SLIDE 1 Crystallography/Wallpaper Groups/Plane Isometries
Mackenzie Pazos, Matthew Eliot, Brendon LeLievre
SLIDE 2 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
SLIDE 3 Plane Isometries
⍺ : C→C is a isometry for any two points a and b such that
⎹ ⍺(a) - ⍺(b) ⎸= ⎹ a - b ⎸
SLIDE 4 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
SLIDE 5 Crystal Systems
- Symmetry of a periodic pattern (repeated unit cells) of
repeated element or “motifs” is a set of symmetry
- perators 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
SLIDE 6 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
SLIDE 7 Rotations
- 1 Fold (360)
- 2 Fold (180)
- 3 Fold (120)
- 4 Fold (90)
- 6 Fold (60)
SLIDE 8 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
SLIDE 9
The Crystal Systems
SLIDE 10 The Crystal Systems
- 1. Cubic (isometric) Systems
- 2. Tetrahedral System
- 3. Hexagonal System
- 4. Orthorhombic System
- 5. Monoclinic System
- 6. Triclinic System
SLIDE 11
Crystallographic Axes
SLIDE 12 Wallpaper Groups
- A wallpaper group is a mathematical classification of
a two dimensional repetitive pattern, based on the symmetries of the pattern
SLIDE 13 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
SLIDE 14 Lattices
- Basis; the lattice generates everywhere the image can move to
- 5 different lattice groups:
- square
- parallelogram
- rhombic
- rectangular
- hexagonal
SLIDE 15
Lattices
SLIDE 16
Characteristics of the 17 Groups
SLIDE 17
Group 1
Mediaeval Wallpaper
SLIDE 18
Group 2
Ceiling of Egyptian Tomb
SLIDE 19
Group 3
Ancient Indian Metalwork
SLIDE 20
Group 4
Sidewalk
SLIDE 21
Group 5
Bronze Cast in Assyria
SLIDE 22
Group 6
Egyptian Mummy Case
SLIDE 23
Group 7
Floor Tiling in Prague
SLIDE 24
Group 8
Pavement in Hungary
SLIDE 25
Group 9
Persian Tapestry
SLIDE 26
Group 10
Renaissance Tiling
SLIDE 27
Group 11
Storm Drain
SLIDE 28
Group 12
Chinese Painting
SLIDE 29
Group 13
Road in Poland
SLIDE 30
Group 14
Persian Tile
SLIDE 31
Group 15
Chinese Painting
SLIDE 32
Group 16
Spanish Wall Tiles
SLIDE 33
Group 17
Kings Dress Assyria
SLIDE 34 Parallelogram
Rectangular Hexagonal Rhombic Square
