The letter L may be tessellated in a variety of ways, as seen by the number of sites dedicated to it. This book's tessellation designs are all upper-case letters. Lower-case letters tessellate as well, and some of their conceivable forms are shown in the typeface selected for this introduction. A lower-case l is identical in every way to an uppercase L, except that it does not need to be filled.

Upper-case Ls are used in logos and other graphic design work because they make excellent shapes that catch the eye. They have two equal sides and two equal angles, so they are symmetrical. When you see an upper-case L, you should think symmetry!

Lower-case l's are used when strict symmetry isn't necessary, such as with drawings or handwritten notes. They have three equal sides and one unequal angle, so they're asymmetrical. This makes them unique, interesting, and attractive.

There are eight different ways letters can tessellate: AAA, BBB, CCC, DDD, EEE, FFF, GGG. There are also four ways spaces can tessellate: AAA, BBB, OOO, and VVV. Finally, there are three combinations: AAO, BBO, and COG. That's 10 total combinations. If you include **rotated versions** of **these patterns**, there are actually 11 total.

Tessellations can be seen in a variety of contexts. Tesellations present in **our everyday environment** may be found in art, architecture, hobbies, and many other areas. Oriental carpets, quilts, origami, Islamic architecture, and M. C. Escher are some specific examples. Tessellations also play **an important role** in mathematics and physics. They appear in geometry as the solution to various problems such as triangles, polygons, and surfaces.

Tessellations can be defined as the geometric patterns formed by interlocking identical polygonal faces without overlapping any vertices or edges. There are several types of tessellations including triangular, hexagonal, pentagonal, and others. A triangular tessellation is one where each face is composed of three straight sides and each vertex lies on exactly three sides of the object. Hexagonal tessellations have six sides and twelve angles for each component piece. Pentagonal tessellations include five sides and ten angles for each component piece. Other examples include heptagonal (seven sides), octagonal (eight sides), nonagon (nine sides), decagonal (ten sides).

Triangular tessellations can be found in nature and technology. In geography, for example, they are used to describe the pattern of mountains, hills, valleys, and rivers across a surface like a continent. Triangles also form the basis of **both Euclideine and Non-Euclidean geometries**.

Tessellation is the process through which forms fit together in a pattern without gaps or overlaps. The word comes from Latin tessera, which means "little cube." When used in reference to graphics hardware and software, it has a more specific meaning: the process of dividing each triangle in a polygon into many small triangles, typically four per vertex but any number may be used.

Because polygons are made up of many sides, they can be thought of as being composed of many triangles. In fact, there are only a few types of shapes in nature that cannot be represented as **some type** of polygon (circles and spheres). Polygons can also be described as closed surfaces; that is, surfaces with one point identified as the beginning and another as the end. Closed planes are examples of polygons, as are rings (such as a rope) and cones (such as candle flames). A polyhedron is any solid whose every surface is divided by itself or some other surface into **several convex pieces**. All cubes, pyramids, and tetrahedrons are examples of polyhedrons. A mesh is a structure used in finite element analysis to divide a computational domain into smaller elements for modeling purposes.