The mineral Andalusite belongs to a mathematically unique space group, centrosymmetric, and orthorhombic – a three-dimensional geometric arrangement having three unequal axes at right angles.

The symmetry elements of these operations all pass through a single point of the object. The rules that govern symmetry are found in the mathematics of group theory. Group theory addresses the way in which a certain collection of mathematical “objects” are related to each other. Such groups always have the following properties: identity, inverse, associativity, closure, and commutativity. [1].

Beryl is a relatively rare hexagonal, a six-sided polygon silicate mineral.

In Elements (proposition IV.15), Euclid showed how to inscribe a regular hexagon in a circle. To construct a regular hexagon with a compass and straightedge, draw an initial circle A. Picking any point on the circle as the center, draw another circle B of the same radius. From the two points of intersection, draw circles C and D. Finally, draw E centered on the intersection of circles A and C. The six circle-circle intersections then determine the vertices of a regular hexagon [1].

Chrysoberyl belongs to the orthorhombic system. Orthorhombic Space groups are divided into 3 main groups because of the large number of ways in which twofold rotation, two-one screw axes, mirror and glide planes can interact together in three dimensions.

The corundum crystal looks like a bipyramidal hexagon wider in the center and tapers thinly on the ends. It belongs to the Trigonal system and has a symmetry of R -3 c, arranged in rhombohedral and hexagonal axes.

The diamond crystal belongs to the cubic system and has the shape of an octahedron, a polyhedron with eight faces, twelve edges, and six vertices.

A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex.

Elbaite and other tourmaline minerals crystallize in the trigonal system, a subsystem of the hexagonal system with three-fold symmetry and four crystal axes. Its six-sided geometry is modified in the cross section of the crystals, and three alternating prism edges are rounded to resemble a triangle [1]

Fluorite is an isometric, octahedral, colorless, soft, and almost translucent crystal.

Its hexoctahedral structure–48 equal triangular faces, makes of this mineral one of the most symmetrical geometric solid found in three-dimensional space

Cordierite is a tetrahedral framework silicate that belongs to the dipyramidal, orthorhombic family. It is pseudohexagonal, meaning its geometry is slightly deformed when compared to higher symmetry. Its mesh has a geometry more regular than required by the symmetry of the pattern [1]

Muscovite is a transparent monoclinic crystal where two vectors are perpendicular to each other, and the third vector meets the other two at an angle other than 90 degrees.

The tanzanite is part of the orthorhombic group–a three-dimensional geometric arrangement having three unequal axes at a right angle.

The Tanzanite crystal has a dipyramidal, Pnma symmetry. It is centrosymmetric with 8 points of inversion per unit cell

The vertices in this composition show the atomic arrangement of the crystal. The mesh in the background reflects the binding between the vertices in a three-dimensional model.