1. Point groups

2. Monoclinic point groups I
Symmetry element: two-fold axis parallel to b
Characteristic forms: Pedions, sphenoids b
stereographic projection c b a
monoclinic sphenoid

3. Monoclinic point groups II
Point group: m
Symmetry element: mirror perpendicular to b
Characteristic forms: Pedions, pinacoidsdomes b
stereographic projection c b a
monoclinic dome

4. Monoclinic point groups III
Example point group: 2/m
Symmetry element: two-fold axis parallel to b, mirror perpendicular to b, inversion center.
Characteristic forms: Pinacoids, prisms b
stereographic projection c b a
monoclinic
prism

5. Monoclinic minerals
Transparent, bluish crystal aggregate of vivianite on matrix. Santa Eulalia, Chihuahua, Mexico.
Several twinned (two crystals “glued” together in an oriented fashion) green crystals of titanite (sphene) on matrix. Capelinha, Minas Gerais, Brazil. 5.5x5x3.5 cm.
Faces of the monoclinic prism

6. Tetragonal point groups I
Example point group:422
Symmetry element: four-fold axis parallel to the c axis and 2 sets of 2 two-fold axis perpendicular to it.
Characteristic forms: trapezohedron
Other forms: pinacoid, tetragonal prism ditetragonal prisms, tetragonal dipyramid, and ditetragonal dipyramids
h k l a2 a2 a1 a1
tetragonal trapezohedron

7. Tetragonal point groups II
Example point group: 4/m2/m2/m
Symmetry element: four-fold axis parallel to the c axis, 2 sets of mirrors parallel to it, a mirror and 2 sets of two-fold axes perpendicular to it.
Characteristic forms: ditetragonal dipyramid
Other forms: pinacoid, tetragonal prisms ditetragonal prism, tetragonal dipyramids c a2 a1 a1 a2
ditetragonal dipyramid

8. Tetragonal crystals Narsarsukite Na2(Ti,Fe)Si2 (O,F)11
Mont Saint-Hilaire, Quebec, Canada.
Point group: 4/m2/m2/m
Mackayite FeTe2 O5(OH)
McGuinnity shaft, Goldfield, Esmeralda County, Nevada, USA. Point group: 4/m2/m2/m
Schematic
projection of the
narsarsukite crystal

9. Tetragonal crystals
Zircon ZrSiO4 Finnmark, Norway. Crystal diameter 6 cm. Point group 4/m2/m2/m. The crystal shape is composed of a dipyramid and a prism.
Scheelite CaWO4 Traversella, Piemont, Italy. Crystal diameter 4.5 cm. Point group 4/m. The crystal shape is composed of a dipyramid and if only the macroscopic shape is taken the resulting point group would be the same as for zircon. The lowering of the point group is due to the microscopic symmetry.

10. Triclinic, monoclinic and orthorhombic forms
Dihedrons
Pedion
Pinacoid
Dome
Sphenoid
Orthorhombic prism
Orthorhombic pyramid
Orthorhombic dipyramid
Orthorhombic disphenoid

11. Tetragonal, trigonal and hexagonal prisms

12. Tetragonal, trigonal and hexagonal pyramids

13. Tetragonal, trigonal and hexagonal dipyramids

14. Tetragonal, trigonal and hexagonal salenohedron and trapezohedrons
Tetragonal trapezohedron
Tetragonal disphenoid
Dietragonal scalenohedron
Trigonal disphenoid
Trigonal trapezohedron
Hexagonal
scalenohedron
Hexagonal
trapezohedron

15. Cubic point groups a2 a3 a1 Example point group: 4/m32/m
Symmetry element: three four-fold axes parallel to the coordinate axes, four three-fold rotoinversion axes parallel to room diagonal directions, six two-fold axes, mirrors perpendicular to all two- and four-fold rotation axes
Characteristic forms: cube, octahedron, dodecahedron, tetrahexahedron, trapezohedron, trisoctahedron , hexoctahedron h k l, (s. sep. page) a3 a2 a1

16. Isometric (cubic) crystal form I
cube tetrahedron octahedron dodecahedron
Trisoctahedron tetrahexahedron trapezohedrons

17. Isometric (cubic) crystal form II
hexoctahedron tristetrahedron hextetrahedron deltoid dodecahedron
gyroids pentagondodecahedron diploid or pyritohedron

18. Cubic crystals I
Spessartine garnet Mn3Al2Si3O12. Haramosh Mountains Baltistan, Northern Areas, Pakistan. The crystal shape is a trapezohedron.
Diamonds, C, South Africa. Point group: 4/m3bar2/m. Average crystal size: 1cm. Crystal shape: top row and first crystal in the bottom row: octahedron; middle crystal in bottom row: tetrahedron,; third crystal bottom row: two intergrown octahedra.

19. Cubic crystals II
Grossular garnet Ca3Al2Si3O12, Jeffrey Quarry, Asbestos, Quebec, Canada. Crystal size: 12 mm. Point group: 4/m3bar2/m. The crystal shape is a dodecahedron. The faces show growth steps.
Pyrite FeS2 encased in a very fine-grained sedimentary marlstone (clay and calcite). Navajun-(La Rioja)-Spain. Crystal size: 1-2cm. Point group: 2/m3bar. The striation on the cube faces indicate that the microsciopic symmetry is less than 4/m3bar2/m

20. Steno's law
One property of crystals, that remains constant in all specimens, is the angle between two defined faces. The law of contant interfacial angles is called Steno’s law. The presence or absence of specific faces and the relative size of the indivual faces, however, can be different from specimen to specimen of the same crystal. Shapes typical for a mineral (and location) are called habitus.
face missing
The first four shapes are hexagons, with an angle of 120° between adjacent faces. Different is the size of each of the individual faces. The last shape is a pentagon, which can be interpreted as hexagon with a missing face.

21. Quartz crystals
Faces (forms) which may be developped are: two trigonal prisms = hexagonal prism (m1, m2), the principal rhombohedron r, the inverse rhombohedron z, a trigonal bipyramid s and a trigonal trapezohedron x, an elongated (acute) rhombohedron (green, f), an acute inverse rhombohedron (yellow, -f), and an acute (negative) trapezohedron (violet, u). The number of forms and their relative size (=habitus) differs according to the physico-chemical conditions in which the crystals were formed.

22. Habitus in quartz Smoky quartz with prismatic habitus
The forms r, z and m1 and m2 are developped (Dôme de Miage)
Citrine (yellow quartz) with Tessin habitus
The acute rhombohedron f is strongly developped, the rhombohedra r and z are strongly reduced (Arbola).