diff --git a/euclid.cpp b/euclid.cpp
index 1de96c61..ab2266ce 100644
--- a/euclid.cpp
+++ b/euclid.cpp
@@ -497,13 +497,13 @@ namespace euclid3 {
     
     heptagon *build(heptagon *parent, int d, coord at) {
       auto h = get_at(at);
-      h->c.connect(d^1, parent, d, false);
+      h->c.connect((d+3)%6, parent, d, false);
       return h;
       }
   
     heptagon *createStep(heptagon *parent, int d) {
       int at = ispacemap[parent];
-      coord shifttable[6] = { +1, -1, +1000, -1000, +1000000, -1000000 };
+      coord shifttable[6] = { +1, +1000, +1000000, -1, -1000, -1000000 };
       return build(parent, d, at + shifttable[d]);
       }
     };
@@ -556,7 +556,7 @@ namespace euclid3 {
       drawcell(c, V, 0, false);
 
       for(int i=0; i<6; i++)
-        dq::enqueue(h->move(i), V * cpush(i>>1, (i&1) ? -1 : 1));
+        dq::enqueue(h->move(i), V * cpush(i%3, (i>=3) ? -1 : 1));
       }
     }
   
diff --git a/game.cpp b/game.cpp
index 08a8b6dd..5eda1b6a 100644
--- a/game.cpp
+++ b/game.cpp
@@ -4163,6 +4163,16 @@ cell *determinePush(cellwalker who, cell *c2, int subdir, const T& valid, int& p
     }
   cellwalker push = who;
   push += wstep;
+  if(DIM == 3 && binarytiling) {
+    for(int a=0; a<4; a++) {
+      if(push.spin < 4) push.spin = 8;
+      else if(push.spin >= 8) push.spin = a;
+      else push.spin ^= 1;
+      push += wstep;
+      if(valid(push.at)) return push.at;
+      }
+    return c2;
+    }
   int pd = push.at->type/2;
   push += pd * -subdir;
   push += wstep;
diff --git a/hyper.h b/hyper.h
index 461a8fbf..6624e26f 100644
--- a/hyper.h
+++ b/hyper.h
@@ -506,6 +506,11 @@ template<class T> struct walker {
     }
   walker<T>& operator += (rev_t) {
     int d = at->degree();
+    if(DIM == 3 && binarytiling) {
+      if(spin < 4) spin = 8;
+      else if(spin >= 8) spin = 0;
+      else spin ^= 1;
+      }
     return (*this) += d/2 + ((d&1)?hrand(2):0);
     }
   walker<T>& operator += (revstep_t) {
diff --git a/polygons.cpp b/polygons.cpp
index 07d4fdf5..af088b27 100644
--- a/polygons.cpp
+++ b/polygons.cpp
@@ -2516,10 +2516,10 @@ void buildpolys() {
       bshape(shBinaryWall[w], PPR::WALL);
       for(int a=0; a<=4; a++) {
         int t[3];
-        t[0] = (w&1) ? -1 : 1;
+        t[0] = (w>=3) ? -1 : 1;
         t[1] = among(a, 0, 3, 4) ? -1 : 1;
         t[2] = among(a, 2, 3) ? -1 : 1;
-        int x = w/2;
+        int x = w%3;
         int y = (x+2)%3;
         int z = (y+2)%3;
         hpcpush(hpxy3(t[x]/2., t[y]/2., t[z]/2.));
diff --git a/sphere.cpp b/sphere.cpp
index 4143770b..debc0ea4 100644
--- a/sphere.cpp
+++ b/sphere.cpp
@@ -277,6 +277,10 @@ void gen600() {
   
   adj0.clear();
   for(int i=0; i<120; i++) if(inedge[root][i]) adj0.push_back(i);
+  
+  using namespace hyperpoint_vec;
+  for(int i=0; i<6; i++) for(int j=i+1; j<12; j++) if(zero_d(vertices120[adj0[i]] + vertices120[adj0[j]], 3))
+    swap(adj0[j], adj0[i+6]);
 
   for(int i=0; i<120; i++) for(int j=0; j<120; j++)
     if(hdist(vertices120[i], vertices120[j]) > 3)