diff --git a/cosets/src/main.cpp b/cosets/src/main.cpp
index 0ac8492..5a46978 100644
--- a/cosets/src/main.cpp
+++ b/cosets/src/main.cpp
@@ -32,12 +32,9 @@ public:
 
       program = build_program("main", vs, fs);
 
-      verts_data = vertices<3>({}, Multiplicites<3>({
-         {0, 1, 5},
-         {1, 2, 3}
-      }), {
-         10, 1, 1
-      });
+      verts_data = vertices<3>(
+         schlafli<3>({5, 3}),
+         {10, 1, 1});
 
       glGenBuffers(1, &verts);
       glBindBuffer(GL_ARRAY_BUFFER, verts);
@@ -81,9 +78,7 @@ public:
 
       glBindVertexArray(tri);
       glUniform4f(0, 1, 1, 1, 1);
-      glDrawArrays(GL_POINTS, 0, verts_data.size() - 3);
-      glUniform4f(0, 1, 1, 1, .1);
-      glDrawArrays(GL_TRIANGLES, verts_data.size() - 3, 3);
+      glDrawArrays(GL_POINTS, 0, verts_data.size());
 
       swapbuffers();
    }
diff --git a/cosets/src/mesh.cpp b/cosets/src/mesh.cpp
index eb26070..4eb26ab 100644
--- a/cosets/src/mesh.cpp
+++ b/cosets/src/mesh.cpp
@@ -1,16 +1,20 @@
 #include "util/mesh.hpp"
 
 int main(int argc, char *argv[]) {
-   auto vs = vertices<3>({}, Multiplicites<3>({
-      {0, 1, 4},
-      {1, 2, 3}
-   }), {
-      1.0f, 1.0f, 1.0f
-   });
+   const auto &mults = schlafli<4>({4, 3, 3});
+   const std::vector<glm::vec4> &vs = vertices<4>(mults, {10, 1, 1, 1});
 
-   for (const auto &v : vs) {
+   for (const auto &v:vs) {
       std::cout << glm::to_string(v) << std::endl;
    }
 
-   return 0;
+//   Table *table = solve({}, mults);
+//   std::cout << table->size() << std::endl;
+//
+//   for (const auto &rel : mults.relations()) {
+//      for (const auto &e : rel) {
+//         std::cout << e << " ";
+//      }
+//      std::cout << std::endl;
+//   }
 }
\ No newline at end of file
diff --git a/cosets/src/mirrors.cpp b/cosets/src/mirrors.cpp
index 51ba1d6..d6281bf 100644
--- a/cosets/src/mirrors.cpp
+++ b/cosets/src/mirrors.cpp
@@ -1,7 +1,7 @@
 #include "util/mirrors.hpp"
 
 int main(int argc, char *argv[]) {
-   auto normals = mirror<3>(Multiplicites<3>({
+   auto normals = mirror<3>(Multiplicities<3>({
       {0, 1, 4},
       {1, 2, 3}
    }));
diff --git a/cosets/src/util/coxeter.hpp b/cosets/src/util/coxeter.hpp
index 054a49f..54d39d6 100644
--- a/cosets/src/util/coxeter.hpp
+++ b/cosets/src/util/coxeter.hpp
@@ -5,39 +5,60 @@
 #include <iomanip>
 #include <algorithm>
 #include <cmath>
-
-struct Mult {
-   int a, b, mult;
-};
+#include <map>
+#include <tuple>
 
 template<int N>
-struct Multiplicites {
-   std::vector<int> mults;
-
-   explicit Multiplicites(const std::vector<Mult> &vals) {
-      mults = std::vector<int>(N * (N - 1) / 2, 2);
-      for (const auto &mult : vals) {
-         set(mult.a, mult.b, mult.mult);
-      }
-   }
+struct Mults {
+   std::map<std::tuple<int, int>, int> mults{};
 
    void set(int a, int b, int mult) {
-      if (a > N or b > N) throw std::logic_error("mirror does not exist");
-      if (a == b) throw std::logic_error("cannot compare mirror to itself");
-      if (a < b) std::swap(a, b); // a is bigger
-
-      mults[a + b] = mult;
+      if (a > b) std::swap(a, b);
+      mults[std::make_tuple(a, b)] = mult;
    }
 
    [[nodiscard]] int get(int a, int b) const {
-      if (a > N or b > N) throw std::logic_error("mirror does not exist");
-      if (a == b) throw std::logic_error("cannot compare mirror to itself");
-      if (a < b) std::swap(a, b); // a is bigger
+      if (a == b) return 1;
+      if (a > b) std::swap(a, b);
 
-      return mults[a + b];
+      const auto &tup = std::make_tuple(a, b);
+      const auto &res = mults.find(tup);
+
+      if (res == mults.end()) return 2;
+      return res->second;
+   }
+
+   [[nodiscard]] std::vector<int> relation(int a, int b) const {
+      std::vector<int> res{};
+      int mult = get(a, b);
+
+      for (int i = 0; i < mult; ++i) {
+         res.push_back(a);
+         res.push_back(b);
+      }
+
+      return res;
+   }
+
+   [[nodiscard]] std::vector<std::vector<int>> relations() const {
+      std::vector<std::vector<int>> res{};
+      for (int a = 0; a < N; ++a) {
+         for (int b = a; b < N; ++b) {
+            res.push_back(relation(a, b));
+         }
+      }
+      return res;
    }
 };
 
+template<int N>
+Mults<N> schlafli(const int (&symbol)[N - 1]) {
+   Mults<N> mults{};
+   for (int i = 0; i < N; ++i) {
+      mults.set(i, i + 1, symbol[i]);
+   }
+   return mults;
+}
 
 struct Table {
    int N;
@@ -171,7 +192,10 @@ std::ostream &operator<<(std::ostream &out, const Row &row) {
    return out;
 }
 
-Table *solve(int gens, const std::vector<int> &subgens, const std::vector<std::vector<int>> &rels) {
+template<int N>
+Table *solve(const std::vector<int> &subgens, const Mults<N> &mults) {
+   const auto rels = mults.relations();
+   int gens = N;
    auto *table = new Table(gens);
 
    for (int gen : subgens)
diff --git a/cosets/src/util/mesh.hpp b/cosets/src/util/mesh.hpp
index 9380bb6..23197d4 100644
--- a/cosets/src/util/mesh.hpp
+++ b/cosets/src/util/mesh.hpp
@@ -7,58 +7,44 @@
 #include <vector>
 
 template<int N>
-std::vector<glm::vec4>
-vertices(const std::vector<int> &subgens, const Multiplicites<N> &mults, const float (&coords)[N]) {
+std::vector<std::vector<int>> coxeter_rels() {
    std::vector<std::vector<int>> rels{};
-   rels.reserve(N + N * (N - 1) / 2);
 
-   for (int i = 0; i < N; ++i) {
-      rels.push_back({i, i});
-   }
-   for (int a = 0; a < N; ++a) {
-      for (int b = 0; b < a; ++b) {
-         int mult = mults.get(a, b);
+   return rels;
+}
 
-         std::vector<int> rel{};
-         for (int i = 0; i < mult; ++i) {
-            rel.push_back(a);
-            rel.push_back(b);
-         }
-         rels.push_back(rel);
-      }
-   }
-
-   Table *table = solve(N, subgens, rels);
-
-//   std::cout << table->size() << std::endl;
-//   std::cout << *table << std::endl;
-
-//   for (const auto &v : table->words()) {
-//      std::cout << "[ ";
-//      for (auto e : v) {
-//         std::cout << e << " ";
-//      }
-//      std::cout << "]\n";
-//   }
-
-   const std::vector<glm::vec4> &normals = mirror<3>(mults);
-   const std::vector<glm::vec4> &corners = plane_intersections(normals);
+template<int N>
+glm::vec4 identity(const std::vector<glm::vec4> &normals, const float(&coords)[N]) {
+   const std::vector<glm::vec4> corners = plane_intersections(normals);
    const std::vector<float> coords_vec(coords, coords + N);
-   const glm::vec4 &identity = glm::normalize(barycentric(corners, coords_vec));
+   const glm::vec4 identity = barycentric(corners, coords_vec);
+   return glm::normalize(identity);
+}
+
+template<int N>
+std::vector<glm::vec4>
+vertices(const Mults<N> &mults, const float (&coords)[N]) {
+   Table *table = solve({}, mults);
+
+   const std::vector<glm::vec4> normals = mirror<N>(mults);
+   glm::vec4 ident = identity(normals, coords);
 
    std::vector<glm::vec4> verts{};
-
    for (const auto &word : table->words()) {
-      glm::vec4 vert = identity;
+      glm::vec4 vert = ident;
       for (const auto &gen : word) {
          vert = reflect(vert, normals[gen]);
       }
       verts.push_back(vert);
    }
 
-   for (const auto &e : corners) {
-      verts.push_back(e * 1.1f);
-   }
-
    return verts;
-}
\ No newline at end of file
+}
+
+template<int N>
+std::vector<unsigned> lines(const Mults<N> &mults) {
+//   Table *table = solve(N, {}, coxeter_rels(mults));
+
+   return {};
+}
+
diff --git a/cosets/src/util/mirrors.hpp b/cosets/src/util/mirrors.hpp
index 6a629bd..ce107bd 100644
--- a/cosets/src/util/mirrors.hpp
+++ b/cosets/src/util/mirrors.hpp
@@ -11,7 +11,7 @@
 #include "coxeter.hpp"
 
 template<int N>
-std::vector<glm::vec4> mirror(const Multiplicites<N> &mults) {
+std::vector<glm::vec4> mirror(const Mults<N> &mults) {
    static_assert(1 <= N and N <= 4, "Vector size is unsupported");
 
    std::vector<glm::vec4> mirrors{};