diff --git a/app-proto/inversive-display/src/inversive.frag b/app-proto/inversive-display/src/inversive.frag
index da2f415..c469edc 100644
--- a/app-proto/inversive-display/src/inversive.frag
+++ b/app-proto/inversive-display/src/inversive.frag
@@ -138,16 +138,34 @@ void main() {
     const int SPHERE_MAX_INTERNAL = 6;
     vec2 depth_pairs [SPHERE_MAX_INTERNAL];
     taggedFrag frags [2*SPHERE_MAX_INTERNAL];
-    int frag_cnt = 0;
-    for (int i = 0; i < sphere_cnt; ++i) {
-        vec2 hit_depths = sphere_cast(sphere_list[i], dir);
-        if (hit_depths[0] > 0.) {
-            frags[frag_cnt] = sphere_shading(sphere_list[i], hit_depths[0] * dir, color_list[i], i);
-            ++frag_cnt;
-        }
-        if (hit_depths[1] > 0.) {
-            frags[frag_cnt] = sphere_shading(sphere_list[i], hit_depths[1] * dir, color_list[i], i);
-            ++frag_cnt;
+    int layer_cnt = 0;
+    for (int id = 0; id < sphere_cnt; ++id) {
+        // find out where the ray hits the sphere
+        vec2 hit_depths = sphere_cast(sphere_list[id], dir);
+        
+        // insertion-sort the fragments we hit into the fragment list
+        for (int side = 0; side < 2; ++side) {
+            if (hit_depths[side] > 0.) {
+                for (int layer = layer_cnt; layer >= 0; --layer) {
+                    if (layer < 1 || frags[layer-1].pt.z >= -hit_depths[side]) {
+                        // we're not as close to the screen as the fragment
+                        // before the empty slot, so insert here
+                        frags[layer] = sphere_shading(
+                            sphere_list[id],
+                            hit_depths[side] * dir,
+                            color_list[id],
+                            id
+                        );
+                        break;
+                    } else {
+                        // we're closer to the screen than the fragment before
+                        // the empty slot, so move that fragment into the empty
+                        // slot
+                        frags[layer] = frags[layer-1];
+                    }
+                }
+                ++layer_cnt;
+            }
         }
     }
     
@@ -156,29 +174,16 @@ void main() {
     if (debug_mode) {
         // at the bottom of the screen, show the color scale instead of the
         // layer count
-        if (gl_FragCoord.y < 10.) frag_cnt = int(8. * gl_FragCoord.x / resolution.x);
+        if (gl_FragCoord.y < 10.) layer_cnt = int(8. * gl_FragCoord.x / resolution.x);
         
         // convert number to color
-        ivec3 bits = frag_cnt / ivec3(1, 2, 4);
+        ivec3 bits = layer_cnt / ivec3(1, 2, 4);
         outColor = vec4(mod(vec3(bits), 2.), 1.);
         return;
     }
     
-    // sort the fragments by depth, using an insertion sort
-    for (int take = 1; take < frag_cnt; ++take) {
-        taggedFrag pulled = frags[take];
-        for (int put = take; put >= 0; --put) {
-            if (put < 1 || frags[put-1].pt.z >= pulled.pt.z) {
-                frags[put] = pulled;
-                break;
-            } else {
-                frags[put] = frags[put-1];
-            }
-        }
-    }
-    
     // highlight intersections and cusps
-    for (int i = frag_cnt-1; i >= 1; --i) {
+    for (int i = layer_cnt-1; i >= 1; --i) {
         // intersections
         taggedFrag frag0 = frags[i];
         taggedFrag frag1 = frags[i-1];
@@ -201,7 +206,7 @@ void main() {
     
     // composite the sphere fragments
     vec3 color = vec3(0.);
-    for (int i = frag_cnt-1; i >= layer_threshold; --i) {
+    for (int i = layer_cnt-1; i >= layer_threshold; --i) {
         if (frags[i].pt.z < 0.) {
             vec4 frag_color = frags[i].color;
             color = mix(color, frag_color.rgb, frag_color.a);