diff --git a/geometry/BUILD.bazel b/geometry/BUILD.bazel
index e9c04e7..1a5ff47 100644
--- a/geometry/BUILD.bazel
+++ b/geometry/BUILD.bazel
@@ -4,7 +4,9 @@ cc_library(
     name = "geometry",
     srcs = [
         "src/algebra.hpp",
+        "src/detail/all_same.hpp",
         "src/detail/contract_dimensions.hpp",
+        "src/detail/ordered.hpp",
         "src/detail/sort_dimensions.hpp",
         "src/detail/strictly_increasing.hpp",
     ],
diff --git a/geometry/src/algebra.hpp b/geometry/src/algebra.hpp
index 9366446..4ad1bb5 100644
--- a/geometry/src/algebra.hpp
+++ b/geometry/src/algebra.hpp
@@ -1,6 +1,8 @@
 #pragma once
 
+#include "geometry/src/detail/all_same.hpp"
 #include "geometry/src/detail/contract_dimensions.hpp"
+#include "geometry/src/detail/ordered.hpp"
 #include "geometry/src/detail/sort_dimensions.hpp"
 #include "geometry/src/detail/strictly_increasing.hpp"
 
@@ -33,6 +35,20 @@ struct algebra
   template <std::size_t... Is>
   struct blade;
 
+  /// determines if a type is a blade
+  /// @tparam T type
+  ///
+  /// @{
+  template <class T>
+  struct is_blade : std::false_type
+  {};
+  template <std::size_t... Is>
+  struct is_blade<blade<Is...>> : std::true_type
+  {};
+  template <class T>
+  static constexpr auto is_blade_v = is_blade<T>::value;
+  /// @}
+
 private:
   template <std::size_t... Is>
   static auto rebind(std::index_sequence<Is...>) -> blade<Is...>;
@@ -217,6 +233,51 @@ struct algebra
       return x.coefficient * y.coefficient * e<Is..., Js...>;
     }
   };
+
+  /// multivector
+  /// @tparam Bs blade types
+  ///
+  /// A linear combination of blades.
+  ///
+  /// @note a multivector may be composed of only blades with the same grade
+  ///
+  template <class... Bs>
+  struct multivector : Bs...
+  {
+    static_assert(
+        (is_blade_v<Bs> and ...), "`Bs` must be a specialization of blade");
+    static_assert(
+        detail::all_same_v<typename Bs::scalar_type...>,
+        "`Bs` must have the same scalar type");
+    static_assert(
+        detail::strictly_increasing(detail::ordered<Bs>{}...),
+        "`Bs` must be lexicographically sorted");
+
+    /// algebra type
+    ///
+    using algebra_type = algebra;
+
+    /// construct a multivector from blades
+    ///
+    constexpr multivector(Bs... bs) : Bs{bs}... {}
+
+    /// equality comparison
+    ///
+    /// @{
+    [[nodiscard]]
+    friend constexpr auto
+    operator==(const multivector& x, const multivector& y) -> bool
+    {
+      return ((static_cast<const Bs&>(x) == static_cast<const Bs&>(y)) and ...);
+    }
+    [[nodiscard]]
+    friend constexpr auto
+    operator!=(const multivector& x, const multivector& y) -> bool
+    {
+      return not(x == y);
+    }
+    /// @}
+  };
 };
 
 }  // namespace geometry
diff --git a/geometry/src/detail/all_same.hpp b/geometry/src/detail/all_same.hpp
new file mode 100644
index 0000000..79f3499
--- /dev/null
+++ b/geometry/src/detail/all_same.hpp
@@ -0,0 +1,24 @@
+#pragma once
+
+#include <type_traits>
+
+namespace geometry::detail {
+
+template <class...>
+struct all_same : std::false_type
+{};
+
+template <>
+struct all_same<> : std::true_type
+{};
+template <class T>
+struct all_same<T> : std::true_type
+{};
+template <class T, class... Ts>
+struct all_same<T, Ts...> : std::bool_constant<((std::is_same_v<T, Ts>)and...)>
+{};
+
+template <class... Ts>
+inline constexpr auto all_same_v = all_same<Ts...>::value;
+
+}  // namespace geometry::detail
diff --git a/geometry/src/detail/ordered.hpp b/geometry/src/detail/ordered.hpp
new file mode 100644
index 0000000..68aab64
--- /dev/null
+++ b/geometry/src/detail/ordered.hpp
@@ -0,0 +1,33 @@
+#pragma once
+
+#include <cstddef>
+#include <tuple>
+
+namespace geometry::detail {
+
+template <class B>
+class ordered
+{
+  template <
+      template <std::size_t...>
+      class blade,
+      std::size_t... Is,
+      std::size_t... Js>
+  static constexpr auto
+  compare_dimensions(ordered<blade<Is...>>, ordered<blade<Js...>>)
+  {
+    return std::tuple{Is...} < std::tuple{Js...};
+  }
+
+  template <class B2>
+  friend constexpr auto operator<(ordered x, ordered<B2> y) -> bool
+  {
+    if constexpr (B::grade == B2::grade) {
+      return compare_dimensions(x, y);
+    } else {
+      return B::grade < B2::grade;
+    }
+  }
+};
+
+}  // namespace geometry::detail
diff --git a/geometry/src/detail/strictly_increasing.hpp b/geometry/src/detail/strictly_increasing.hpp
index ba6e965..391e0bb 100644
--- a/geometry/src/detail/strictly_increasing.hpp
+++ b/geometry/src/detail/strictly_increasing.hpp
@@ -1,38 +1,31 @@
 #pragma once
 
-#include <array>
 #include <cstddef>
+#include <tuple>
 #include <utility>
 
 namespace geometry::detail {
 
 inline constexpr class
 {
-  template <class T>
-  static constexpr auto impl(std::index_sequence<>, const std::array<T, 1>&)
+  template <std::size_t... Is, class T>
+  static constexpr auto impl(std::index_sequence<Is...>, const T& values)
   {
-    return true;
-  }
-  template <
-      std::size_t... Is,
-      class T,
-      class = std::enable_if_t<(sizeof...(Is) != 0)>>
-  static constexpr auto
-  impl(std::index_sequence<Is...>,
-       const std::array<T, sizeof...(Is) + 1>& values)
-  {
-    return ((std::get<Is>(values) < std::get<Is + 1>(values)) and ...);
+    if constexpr (sizeof...(Is) == 0) {
+      return true;
+    } else {
+      return (((std::get<Is>(values)) < (std::get<Is + 1>(values))) and ...);
+    }
   }
 
 public:
   // 0-args
   constexpr auto operator()() const { return true; }
   // 1+-args
-  template <class... Ts>
-  constexpr auto operator()(std::size_t t0, Ts... ts) const
+  template <class T0, class... Ts>
+  constexpr auto operator()(T0 t0, Ts... ts) const
   {
-    static_assert((std::is_same_v<std::size_t, Ts> and ...));
-    return impl(std::index_sequence_for<Ts...>{}, std::array{t0, ts...});
+    return impl(std::index_sequence_for<Ts...>{}, std::tuple{t0, ts...});
   }
 } strictly_increasing{};
 
diff --git a/test/BUILD.bazel b/test/BUILD.bazel
index 7009c30..ee7904d 100644
--- a/test/BUILD.bazel
+++ b/test/BUILD.bazel
@@ -1,8 +1,17 @@
 load("@rules_cc//cc:defs.bzl", "cc_test")
 
 cc_test(
-    name = "algebra_test",
-    srcs = ["algebra_test.cpp"],
+    name = "algebra_blade_test",
+    srcs = ["algebra_blade_test.cpp"],
+    deps = [
+        "//:geometry",
+        "@skytest",
+    ],
+)
+
+cc_test(
+    name = "algebra_multivector_test",
+    srcs = ["algebra_multivector_test.cpp"],
     deps = [
         "//:geometry",
         "@skytest",
diff --git a/test/algebra_test.cpp b/test/algebra_blade_test.cpp
similarity index 100%
rename from test/algebra_test.cpp
rename to test/algebra_blade_test.cpp
diff --git a/test/algebra_multivector_test.cpp b/test/algebra_multivector_test.cpp
new file mode 100644
index 0000000..cabeaa8
--- /dev/null
+++ b/test/algebra_multivector_test.cpp
@@ -0,0 +1,48 @@
+#include "geometry/geometry.hpp"
+#include "skytest/skytest.hpp"
+
+#include <cstddef>
+#include <tuple>
+#include <type_traits>
+
+template <std::size_t... Is>
+constexpr auto e = ::geometry::algebra<double, 2>::e<Is...>;
+
+/// construct a multivector from blades
+///
+inline constexpr struct
+{
+  template <
+      class... Bs,
+      class = std::enable_if_t<
+          sizeof...(Bs) != 0 and
+          ::geometry::detail::all_same_v<typename Bs::algebra_type...>>>
+  [[nodiscard]]
+  constexpr auto
+  operator()(Bs... bs) const
+  {
+    using Algebra = std::common_type_t<typename Bs::algebra_type...>;
+    return typename Algebra::template multivector<Bs...>{bs...};
+  }
+} multivector{};
+
+auto main() -> int
+{
+  using namespace ::skytest::literals;
+  using ::skytest::eq;
+  using ::skytest::expect;
+  using ::skytest::param_ref;
+
+  static constexpr auto multivectors = std::tuple{
+      multivector(e<>),
+      multivector(e<>, e<0>),
+      multivector(e<>, e<1>),
+      multivector(e<>, e<2>),
+      multivector(e<>, e<0>, e<1, 2>),
+      multivector(e<>, e<0, 1>, e<1, 2>),
+      multivector(e<>, e<0, 1, 2>)};
+
+  "multivector with same elements comparable"_ctest *
+      param_ref<multivectors> =  //
+      [](auto v) { return expect(eq(v, v)); };
+}