diff --git a/main.py b/main.py
index 5e7a1de..306c3bb 100644
--- a/main.py
+++ b/main.py
@@ -48,8 +48,8 @@ class Premise(TitledScene):
         self.play(Create(arrow_1), Create(arrow_2))
         self.wait()
 
-        for i in range(-7, 8):
-            for j in range(-6, 7):
+        for i in range(-50, 50):
+            for j in range(-25, 25):
                 lattice_1.add(Dot([i, j, 0]))
 
         self.play(Create(lattice_1))
@@ -102,3 +102,86 @@ class Premise(TitledScene):
             ]
         )
         self.wait()
+
+        self.play(
+            Transform(arrow_1, Arrow(ORIGIN, [2, 0, 0], buff=0)),
+            *[
+                Transform(
+                    dot,
+                    Dot(
+                        dot.get_center()
+                        * np.matrix([[1 / 2, 1 / 2, 0], [-1 / 2, 1 / 2, 0], [0, 0, 1]])
+                        * np.matrix([[2, 0, 0], [1, 1, 0], [0, 0, 1]])
+                    ),
+                )
+                for dot in lattice_1
+            ]
+        )
+        self.wait()
+
+
+class MatrixRep(TitledScene):
+    def construct(self):
+        self.add_title("Lattices")
+
+        matrix_comp = MathTex(
+            r"\begin{bmatrix}1 & 1\\ 1 & -1\end{bmatrix} \sim \begin{bmatrix}1 & 2\\ 1 & 0\end{bmatrix}"
+        )
+        lattice_comp = MathTex(
+            r"L\left(\begin{bmatrix}1 & 1\\ 1 & -1\end{bmatrix}\right) = L\left(\begin{bmatrix}1 & 2\\ 1 & 0\end{bmatrix}\right)"
+        )
+
+        self.play(Create(matrix_comp))
+        self.wait()
+
+        self.play(Transform(matrix_comp, lattice_comp))
+        self.wait()
+
+        self.play(ApplyMethod(matrix_comp.shift, 2 * UP))
+        self.wait()
+
+        l_def = MathTex(
+            r"L(\begin{bmatrix}\mathbf{b_1} & \mathbf{b_2}\end{bmatrix}) = \{ k_1 \mathbf{b_1} + k_2 \mathbf{b_2} \mid k_i \in \mathbb{Z} \}"
+        )
+
+        l_def.shift(DOWN)
+
+        self.play(Create(l_def))
+        self.wait()
+
+
+class OrthoDefect(TitledScene):
+    def construct(self):
+        text = Tex("How do we decide which bases are ``good''?", font_size=LARGE_FONT)
+
+        self.play(Create(text))
+        self.wait()
+
+        self.play(ApplyMethod(text.to_edge, UP))
+        self.wait()
+
+        diagram_1 = VGroup()
+        diagram_1.add(Dot(ORIGIN))
+        diagram_1.add(Arrow(ORIGIN, [1, 0, 0], buff=0))
+        diagram_1.add(Arrow(ORIGIN, [0, 1, 0], buff=0))
+
+        diagram_1.shift(2 * LEFT)
+        diagram_1.set_color(GREEN)
+        label_1 = Tex("Defect 1.0", font_size=MEDIUM_FONT)
+        label_1.next_to(diagram_1, DOWN)
+
+        diagram_2 = VGroup()
+        diagram_2.add(Dot(ORIGIN))
+        diagram_2.add(Arrow(ORIGIN, [0.25, 1, 0], buff=0))
+        diagram_2.add(Arrow(ORIGIN, [0.5, 1, 0], buff=0))
+
+        diagram_2.shift(2 * RIGHT)
+        diagram_2.set_color(RED)
+        label_2 = Tex(r"Defect $\approx$ 4.6", font_size=MEDIUM_FONT)
+        label_2.next_to(diagram_2, DOWN)
+
+        self.play(Create(diagram_1), Create(diagram_2))
+        self.wait()
+
+        self.play(Create(label_1), Create(label_2))
+        self.wait()