid3, {{10, 10}}, eVariation::pure},
   {"rh{4,3,4}","none",  "Sierpiński tetrahedron",                     "S4b",     12, 3, qFRACTAL,  giEuclid3, {{10, 10}}, eVariation::pure},
+  {"spectre","none",    "aperiodic spectre",                          "spectre", 14, 3, qAPERIODIC | qHAT,  giEuclid2, {{7, 7}}, eVariation::pure},
   };
   // bits: 9, 10, 15, 16, (reserved for later) 17, 18
 

config.cpp

@@ -834,14 +834,18 @@ EX void initConfig() {
   param_f(hat::hat_param, "hat_param", "hat_param", 1)
   -> editable(0, 2, 0.1, "hat parameter",
     "Apeirodic hat tiling based on: https://arxiv.org/pdf/2303.10798.pdf\n\n"
-    "This controls the parameter discussed in Section 6. Parameter p is Tile(p, (2-p)√3), scaled so that the area is the same for every p.", 'v'
+    "This controls the parameter discussed in Section 6. Parameter p is Tile(p, (2-p)√3), scaled so that the area is the same for every p."
+    "Aperiodic spectre tiling based on: https://arxiv.org/abs/2305.17743\n\n"
+    "Set the parameter to 'spectre' value to make all tiles have the same shape."
+    ,
+    'v'
     )
   -> set_extra([] {
       dialog::addSelItem(XLAT("chevron (periodic)"), "0", 'C');
       dialog::add_action([] { dialog::ne.s = "0"; dialog::apply_edit(); });
       dialog::addSelItem(XLAT("hat"), "1", 'H');
       dialog::add_action([] { dialog::ne.s = "1"; dialog::apply_edit(); });
-      dialog::addSelItem(XLAT("all equal (periodic)"), "3-√3", 'T');
+      dialog::addSelItem(XLAT("spectre"), "3-√3", 'T');
       dialog::add_action([] { dialog::ne.s = "3 - sqrt(3)"; dialog::apply_edit(); });
       dialog::addSelItem(XLAT("turtle"), "1.5", 'T');
       dialog::add_action([] { dialog::ne.s = "1.5"; dialog::apply_edit(); });
@@ -852,8 +856,7 @@ EX void initConfig() {
 
   param_f(hat::hat_param_imag, "hat_param_imag", "hat_param_imag", 0)
   -> editable(0, 2, 0.1, "hat parameter (imaginary)",
-    "Apeirodic hat tiling based on: https://arxiv.org/pdf/2303.10798.pdf\n\n"
+    "Imaginary part of the hat parameter. This corresponds to the usual interpretation of complex numbers in Euclidean planar geometry: rather than shortened or lengthened, the edges are moved in the other dimension.", 'v'
-    "This controls the parameter discussed in Section 6. Parameter p is Tile(p, (2-p)√3), scaled so that the area is the same for every p.", 'v'
     )
   -> set_reaction(hat::reshape);