XtereO is a playful-learning app to practice stereographic projection of crystallographic models. This app is useful for students of Crystallography of different university degrees, especially Chemistry and Geology. Students consider Crystallography a difficult subject, for this reason, the authors intend XtereO to be a tool that helps users understand and study the subjects with crystallographic content in a fun way according to their interests in technology and video games.

Stereographic projection requires imagining or mentally visualizing reflections and rotations in three dimensions and representing these results in a 2D space, therefore, we think that XtereO is an adequate tool to improve the visuospatial skills, which are fundamental in the comprehension of many concepts within STEM disciplines.

We consider that the information collected in the "Theory" entry of the start menu, together with the instructions of the game, is enough to be able to play with the app without having received a deep instruction on Crystallography.

XtereO is a practical application on stereographic projection developed for Android devices.

The stereographic projection is used in crystallography and other scientific disciplines to represent in a 2D space the orientation of the planes and directions located in a 3D space.

XtereO is based on the application of stereographic projection for the representation of the shape and symmetry of crystals.

*With XtereO, geologists, crystallographers and students can exercise and practice easily with the stereographic projection*

Follow XtereeOApp Youtube channel for online resources and tutorials on stereographic projection and crystallography

Start XtereO choosing a specific System and Point Group.

Plot on the stereogram all symbols symmetrically related to the first one.

The empty sectors, the dashed and solid lines and their intersections, represent positions in which a symbol can be placed.

The symbol represents faces(polyhedron faces) located above the stereogram or vertical faces, and the symbol represents faces below the stereogram. In each position you can select , , or leave the position empty, through successive "clicks". The solid lines represent planes of symmetry

- The symbol represents a 2-fold rotation axis, which operates by turning 180º in the 3D space.

- The symbol represents a 4-fold rotation axis, which operates by turning 90º in the 3D space.

- The symbol represents a 3-fold rotation axis, which operates by turning 120º in the 3D space.

- The symbol represents a 6-fold rotation axis, which operates by turning 60º in the 3D space.

- The symbol represents a 4-fold rotoinversion axis, which operates by turning 90º in the 3D space, followed by an inversion through the center of the stereogram (the center of the polyhedron.

- The symbol represents a 3-fold rotoinversion axis, which operates by turning 120º in the 3D space, followed by an inversion through the center of the stereogram (the center of the polyhedron.

- The symbol represents a 6-fold rotoinversion axis, which operates by turning 60º in the 3D space, followed by an inversion through the center of the stereogram (the center of the polyhedron.

Once all the symmetry operations represented in the stereogram have been done, you can press ** "validate"** to check if the result is correct or incorrect.

The stereographic projection is used in Crystallography and other scientific disciplines to represent, in a 2D space, the orientation of planes and directions located in a 3D space. **XtereO** is based on the application of the 2D stereographic projection of the geometric shapes of 3D polyhedrons, taking into account their symmetry.

The stereographic projection of a polyhedron requires to imagine that it is contained in a sphere and that its center coincides with the center of the sphere. In this sphere, three axes are defined: vertical axis **c**, with the positive side up; transversal axis **b**, with the positive side to the right, and the front-to-back axis **a**, with the positive side facing forward.

The stereographic projection is a projection of points from the surface of a sphere onto its equatorial plane. We must imagine that normals to each faces intersect the surface of the sphere at points called face poles. The face normals can be considered to radiate from a single point at the centre. Faces whose normals intersect the upper hemisphere are projected onto the equatorial plane by drawing a line from the face pole to the “south pole” of the stereographic sphere. The intersection of this line with the equatorial plane of the sphere is the stereographic projection of the face. Those which intersect onto the lower hemisphere are projected by drawing a line to the “north pole”.

The vertical faces, parallel to the c axis, are projected on the equatorial circumference; a horizontal face is projected in the center and the other faces are projected in the interior in a position that depends on its orientation with respect to the axes. Symbol X is used for faces parallel to the c axis and those that cut it on the positive side and symbol O is used for the faces that cut the c axis on the negative side. The stereographic projection represents both the faces of the polyhedron and the elements of symmetry that it present.
**Symmetry elements**

Rotation axes are imaginary lines that cross a polyhedron passing through its center. When the polyhedron rotate around a rotation axis, all its geometric elements (faces, edges and vertices) are repeated at equal and fixed angular intervals that depend on the type of axis and the symmetry. **XtereO** includes symmetries with seven possible types of rotation axes:

- The **2-fold rotation** axes, which operate by turning 180° in the 3D space. They can coincide with the c axis or be perpendicular to it.

- The **4-fold rotation** axes, which operate by turning 90° in the 3D space and coincide with the c axis.

- The **3-fold rotation** axes, which operate by turning 120° in the 3D space and coincide with the c axis.

- The **6-fold rotation** axes, which operate by turning 60° in the 3D space and coincide with the c axis.

- The **4-fold rotoinversion** axes, which operate by turning 90° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis.

- The **3-fold rotoinversion** axes, which operate by turning 120° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis.

- The **6-fold rotoinversion** axes, which operate by turning 60° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis.

A **plane of symmetry** is an imaginary plane that bisects a polyhedron into halves that are mirror images of each other. They are represented with the letter **“m”** (for “mirror”). Symmetry planes in **XtereO** can be perpendicular to the **c** axis or be parallel to it (contain it). A plane of symmetry is drawn in a stereographic projection with a continuous line, while the surfaces in which there are no planes of symmetry are represented in a dashed line.

**Point Groups. **
The crystalline matter has a periodic and symmetrical internal structure that manifests externally generating polyhedral bodies. Up to 32 types of different combinations of rotation axes and symmetry planes can be identified in the polyhedra that represent crystals. These 32 combinations are called "Point Groups" and are grouped into 7 crystalline systems. All the symmetry elements of a Point Group coincide in a point, exactly the center of the crystal and, in stereographic projection, the center of the stereogram.

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