diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index a0a59be..a874485 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -33,6 +33,19 @@ rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number =
 
 rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), shells)
 
+# === elements ===
+
+plane(normal, offset) = [normal; offset; offset]
+
+function sphere(center, radius)
+  dist_sq = dot(center, center)
+  return [
+    center / radius;
+    0.5 * ((dist_sq - 1) / radius - radius);
+    0.5 * ((dist_sq + 1) / radius - radius)
+  ]
+end
+
 # === Gram matrix realization ===
 
 # the Lorentz form