Crystals and geometry

The crystals grow like convex polyedra. But the shapes of crystals are limited because they can have only particular axial symmetries.

In a molecule, or in a complex ione, or in a crystal, the atoms around an other make up a polyedric coordination group, the number of vertices gives the coordination number (c.n) of the atom.

The coordination polyedra with triangular faces are the most commun because they constitute the more solid structure.

Tetrahedron and octahedron are the most important coordination polyedra because their vertices have the most symmetric dispositions of 4 or 6 points around a central point.

A short introduction about polyedra geometry.

The Ancient Greeks had a good knolewdge of solids, but after the Euler's book (1758) "Elementa doctrinae solidorum", the mathematicians went into the polyedra properties in depth.

The Euler's relationship, beetwen number of vertices (N0), of sides (N1), and of faces  (N2)

N0 + N2  = N1 + 2

The Platonic solids are those polyedra in that all the faces are constituted by regular polygons and the same number of sides (p) gathers in every vertix: tetrahedron (p=3), cube (p=3), dodecahedron (p=3), octahedron (p=4), icosahedron (p=5).The symbol (n,p) or (np) is more full, it shows a solid with p faces of n sides that meet in every vertix.

We can meet all the regular solids in the structuristic chemistry, but the most importan are the tetrahedron and the octahedron. The cube is the coordination polyhedron in CsCl (both ions) and CaF2 (only Ca2+). The icosahedron is present in these complexes: Ce(NO3)63-, Fe3(CO)12 , Co4(CO)12

The Platonic solids

 

n, p

  Vertices (N0)

Sides (N1)

Faces (N2)

Dihedral angles   

Tetrahedron
3,3
4
6
4
70° 31'

Cube
3,4
6
12
8
109° 28'

Octahedron
4,3
8
12
6
90°

Dodecahedron
5,3
20
30
12
116° 34'

Icosahedron
3,5
12
30
20
138° 12'

The Archimedean solids, prisms, anti-prisms

 Symbol Name
faces
vertices
sides
1
 3,62 truncated tetrahedron
8
12
18
2
 3,82 truncated cube
14
24
36
3
 4,62 truncated octahedron
14
24
36
4
 32,42   cuboctahedron
14
12
24
5
 4,6,8 truncated cuboctahedron
26
48
72
6
 3,43 rhombicuboctahedron
26
24
48
7
 34,4 snub cube 
38
24
60
8
 3,102 truncated dodecahedron
32
60
90
9
 32,52 icosidodecahedron
32
30
60
10
 5,62 truncated icosahedron
32
60
90
11
 4,6,10  truncated icosidodecahedron
62
120
180
12
 3,4,5,4  rhombicosidodecahedron
62
60
120
13
 34,5  snub dodecahedron
92
60
150
14
 n,42  regular prisms
n + 2
2n
3n
15
 n,33  regular Anti prisms 
2n + 2
2n
4n

While the Platonic solida have equivalent all the vertices and equal all the faces, the Archimedean solids have equivalent all the vertices but not egual all the faces; they have some different faces. (table) The first column gives the kinds of polygons.

In the table there are three kinds of solids:

  1. The Archimedian solids (13), only teh first are important in chemistry;
  2. The regular prisms;
  3. The regular anti-prisms.

Text from Italian site: http://www.chimdocet.it/index.htm

Maggio 2022 - Contatti - Graziano Scotto di Clemente