Math 1230, Notes 6 Sep. 9 , 2014 Math 1230, Notes 6 Sep. 9 , 2014 1 / 1 Does the set of reflections of a triangle form a group by itself? Does the set of reflections of a tetrahedron form a group by itself? Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron. Questions to think about. Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Does the set of reflections of a tetrahedron form a group by itself? Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron. Questions to think about. Does the set of reflections of a triangle form a group by itself? Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron. Questions to think about. Does the set of reflections of a triangle form a group by itself? Does the set of reflections of a tetrahedron form a group by itself? Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Questions to think about. Does the set of reflections of a triangle form a group by itself? Does the set of reflections of a tetrahedron form a group by itself? Glue two copies of a regular tetrahedron together so that they have a face in common, and work out all the rotational symmetries of this new solid. Compare with an octahedron. Math 1230, Notes 6 Sep. 9 , 2014 2 / 1 Definition: A crystal is a solid in which the atoms form a periodic structure. SYMMETRY AND ITS RELATION TO CRYSTALLOGRAPHY Math 1230, Notes 6 Sep. 9 , 2014 3 / 1 SYMMETRY AND ITS RELATION TO CRYSTALLOGRAPHY Definition: A crystal is a solid in which the atoms form a periodic structure. Math 1230, Notes 6 Sep. 9 , 2014 3 / 1 Different kinds found in crystals: translational, rotational, reflectional A lattice of points form a periodic structure" if and only if it exhibits translational symmetry. : Key concept for studying crystals: SYMMETRY Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 A lattice of points form a periodic structure" if and only if it exhibits translational symmetry. : Key concept for studying crystals: SYMMETRY Different kinds found in crystals: translational, rotational, reflectional Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 Key concept for studying crystals: SYMMETRY Different kinds found in crystals: translational, rotational, reflectional A lattice of points form a periodic structure" if and only if it exhibits translational symmetry. : Math 1230, Notes 6 Sep. 9 , 2014 4 / 1 translational symmetry (all points move) Math 1230, Notes 6 Sep. 9 , 2014 5 / 1 translational symmetry (all points move) Math 1230, Notes 6 Sep. 9 , 2014 6 / 1 translational symmetry (all points move) Math 1230, Notes 6 Sep. 9 , 2014 7 / 1 translational symmetry (all points move) Math 1230, Notes 6 Sep. 9 , 2014 8 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 9 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 10 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 11 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 12 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 13 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 14 / 1 Rotational Symmetry 6 fold rotational symmetry around a fixed point. Math 1230, Notes 6 Sep. 9 , 2014 15 / 1 Reflectional Symmetry Reflection across a line Math 1230, Notes 6 Sep. 9 , 2014 16 / 1 Reflectional Symmetry Reflection across a line Math 1230, Notes 6 Sep. 9 , 2014 17 / 1 Reflectional Symmetry Reflection across a line Math 1230, Notes 6 Sep. 9 , 2014 18 / 1 Reflection through a point Math 1230, Notes 6 Sep. 9 , 2014 19 / 1 Reflection through a point Math 1230, Notes 6 Sep. 9 , 2014 20 / 1 Reflection through a point Math 1230, Notes 6 Sep. 9 , 2014 21 / 1 Reflection through a point Math 1230, Notes 6 Sep. 9 , 2014 22 / 1 Reflection through a point Math 1230, Notes 6 Sep. 9 , 2014 23 / 1 Applications to Crystallography Definition: A crystal is a substance in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating three-dimensional pattern. (From \About.com Chemistry") Math 1230, Notes 6 Sep. 9 , 2014 24 / 1 Theorem: Suppose that a rotation by an angle a leaves the lattice of 2p points unchanged. Then a must be n for some n 2 f3, 4, 6g . In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry. For example, a lattice cannot have five-fold rotational symmetry. (Coxeter, An Introduction to Geometry, 1989) Corollary: Dodecahedra and Icosahedra cannot be part of a crystal. Crystallographic Restriction Theorem: Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged. Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 (Coxeter, An Introduction to Geometry, 1989) Corollary: Dodecahedra and Icosahedra cannot be part of a crystal. Crystallographic Restriction Theorem: Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged. Theorem: Suppose that a rotation by an angle a leaves the lattice of 2p points unchanged. Then a must be n for some n 2 f3, 4, 6g . In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry. For example, a lattice cannot have five-fold rotational symmetry. Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Corollary: Dodecahedra and Icosahedra cannot be part of a crystal. Crystallographic Restriction Theorem: Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged. Theorem: Suppose that a rotation by an angle a leaves the lattice of 2p points unchanged. Then a must be n for some n 2 f3, 4, 6g . In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry. For example, a lattice cannot have five-fold rotational symmetry. (Coxeter, An Introduction to Geometry, 1989) Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Crystallographic Restriction Theorem: Definition: A lattice is an infinite set of points in Rn such that for some non-zero vector v, translation of all the points by v leaves the set of points unchanged. Theorem: Suppose that a rotation by an angle a leaves the lattice of 2p points unchanged. Then a must be n for some n 2 f3, 4, 6g . In other words, a crystal can only have 3, 4, or 6 fold rotational symmetry. For example, a lattice cannot have five-fold rotational symmetry. (Coxeter, An Introduction to Geometry, 1989) Corollary: Dodecahedra and Icosahedra cannot be part of a crystal. Math 1230, Notes 6 Sep. 9 , 2014 25 / 1 Proof: We can assume that 0 < a < 2p. If a rotation by a leaves the lattice unchanged, then rotations by 2a, 3a, etc. also leave it unchanged. In particular, let the rotations be around p, and rotate the point q = p + v. Let r be the rotation around p. The points q, rq, r 2q, etc. all lie on a circle around p. 2p n Case (1); a = n for some positive integer n. Then r q = q. The points q, rq, ..., r n−1q are vertices of a regular n -gon. By standard Euclidean geometry we find that sum of the interior angles (all equal) of this n - gon is (n − 2) p. So each individual interior angle is (n−2)p n . On the other hand, treating the edges as vectors shows that 2p interior angle must be some fraction m . But the only values of n and m which can satisfy this are n = 3, m = 6, n = 4, m = 4 and n = 6, m = 3. Theorem: Suppose that a rotation by an angle a leaves the lattice of points unchanged. Choose the smallest possible a for which this is 2p true. Then a must be n for some n 2 f3, 4, 6g . Math 1230, Notes 6 Sep. 9 , 2014 26 / 1 2p n Case (1); a = n for some positive integer n. Then r q = q. The points q, rq, ..., r n−1q are vertices of a regular n -gon. By standard Euclidean geometry we find that sum of the interior angles (all equal) of this n - gon is (n − 2) p. So each individual interior angle is (n−2)p n . On the other hand, treating the edges as vectors shows that 2p interior angle must be some fraction m . But the only values of n and m which can satisfy this are n = 3, m = 6, n = 4, m = 4 and n = 6, m = 3. Theorem: Suppose that a rotation by an angle a leaves the lattice of points unchanged. Choose the smallest possible a for which this is 2p true.