Symmetry of Solids

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Symmetry of Solids

• We consider the symmetry of some basic geometric
  solids (convex polyhedra).
• Important amongst these are the 5 Platonic solids
  (the only possible regular solids in 3D) ?
  Tetrahedron? Cube ? Octahedron (identical
  symmetry)? Dodecahedron ? Icosahedron (identical
  symmetry) The symbol ? implies the dual
  of.
• Only simple rotational symmetries are considered
  (roto-inversion axes are not shown).
• These symmetries are best understood by taking
  actual models in hand and looking at these
  symmetries.
• Certain semi-regular solids are also frequently
  encountered in the structure of materials (e.g.
  rhombic dodecahedron). Some of these can be
  obtained by the truncation (cutting the edges in
  a systematic manner) of the regular solids (e.g.
  Tetrakaidecahedron, cuboctahedron)

Regular solids are those with one type of
vertex, one type of edge and one type of face
(i.e. ever vertex is identical to every other
vertex, every edge is identical to every other
edge and every face is identical to every other
face)
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Symmetry of the Cube 4,3
4- fold axes pass through the opposite set of
face centres3 numbers
Pink2-fold
Blue3-fold
Yellow4-fold
The schläfli symbol for the cube is 4,3 ?
4-sided squares are put together in 3 numbers at
each vertex
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The body diagonals are 3-fold axes (actually a
?3 axis) 4 numbers
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2-fold axes pass through the centres of opposite
edges 6 numbers
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3 mirrors
Centre of inversion at the body centre of the cube
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• Important Note
• These are the symmetries of the cube (which are
  identical to those present in the cubic lattice)
• A crystal based on the cubic unit cell could have
  lower symmetry as well
• A crystal would be called a cubic crystal if the
  3-folds are NOT destroyed

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Symmetry of the Octahedron 3,4

• Octahedron has symmetry identical to that of the
  cube
• Octahedron is the dual of the cube (made by
  joining the faces of the cube as below)

3-fold is along centre of opposite faces 2-fold
is along centre of opposite edges 4-fold is along
centre of opposite vertices Centre of inversion
at the body centre
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Yellow4-fold
Pink2-fold
Blue3-fold
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Symmetry of the Tetrahedron 3,3
3-fold connects vertex to opposite face 2-fold
connects opposite edge centres
No centre of inversion No 4-fold axis
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Yellow3-fold
Pink2-fold
Blue3-fold
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Symmetry of the Dodecahedron 5,3
Yellow5-fold
Blue3-fold
Pink2-fold
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Symmetry of the Icosahedron 3,5
Yellow5-fold
Blue3-fold
Pink2-fold
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Truncated solids

• Certain semi-regular solids can be obtained by
  the truncation of the regular solids.
• Usually truncation implies cutting of all
  vertices in a systematic manner (identically)
• E.g. Tetrakaidecahedron, cuboctahedron can be
  obtained by the truncation of the cube. In these
  polyhedra the rotational symmetry axes are
  identical to that in the cube or octahedron.
• Tetrakaidecahedron 4,6,6 ? Two types of
  faces square and hexagonal faces? Two types of
  edges between square and hexagon between
  hexagon and hexagon
• Cuboctahedron 3,4,3,43,42? Two types of
  faces square and triangular faces

Cuboctahedron
Tetrakaidecahedron
Cuboctahedron formed by truncating a CCP crystal
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Cuboctahedron
Pink2-fold
Yellow4-fold
Blue3-fold
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Space filling solids

• Space filling solids are those which can
  monohedrally tile 3D space (i.e. can be put
  together to fill 3D space such that there is no
  overlaps or no gaps).
• In 2D the regular shapes which can monohedrally
  tile the plane are triangle 3, Square 4 and
  the hexagon 5.
• The non regular pentagon can tile the 2D plane
  monohedrally in many ways.
• The cube is an obvious space filling solid. None
  of the other platonic solids are space filling.
• The Tetrakaidecahedron and the Rhombic
  Dodecahedron are examples of semi-regular space
  filling solids.

Tetrahedral configuration formed out the space
filling units
Video Space filling in 2D
Video Space filling in 3D
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