Crystallography/Wallpaper Groups/Plane Isometries Mackenzie Pazos, Matthew Eliot, Brendon LeLievre
Plane Isometries • A transformation of a plane that preserves distance • moves the plane but does not change • 4 types of transformations • Translation • Rotation • Reflection • Glide Reflection
Plane Isometries ⍺ : C → C is a isometry for any two points a and b such that ⎹ ⍺ ( a ) - ⍺ ( b ) ⎸ = ⎹ a - b ⎸
Crystallography • A branch of science concerned with the structure and properties of crystals • Unit Cell — smallest unit volume that permits identical cells that will fill the space • Crystal Lattice is constructed by repeating the unit cell in all directions
Crystal Systems • Symmetry of a periodic pattern (repeated unit cells) of repeated element or “motifs” is a set of symmetry operators allowed by the pattern • The total sets of symmetry operations applicable to the pattern is the pattern symmetry and is mathematically described as a space group • Space group of a crystal describes the symmetries of that crystal which is an important aspect of that crystals internal structure
4 Operations of Crystallography • Reflection on a point (inversion) — Centre of Symmetry • Reflection in a Plane — Mirror Symmetry • Rotation about a Imaginary Axis — Rotational Symmetry • Rotation and After it Inversion — Roto-Inversion
Rotations • 1 Fold (360) • 2 Fold (180) • 3 Fold (120) • 4 Fold (90) • 6 Fold (60)
The Crystal Systems • ALL classes (rotations/reflection) combine with the folds to make specific crystals: 32 Crystal Symmetry Classes • Example: Highest Symmetrical Crystal • Three 4 fold rotation axes • Six 2 fold rotation axes • Six secondary mirror planes • Four 3 fold rotation axes • Three primary mirror planes • Centre of symmetry
The Crystal Systems
The Crystal Systems 1. Cubic (isometric) Systems 2. Tetrahedral System 3. Hexagonal System 4. Orthorhombic System 5. Monoclinic System 6. Triclinic System
Crystallographic Axes
Wallpaper Groups • A wallpaper group is a mathematical classification of a two dimensional repetitive pattern, based on the symmetries of the pattern • 17 groups total
Mathematically, a wallpaper group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Frieze Wallpaper
Lattices • Basis; the lattice generates everywhere the image can move to • 5 different lattice groups: • square • parallelogram • rhombic • rectangular • hexagonal
Lattices
Characteristics of the 17 Groups
Group 1 Mediaeval Wallpaper
Group 2 Ceiling of Egyptian Tomb
Group 3 Ancient Indian Metalwork
Group 4 Sidewalk
Group 5 Bronze Cast in Assyria
Group 6 Egyptian Mummy Case
Group 7 Floor Tiling in Prague
Group 8 Pavement in Hungary
Group 9 Persian Tapestry
Group 10 Renaissance Tiling
Group 11 Storm Drain
Group 12 Chinese Painting
Group 13 Road in Poland
Group 14 Persian Tile
Group 15 Chinese Painting
Group 16 Spanish Wall Tiles
Group 17 Kings Dress Assyria
Rectangular Parallelogram Square Rhombic Hexagonal
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