Upright square tiling. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale √2 times as large as the upright square lattice of vertices. The vertices of all squares together with their centers form a diagonal square lattice which is in linear scale √2 times as small as the upright square lattice of vertices.

The square lattice is one of the five 2D lattice types. Image File history File links Tile4444bc. ... Image File history File links Tile4444bc. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... See lattice for other meanings of this term, both within and without mathematics. ...

Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as "upright square lattice" and "diagonal square lattice". They differ by an angle of 45°.

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Upright square lattice and diagonal square lattice. (Depending on the browser these may look rectangular and rhombic, respectively.)

Its symmetry category is wallpaper group p4m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice we have a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice. A pattern with 4-fold rotational symmetry has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of translational symmetry. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... In physics and mathematics, translational symmetry is the invariance of an object or a system of equations under the translations - operations that change the coordinates of all objects by a constant. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... In physics and mathematics, translational symmetry is the invariance of an object or a system of equations under the translations - operations that change the coordinates of all objects by a constant. ...

With respect to reflection axes there are three possibilities:

• None. This is wallpaper group p4.
• In four directions. This is wallpaper group p4m.
• In two perpendicular directions. This is wallpaper group p4g. The points of intersection of the reflexion axes form a square grid which is as fine as, and oriented the same as, the square lattice of 4-fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection axes.

Wallpaper group p4, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for p4g and p4m). A fundamental domain is indicated in yellow.

Wallpaper group p4g. There are reflection axes in two directions, not through the 4-fold rotocenters.

Wallpaper group p4m. There are reflection axes in four directions, through the 4-fold rotocenters. In two directions the reflection axes are oriented the same as, and as dense as, those for p4g, but shifted. In the other two directions they are linearly a factor √2 denser.

Image File history File links File links The following pages link to this file: Symmetry User:Patrick Wallpaper group ... Image File history File links File links The following pages link to this file: Symmetry User:Patrick Wallpaper group ... Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... Image File history File links File links The following pages link to this file: Symmetry User:Patrick Wallpaper group ... Image File history File links File links The following pages link to this file: Symmetry User:Patrick Wallpaper group ... Image File history File links http://commons. ... Image File history File links http://commons. ...

See also

• square tiling
• hexagonal lattice
• symmetry combinations
• centered square number