Algebraic (Group Theory). Jim Morey. Cognitive Engineering Lab. Department of Computer Science. The University of Western Ontario www.csd.uwo.ca/~morey/ ...
Representation and History: Algebraic (Group Theory)
Jim Morey Cognitive Engineering Lab Department of Computer Science The University of Western Ontario www.csd.uwo.ca/~morey/CogEng
Outline Ê Ê
Regularity Transformations of Space • Chemistry description • Matrix description • Strip patterns
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Groups • Actions • Orbits • Structure • Subgroups, cosets
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Tilings Bravais Lattices Nets 2
Regularity
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Artifacts with indistinguishable features 3
Transformations of Space Ê
Chemistry description (physical) • • • • • •
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Translation Reflection Rotation Glide (translations + reflection) Screw (translations + rotation) Inversion
Matrix description (notational) • Orthogonal matrix + translation (affine transformation) • An operational way to compose transformations
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Sets of Transformations
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Regularities in the set transformations have certain properties: • • • •
Composing transformations creates only transformations in the set Composing transformations is associative (matrix property) There is an identity transformation. For every transformation, there is a inverse transformation, which undoes the original
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Strip patterns (Friezes)
Ê Ê Ê Ê Ê Ê Ê Ê
(p11) ppppppppppp (p1g) pbpbpbpbpbp (pm1) bdbdbdbdbdb (p12) bqbqbqbqbqb (pmg) bdpqbdpqbdpq (p1m) EEEEEEEEE (pmm) XXXXXXXXX
Iv + (1 0) Av + (1 0) Bv, Iv + (1 0) Cv, Iv + (1 0) Cv, Av + (1 0) Av, Iv + (1 0) Av, Bv, Iv + (1 0)
Why not three-fold symmetry?
I=
1 0 0 1
A=
1 0 0 -1
B=
-1 0 0 1
C=
-1 0 0 -1
▲▲▲▲▲▲▲▲
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Affine Transformations Ê
Why represent the transformation like this? Av + (1 0)
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Why not use an extra dumby coordinate? M (vx vy 1)
M=
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1 0 0 0 1 0 1 0 1
What is M-1?
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Groups Ê
A group is an abstract structure with the desired properties
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Definition: A set with an operation denoted (G,*) is a called a group iff • (Closure) For all x and y in G, x*y is also in G • (Associative) For all x,y, and z in G, x*(y*z) = (x*y)*z • (Identity) There exists an identity I s.t. for all x in G, x*I = I*x = x • (Inverses) For all x in G, there exists x-1 s.t. x* x-1 = x-1 *x = x
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Every group can be represented as a matrix group • For instance permutation groups and abstract groups 8
Example Groups Ê
Finite •
•
•
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Dihedral groups, D2n • Symmetries of regular n-polygons – Can be generated by two reflections • Symmetries of flowers Cyclic groups, Zn (integer addition mod n) • Rotational symmetries of regular n-polygons – Can be generated by one rotation • Symmetries of propellers Symmetry groups of the Platonic solids • Can be generated by three reflections
Infinite •
•
Z (integer addition) • Symmetry of an infinite row of F’s – Generated by one translation ZxZ (integer coordinates) • Symmetry of a grid of F’s – Generated by two translations 9
Group Actions Ê
If a group G acts on a set A, then each g in G defines a map, mg: a → ga s.t. mg permutes the elements of A
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Implicit in matrix groups is the idea that the group acts on vectors
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Oa, the orbit of G containing a (in A), is {ga | g in G}
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The orbits are useful to consider rather than the whole space.
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Orbits of D10 Ê
For D10, the size of an orbit of a single point depends on the point. • What are the possible sizes of orbits? • Where are the different points located?
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A region of space, called the fundamental region, can be identified with the identity. The other regions of space are then associated with images of the fundamental region.
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The dark face lies in the fundamental region which is defined by the two reflections 11
Group Structure Ê
Some subsets of group elements themselves form groups. These subsets are called subgroups. • What properties need to be checked?
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Each subgroup partitions the group into sets, called cosets, that are similar to the subgroup but do not contain the identity element.
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The subgroups are means of understanding the group.
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Visualizing Cosets Ê Ê
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The orbit of the darkened point form the vertices of the shape The right angled triangle with the darkened point is the region that represents the identity The three sides of the region represent the planes of reflections that generate the symmetry group of the cube Any two reflections planes generate a subgroup Each subgroup corresponds to a face that includes the darkened point The other faces represents cosets of the like faces
Archimedean Kaleidoscope 13
Constraints of Subgroups Ê
The groups are constrained by the coset structure.
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This can be seen in the possible sizes of finite groups • The size of every subgroup divides the size of the group
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Another perspective is to combine groups to form bigger groups (the Hölder program) • Relationships between subgroups and groups • Normal subgroups, simple groups • Simple groups have no normal subgroups (only 1 and G) • Direct products and semi-direct products
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Tiling groups Ê Ê
The constraints of the plane and the subgroups limit the possibilities of groups Seventeen plane group: • Chemistry: (p stands for primitive, c stands for centered) • Oblique (point groups: 1,2) – p1, p2 • Rectangular (point groups: 1m, 2mm) – pm, pg, cm, p2mm, p2mg, p2gg, c2mm • Square (point groups: 4, 4mm) – p4, p4mm, p4gm • Hexagonal (point groups: 3, 3m, 6, 6m) – p3, p3m1, p31m, p6, p6mm
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Identifying Plane Groups
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In theory, it should be easy to correctly identify which symmetry group a particular pattern has but even the experts have a difficult time. (Islamic tilings in Alhambra, Grϋnbaum )
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What are the symmetry groups do these patterns exhibit?
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Fourteen Bravais Lattices Ê
Cubic • Simple, body centered, face centered
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Tetragonal (cubic stretched in one direction) • Simple, body centered
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Orthorhombic (cubic stretched in two direction) • Simple, body centered, end centered, face centered
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Monoclinic (cubic stretched in two direction and a sheer) • Simple, end centered
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Triclinic (cubic stretched in two direction and two sheers) Rhombohedral (cubic with two sheers) Hexagonal (stacked hexagonal prisms) 17
Cubic, BCC, and FCC Ê
Connecting nearest vertices • (left) simple cubic • Can see cubes • (middle) body centered cubic • Can see a rhombic dodecahedron • (right) face centered cubic • Can see tetrahedrons, octahedrons, and cuboctahedrons
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230 Space Groups Ê
The (possibly distorted) cube or hexagonal prism of a Bravias lattice is called the unit cell
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The unit cell is used as a framework for describing other symmetries of a crystal lattice, as well as, the positions of the atoms and the bonds
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Exhaustively considering symmetries within the unit cell and among the unit cells, generate 230 space groups
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Nets (another perspective) Ê
Structures viewed as infinite periodic graph realizations • vertices and edges
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Important features: • Degree of vertices (4 regular graphs are called 4-connected nets) • Circuits, rings, Schläfli symbols
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The language helps: • Articulate local properties • Bring graph techniques to chemistry
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Simple cubic is a 6-connected net BCC is an 8-connected net FCC is a 12-connected net
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References Ê Ê Ê
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Dummit DS, Foote RM, (1991) Abstract Algebra, Prentice Hall, New Jersey. Grünbaum B, Shephard GC (1987), Tilings and Patterns: An Introduction, Freeman, New York. Grünbaum B (1984). The emperor’s new clothes: full regalia, g-string, or nothing? Mathematical Intelligencer, 6(4). Morey J, Sedig K (2004). Archimedean Kaleidoscope: A Cognitive Tool to Support Thinking and Reasoning about Geometric Solids, Geometric Modeling: Techniques, Applications, Systems and Tools, Editor: M. Sarfraz, Kluwer Academic Publisher. O'Keeffe M, Hyde BG (1996). Crystal Structures: I Patterns and Symmetry, Min. Soc. Am., Washingon D.C. 21
