diff --git a/aperiodic-hat.cpp b/aperiodic-hat.cpp
index d1415a9d..3b3b4b18 100644
--- a/aperiodic-hat.cpp
+++ b/aperiodic-hat.cpp
@@ -382,6 +382,8 @@ vector<rule_recursive> rules_recursive = {
   {6, 6, 31, 31, 14},
   };
 
+EX ld hat_param = 1;
+
 struct hrmap_hat : hrmap {
 
   // always generate the same way
@@ -402,17 +404,18 @@ struct hrmap_hat : hrmap {
   void init() {
 
     transmatrix T = Id;
-    hatcorners[0].clear();
+    auto& hc = hatcorners[0];
+    hc.clear();
     hatcorners[1].clear();
 
     auto move = [&] (ld angle, ld dist) {
-      hatcorners[0].push_back(T * C0);
+      hc.push_back(T * C0);
       T = T * spin(angle * degree);
       T = T * xpush(dist);
       };
 
     ld q = 6;
-    ld eshort = 0.5;
+    ld eshort = 0.3;
     ld elong = sqrt(3) * eshort;
     if(fake::in()) q = fake::around;
 
@@ -420,6 +423,13 @@ struct hrmap_hat : hrmap {
       eshort = edge_of_triangle_with_angles(M_PI / q, 60._deg, 90._deg);
       elong = edge_of_triangle_with_angles(60._deg, M_PI / q, 90._deg);
       }
+    else {
+      eshort *= hat_param;
+      elong *= 2 - hat_param;
+      // 0-length edges cause problems...
+      if(eshort == 0) eshort = .0001;
+      if(elong == 0) elong = .0001;
+      }
 
     ld i60 = (M_PI - TAU*2/q)/degree;
     ld n60 = (M_PI - TAU*4/q)/degree;
@@ -430,13 +440,22 @@ struct hrmap_hat : hrmap {
     move(i60, elong);  move(-90, eshort); move( 60, eshort);
     move( 90, elong);  move(i60, elong);
 
-    hyperpoint ctr = Hypc;
-    for(auto h: hatcorners[0]) ctr += h;
-    ctr /= isize(hatcorners[0]);
-    ctr = normalize(ctr);
-    for(auto& h: hatcorners[0]) h = gpushxto0(ctr) * h;
+    if(q == 6) {
+      ld area = 0;
+      for(int i=0; i<14; i++) area += (hc[(i+1)%14] ^ hc[i]) [2];
+      println(hlog, "area = ", area);
+      area = abs(area);
+      ld scale = sqrt(2.5 / area);
+      for(auto& h: hc) h = h * scale + (C0) * (1-scale);
+      }
 
-    hatcorners[1] = hatcorners[0];
+    hyperpoint ctr = Hypc;
+    for(auto h: hc) ctr += h;
+    ctr /= isize(hc);
+    ctr = normalize(ctr);
+    for(auto& h: hc) h = gpushxto0(ctr) * h;
+
+    hatcorners[1] = hc;
     for(auto& h: hatcorners[1]) h = MirrorX * h;
     reverse(hatcorners[1].begin(), hatcorners[1].end());
     }
@@ -608,6 +627,7 @@ EX int get_hat_id(cell *c) {
 EX void reshape() {
   hrmap_hat *hatmap;
   hatmap = FPIU( hat_map() );
+  if(!hatmap) return;
   hatmap->init();
   }
 
diff --git a/config.cpp b/config.cpp
index 6463fe4d..3f03ee1c 100644
--- a/config.cpp
+++ b/config.cpp
@@ -822,7 +822,14 @@ EX void initConfig() {
 
   param_f(vid.binary_width, "bwidth", "binary-tiling-width", 1);
   param_custom(vid.binary_width, "binary tiling width", menuitem_binary_width, 'v');
- 
+
+  param_f(hat::hat_param, "hat_param", "hat_param", 1)
+  -> editable(0, 2, 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.", 'v'
+    )
+  -> set_reaction(hat::reshape);
+
   addsaver(vid.particles, "extra effects", 1);
   param_i(vid.framelimit, "frame limit", 999);
 
diff --git a/geom-exp.cpp b/geom-exp.cpp
index c3de4a3e..1b565fc2 100644
--- a/geom-exp.cpp
+++ b/geom-exp.cpp
@@ -1022,6 +1022,9 @@ EX void showEuclideanMenu() {
     menuitem_binary_width('v');
     add_edit_wall_quality('W');
     }
+  else if(hat::in()) {
+    add_edit(hat::hat_param);
+    }
   else if(nil) {
     menuitem_nilwidth('v');
     }
diff --git a/geometry.cpp b/geometry.cpp
index eade3c9d..5a85a23c 100644
--- a/geometry.cpp
+++ b/geometry.cpp
@@ -720,7 +720,6 @@ void geometry_information::prepare_basics() {
   #if CAP_BT
   if(bt::in()) hexvdist = rhexf = 1, tessf = 1, scalefactor = 1, crossf = hcrossf7;
   if(geometry == gHoroRec || kite::in() || sol || nil || nih) hexvdist = rhexf = .5, tessf = .5, scalefactor = .5, crossf = hcrossf7/2;
-  if(hat::in()) scalefactor *= 1.5, crossf *= 1.5, tessf *= 1.5, hexvdist *= 1.5, rhexf *= 1.5;
   if(bt::in()) scalefactor *= min<ld>(vid.binary_width, 1), crossf *= min<ld>(vid.binary_width, 1);
   #endif
   #if MAXMDIM >= 4
@@ -1262,6 +1261,7 @@ EX string cgi_string() {
   if(nil) V("NIL", its(S7));
   
   if(bt::in()) V("BT", fts(vid.binary_width));
+  if(hat::in()) V("H", fts(hat::hat_param));
 
   if(nil) V("NILW", fts(nilv::nilwidth));