diff --git a/src/kernel/algorithms/sweep.rs b/src/kernel/algorithms/sweep.rs
index 8b006fe7b..cad604db5 100644
--- a/src/kernel/algorithms/sweep.rs
+++ b/src/kernel/algorithms/sweep.rs
@@ -53,7 +53,7 @@ pub fn sweep_shape(
     }
 
     for face in bottom_faces {
-        shape.faces().add(face);
+        shape.faces().add((*face).clone());
     }
     for face in top_faces {
         shape.faces().add(face);
diff --git a/src/kernel/shape/faces.rs b/src/kernel/shape/faces.rs
index 0c177317b..e1c8b3787 100644
--- a/src/kernel/shape/faces.rs
+++ b/src/kernel/shape/faces.rs
@@ -17,19 +17,23 @@ pub struct Faces<'r> {
 impl Faces<'_> {
     /// Add a face to the shape
     pub fn add(&mut self, face: Face) -> Handle<Face> {
-        self.faces.push(face.clone());
-        Storage::new(face).handle()
+        let storage = Storage::new(face);
+        let handle = storage.handle();
+
+        self.faces.push(storage);
+
+        handle
     }
 
     /// Check whether the shape contains a specific face
     #[cfg(test)]
     pub fn contains(&self, face: &Face) -> bool {
-        self.faces.contains(face)
+        self.faces.contains(&Storage::new(face.clone()))
     }
 
     /// Access an iterator over all faces
-    pub fn all(&self) -> impl Iterator<Item = Face> + '_ {
-        self.faces.iter().cloned()
+    pub fn all(&self) -> impl Iterator<Item = Handle<Face>> + '_ {
+        self.faces.iter().map(|storage| storage.handle())
     }
 
     pub fn triangles(
diff --git a/src/kernel/shape/mod.rs b/src/kernel/shape/mod.rs
index 95e1786bd..1c37ddcfb 100644
--- a/src/kernel/shape/mod.rs
+++ b/src/kernel/shape/mod.rs
@@ -92,4 +92,4 @@ impl Shape {
 
 type VerticesInner = Vec<Storage<Vertex>>;
 type CyclesInner = Vec<Storage<Cycle>>;
-type FacesInner = Vec<Face>;
+type FacesInner = Vec<Storage<Face>>;
diff --git a/src/kernel/shapes/difference_2d.rs b/src/kernel/shapes/difference_2d.rs
index 80ff5dfc2..c1eac9f4c 100644
--- a/src/kernel/shapes/difference_2d.rs
+++ b/src/kernel/shapes/difference_2d.rs
@@ -56,22 +56,23 @@ impl ToShape for fj::Difference2d {
                 );
             };
 
-            let (a, b, surface_a, surface_b) = match (a, b) {
-                (
-                    Face::Face {
-                        cycles: a,
-                        surface: surface_a,
-                    },
-                    Face::Face {
-                        cycles: b,
-                        surface: surface_b,
-                    },
-                ) => (a, b, surface_a, surface_b),
-                _ => {
-                    // None of the 2D types still use the triangles representation.
-                    unreachable!()
-                }
-            };
+            let (a, b, surface_a, surface_b) =
+                match ((*a).clone(), (*b).clone()) {
+                    (
+                        Face::Face {
+                            cycles: a,
+                            surface: surface_a,
+                        },
+                        Face::Face {
+                            cycles: b,
+                            surface: surface_b,
+                        },
+                    ) => (a, b, surface_a, surface_b),
+                    _ => {
+                        // None of the 2D types still use the triangles representation.
+                        unreachable!()
+                    }
+                };
 
             assert!(
                 surface_a == surface_b,
diff --git a/src/kernel/shapes/union.rs b/src/kernel/shapes/union.rs
index 5b6a5cee3..8147510fc 100644
--- a/src/kernel/shapes/union.rs
+++ b/src/kernel/shapes/union.rs
@@ -19,10 +19,10 @@ impl ToShape for fj::Union {
         // See issue:
         // https://github.com/hannobraun/Fornjot/issues/42
         for face in a.faces().all() {
-            shape.faces().add(face);
+            shape.faces().add((*face).clone());
         }
         for face in b.faces().all() {
-            shape.faces().add(face);
+            shape.faces().add((*face).clone());
         }
 
         shape