diff --git a/crates/fj-core/src/geometry/geometry.rs b/crates/fj-core/src/geometry/geometry.rs
index 783ff7dc1..6a83194dd 100644
--- a/crates/fj-core/src/geometry/geometry.rs
+++ b/crates/fj-core/src/geometry/geometry.rs
@@ -14,6 +14,8 @@ pub struct Geometry {
     half_edge: BTreeMap<Handle<HalfEdge>, HalfEdgeGeom>,
     surface: BTreeMap<Handle<Surface>, SurfaceGeom>,
 
+    space_2d: Handle<Surface>,
+
     xy_plane: Handle<Surface>,
     xz_plane: Handle<Surface>,
     yz_plane: Handle<Surface>,
@@ -26,6 +28,8 @@ impl Geometry {
             half_edge: BTreeMap::new(),
             surface: BTreeMap::new(),
 
+            space_2d: topology.surfaces.space_2d(),
+
             xy_plane: topology.surfaces.xy_plane(),
             xz_plane: topology.surfaces.xz_plane(),
             yz_plane: topology.surfaces.yz_plane(),
@@ -69,6 +73,10 @@ impl Geometry {
         surface: Handle<Surface>,
         geometry: SurfaceGeom,
     ) {
+        if surface == self.space_2d {
+            panic!("Attempting to define geometry for 2D space");
+        }
+
         if self.surface.contains_key(&surface)
             && (surface == self.xy_plane
                 || surface == self.xz_plane
diff --git a/crates/fj-core/src/operations/build/sketch.rs b/crates/fj-core/src/operations/build/sketch.rs
index 01d5cf2a5..93e8a6840 100644
--- a/crates/fj-core/src/operations/build/sketch.rs
+++ b/crates/fj-core/src/operations/build/sketch.rs
@@ -1,4 +1,4 @@
-use crate::topology::Sketch;
+use crate::topology::{Sketch, Topology};
 
 /// Build a [`Sketch`]
 ///
@@ -7,8 +7,8 @@ use crate::topology::Sketch;
 /// [module-level documentation]: super
 pub trait BuildSketch {
     /// Create a sketch with no regions
-    fn empty() -> Sketch {
-        Sketch::new([])
+    fn empty(topology: &Topology) -> Sketch {
+        Sketch::new(topology.surfaces.space_2d(), [])
     }
 }
 
diff --git a/crates/fj-core/src/operations/replace/curve.rs b/crates/fj-core/src/operations/replace/curve.rs
index 570edd170..1e5a21051 100644
--- a/crates/fj-core/src/operations/replace/curve.rs
+++ b/crates/fj-core/src/operations/replace/curve.rs
@@ -151,7 +151,7 @@ impl ReplaceCurve for Sketch {
         }
 
         if replacement_happened {
-            ReplaceOutput::Updated(Sketch::new(regions))
+            ReplaceOutput::Updated(Sketch::new(self.surface().clone(), regions))
         } else {
             ReplaceOutput::Original(self.clone())
         }
diff --git a/crates/fj-core/src/operations/replace/half_edge.rs b/crates/fj-core/src/operations/replace/half_edge.rs
index 8c6b3687e..ae046861b 100644
--- a/crates/fj-core/src/operations/replace/half_edge.rs
+++ b/crates/fj-core/src/operations/replace/half_edge.rs
@@ -117,7 +117,7 @@ impl ReplaceHalfEdge for Sketch {
         }
 
         if replacement_happened {
-            ReplaceOutput::Updated(Sketch::new(regions))
+            ReplaceOutput::Updated(Sketch::new(self.surface().clone(), regions))
         } else {
             ReplaceOutput::Original(self.clone())
         }
diff --git a/crates/fj-core/src/operations/replace/vertex.rs b/crates/fj-core/src/operations/replace/vertex.rs
index 11715b479..e96f41d51 100644
--- a/crates/fj-core/src/operations/replace/vertex.rs
+++ b/crates/fj-core/src/operations/replace/vertex.rs
@@ -153,7 +153,7 @@ impl ReplaceVertex for Sketch {
         }
 
         if replacement_happened {
-            ReplaceOutput::Updated(Sketch::new(regions))
+            ReplaceOutput::Updated(Sketch::new(self.surface().clone(), regions))
         } else {
             ReplaceOutput::Original(self.clone())
         }
@@ -317,7 +317,7 @@ impl ReplaceVertex for Handle<Sketch> {
         }
 
         if replacement_happened {
-            ReplaceOutput::Updated(Sketch::new(regions))
+            ReplaceOutput::Updated(Sketch::new(self.surface().clone(), regions))
         } else {
             ReplaceOutput::Original(self.clone())
         }
diff --git a/crates/fj-core/src/operations/update/sketch.rs b/crates/fj-core/src/operations/update/sketch.rs
index 4f1bc6a07..843442e47 100644
--- a/crates/fj-core/src/operations/update/sketch.rs
+++ b/crates/fj-core/src/operations/update/sketch.rs
@@ -47,7 +47,7 @@ impl UpdateSketch for Sketch {
     {
         let regions = regions.into_iter().map(|region| region.insert(core));
         let regions = self.regions().iter().cloned().chain(regions);
-        Sketch::new(regions)
+        Sketch::new(self.surface().clone(), regions)
     }
 
     fn update_region<T, R>(
@@ -69,6 +69,6 @@ impl UpdateSketch for Sketch {
                 }),
             )
             .expect("Region not found");
-        Sketch::new(regions)
+        Sketch::new(self.surface().clone(), regions)
     }
 }
diff --git a/crates/fj-core/src/topology/objects/region.rs b/crates/fj-core/src/topology/objects/region.rs
index e9f1fd7c2..9e333e5d0 100644
--- a/crates/fj-core/src/topology/objects/region.rs
+++ b/crates/fj-core/src/topology/objects/region.rs
@@ -5,7 +5,7 @@ use crate::{
     topology::{Cycle, ObjectSet},
 };
 
-/// A single, continuous 2d region, may contain holes
+/// A single, continuous 2d region; may contain holes
 ///
 /// Interior cycles must have the opposite winding of the exterior cycle,
 /// meaning on the front side of the region, they must appear clockwise. This
diff --git a/crates/fj-core/src/topology/objects/sketch.rs b/crates/fj-core/src/topology/objects/sketch.rs
index ba79d5de0..71555a7b4 100644
--- a/crates/fj-core/src/topology/objects/sketch.rs
+++ b/crates/fj-core/src/topology/objects/sketch.rs
@@ -1,22 +1,32 @@
 use crate::{
     storage::Handle,
-    topology::{ObjectSet, Region},
+    topology::{ObjectSet, Region, Surface},
 };
 
 /// A 2-dimensional shape
 #[derive(Clone, Debug)]
 pub struct Sketch {
+    surface: Handle<Surface>,
     regions: ObjectSet<Region>,
 }
 
 impl Sketch {
     /// Construct an empty instance of `Sketch`
-    pub fn new(regions: impl IntoIterator<Item = Handle<Region>>) -> Self {
+    pub fn new(
+        surface: Handle<Surface>,
+        regions: impl IntoIterator<Item = Handle<Region>>,
+    ) -> Self {
         Self {
+            surface,
             regions: regions.into_iter().collect(),
         }
     }
 
+    /// Access the surface of the sketch
+    pub fn surface(&self) -> &Handle<Surface> {
+        &self.surface
+    }
+
     /// Access the regions of the sketch
     pub fn regions(&self) -> &ObjectSet<Region> {
         &self.regions
diff --git a/crates/fj-core/src/topology/stores.rs b/crates/fj-core/src/topology/stores.rs
index 13475548f..47a269cec 100644
--- a/crates/fj-core/src/topology/stores.rs
+++ b/crates/fj-core/src/topology/stores.rs
@@ -50,6 +50,8 @@ impl Topology {
 pub struct Surfaces {
     store: Store<Surface>,
 
+    space_2d: Handle<Surface>,
+
     xy_plane: Handle<Surface>,
     xz_plane: Handle<Surface>,
     yz_plane: Handle<Surface>,
@@ -66,6 +68,15 @@ impl Surfaces {
         self.store.insert(handle, surface);
     }
 
+    /// Access the surface representing 2D space
+    ///
+    /// Every other surface is a 2D subspace within a 3D space. This surface is
+    /// special, in that it represents the 2D space which is not located in a 3D
+    /// space.
+    pub fn space_2d(&self) -> Handle<Surface> {
+        self.space_2d.clone()
+    }
+
     /// Access the xy-plane
     pub fn xy_plane(&self) -> Handle<Surface> {
         self.xy_plane.clone()
@@ -86,6 +97,9 @@ impl Default for Surfaces {
     fn default() -> Self {
         let mut store: Store<Surface> = Store::new();
 
+        let space_2d = store.reserve();
+        store.insert(space_2d.clone(), Surface::new());
+
         let xy_plane = store.reserve();
         store.insert(xy_plane.clone(), Surface::new());
 
@@ -97,6 +111,7 @@ impl Default for Surfaces {
 
         Self {
             store,
+            space_2d,
             xy_plane,
             xz_plane,
             yz_plane,
diff --git a/crates/fj-core/src/validate/sketch.rs b/crates/fj-core/src/validate/sketch.rs
index 794295b74..39170e10f 100644
--- a/crates/fj-core/src/validate/sketch.rs
+++ b/crates/fj-core/src/validate/sketch.rs
@@ -137,14 +137,21 @@ mod tests {
 
         let region = <Region as BuildRegion>::circle([0., 0.], 1., &mut core)
             .insert(&mut core);
-        let valid_sketch = Sketch::new(vec![region.clone()]).insert(&mut core);
+        let valid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![region.clone()],
+        )
+        .insert(&mut core);
         valid_sketch.validate_and_return_first_error(&core.layers.geometry)?;
 
         let shared_cycle = region.exterior();
-        let invalid_sketch = Sketch::new(vec![
-            Region::new(shared_cycle.clone(), vec![]).insert(&mut core),
-            Region::new(shared_cycle.clone(), vec![]).insert(&mut core),
-        ]);
+        let invalid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![
+                Region::new(shared_cycle.clone(), vec![]).insert(&mut core),
+                Region::new(shared_cycle.clone(), vec![]).insert(&mut core),
+            ],
+        );
         assert_contains_err!(
             core,
             invalid_sketch,
@@ -165,7 +172,11 @@ mod tests {
             &mut core,
         )
         .insert(&mut core);
-        let valid_sketch = Sketch::new(vec![region.clone()]).insert(&mut core);
+        let valid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![region.clone()],
+        )
+        .insert(&mut core);
         valid_sketch.validate_and_return_first_error(&core.layers.geometry)?;
 
         let exterior = region.exterior();
@@ -173,10 +184,12 @@ mod tests {
             exterior.half_edges().iter().cloned().collect();
         let interior = Cycle::new(cloned_edges).insert(&mut core);
 
-        let invalid_sketch =
-            Sketch::new(vec![
+        let invalid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![
                 Region::new(exterior.clone(), vec![interior]).insert(&mut core)
-            ]);
+            ],
+        );
         assert_contains_err!(
             core,
             invalid_sketch,
@@ -195,10 +208,10 @@ mod tests {
         let valid_outer_circle = HalfEdge::circle([0., 0.], 1., &mut core);
         let valid_exterior =
             Cycle::new(vec![valid_outer_circle.clone()]).insert(&mut core);
-        let valid_sketch =
-            Sketch::new(vec![
-                Region::new(valid_exterior.clone(), vec![]).insert(&mut core)
-            ]);
+        let valid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![Region::new(valid_exterior.clone(), vec![]).insert(&mut core)],
+        );
         valid_sketch.validate_and_return_first_error(&core.layers.geometry)?;
 
         let invalid_outer_circle = HalfEdge::from_sibling(
@@ -208,10 +221,12 @@ mod tests {
         );
         let invalid_exterior =
             Cycle::new(vec![invalid_outer_circle.clone()]).insert(&mut core);
-        let invalid_sketch =
-            Sketch::new(vec![
+        let invalid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![
                 Region::new(invalid_exterior.clone(), vec![]).insert(&mut core)
-            ]);
+            ],
+        );
         assert_contains_err!(
             core,
             invalid_sketch,
@@ -238,20 +253,20 @@ mod tests {
 
         let valid_interior =
             Cycle::new(vec![cw_inner_circle.clone()]).insert(&mut core);
-        let valid_sketch = Sketch::new(vec![Region::new(
-            exterior.clone(),
-            vec![valid_interior],
-        )
-        .insert(&mut core)]);
+        let valid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![Region::new(exterior.clone(), vec![valid_interior])
+                .insert(&mut core)],
+        );
         valid_sketch.validate_and_return_first_error(&core.layers.geometry)?;
 
         let invalid_interior =
             Cycle::new(vec![inner_circle.clone()]).insert(&mut core);
-        let invalid_sketch = Sketch::new(vec![Region::new(
-            exterior.clone(),
-            vec![invalid_interior],
-        )
-        .insert(&mut core)]);
+        let invalid_sketch = Sketch::new(
+            core.layers.topology.surfaces.space_2d(),
+            vec![Region::new(exterior.clone(), vec![invalid_interior])
+                .insert(&mut core)],
+        );
         assert_contains_err!(
             core,
             invalid_sketch,
diff --git a/models/cuboid/src/lib.rs b/models/cuboid/src/lib.rs
index 6c37de589..e9df50729 100644
--- a/models/cuboid/src/lib.rs
+++ b/models/cuboid/src/lib.rs
@@ -16,7 +16,7 @@ pub fn model(size: impl Into<Vector<3>>, core: &mut fj::core::Core) -> Solid {
     let bottom_surface = core.layers.topology.surfaces.xy_plane();
     let sweep_path = Vector::from([Scalar::ZERO, Scalar::ZERO, z]);
 
-    Sketch::empty()
+    Sketch::empty(&core.layers.topology)
         .add_regions(
             [Region::polygon(
                 [
diff --git a/models/spacer/src/lib.rs b/models/spacer/src/lib.rs
index 89db21660..e214fcc2b 100644
--- a/models/spacer/src/lib.rs
+++ b/models/spacer/src/lib.rs
@@ -20,7 +20,7 @@ pub fn model(
     let bottom_surface = core.layers.topology.surfaces.xy_plane();
     let sweep_path = Vector::from([0., 0., height]);
 
-    Sketch::empty()
+    Sketch::empty(&core.layers.topology)
         .add_regions(
             [Region::circle(Point::origin(), outer, core).add_interiors(
                 [Cycle::circle(Point::origin(), inner, core).reverse(core)],
diff --git a/models/star/src/lib.rs b/models/star/src/lib.rs
index 35615b420..02b6ea537 100644
--- a/models/star/src/lib.rs
+++ b/models/star/src/lib.rs
@@ -43,7 +43,7 @@ pub fn model(
     let bottom_surface = core.layers.topology.surfaces.xy_plane();
     let sweep_path = Vector::from([0., 0., h]);
 
-    Sketch::empty()
+    Sketch::empty(&core.layers.topology)
         .add_regions(
             [Region::polygon(outer_points, core).add_interiors(
                 [Cycle::polygon(inner_points, core).reverse(core)],