A tessellation is a repetitive pattern of forms that are closely fitted together with **no gaps** or overlap. All you need are few pieces of paper, scissors, and tape. You can use any paper size or shape as long as each piece is equal in size. Start by making a template out of paper or cardboard by simply drawing around two adjacent squares and then cutting them out. Place one of the templates on top of another one with the identical shapes cut out, and cover the entire sheet with tape. When removing one template at a time, make sure to keep the remaining templates intact.

The taped-together sheets are now ready for cutting. Remove all but one of the templates and cut across both sizes of papers simultaneously. The smaller pieces of paper will be torn away when removed, and the larger pieces remain in place. Tape the remaining template to a new piece of paper and repeat the process until all sheets have been used.

A tessellation is a pattern made up of **similar objects** that fit together perfectly. Regular polygons tessellate if their inner angles sum up to 360 degrees. A hexagon satisfies this condition because it has six sides and therefore six angles. The same rule applies to triangles, squares, and circles.

Hexagons have **many beautiful patterns** and figures of ground. They are useful in geometry for teaching concepts about congruency and similarity. When put on top of each other, they make **very regular grids**.

There are many ways to explain why hexagons tessellate.

1 "The secret of hexagonal tiles is that they can be placed in three dimensions so that no space is wasted." - This quote comes from an article by **Mary Ann Jensen** called "The magic of hexagons". It can be found online at http://www-cs-students.stanford.edu/~mjensen/hexmagic.html.

2 "The reason hexagons tessellate is that you can cut them into six identical pieces." - This statement was given by **my teacher** Mr. Sean Carroll when he taught a course on contemporary mathematics recently.

Definition of **Tessellation A tessellation** is formed when a shape is repeated indefinitely, covering a plane with no gaps or overlaps. The term comes from **the Latin word** for "little cubes," based on the fact that some crystals form cubes when cut into pieces.

There are two main types of tessellations: regular and irregular. In a regular tessellation, each unit cell contains the same number of sides. Thus, if we were to divide up a cube in half both times, it would be possible to keep dividing it in half forever, producing an infinite array of identical cubes.

An example of a regular tessellation is the one formed by dividing up a cube in half twice, resulting in four cubes. There are many other regular tessellations too, such as six-sided cells divided into three pairs or twelve-sided cells divided into four quadruples.

In an irregular tessellation, the sizes of cells vary. It is not possible to define exactly how large or small a cell must be to be considered irregular; instead, this type of tessellation is described only by its overall symmetry.

A tessellation pattern is one that is made up of forms that do not have any gaps or overlaps. Tessellations can be created using simple geometric forms (such as squares and triangles) or with much more complicated or irregular shapes (such as **stylized birds** or fish) that have been engineered to fit together neatly in a repeating pattern. The patterns are often used for wall coverings, floor tiles, and roof shingles.

Simple tessellations are those in which each subsequent layer is placed directly on top of the last without lifting the pencil. When creating these patterns it is important to keep in mind what was already drawn on the sheet of paper. You should try to avoid covering over existing lines or filling in areas where there are holes. They can be fun to draw but they take time so only draw them if you have a lot of **free time**!

There are two types of simple tessellations: uniform and non-uniform.

In a uniform tessellation, all parts of the design are equal in size and shape. All triangles in this example are identical in size and position. Uniform designs can be easy to create since you just need to make sure that each layer fits smoothly on top of the previous one without leaving **any gaps** or overlapping.

Non-uniform tessellations contain elements of different sizes or shapes. These designs may require more thought process when drawing them but they are more interesting and varied.