What is tiling in geometry

There are three regular tessellations. They are:. A second group of tessellations is classified as semi-regular tessellations. All the polygons are regular, but there are two or more different polygons in the tessellation. The numbers under each tessellation below list in order the number of sides in the polygons that meet at a vertex. For example, the 3, 6, 3, 6 under the first example indicates that the tessellation is formed by placing a triangle, a hexagon, a triangle, and a hexagon together at a vertex.

There are eight semi-regular tessellations. In questions 2 and 3, the students need to experiment with combinations of shapes and then look carefully at their patterns to find reasons why the chosen shapes do or do not tessellate.

In question 4, they discuss this with a classmate. To understand why some shapes tessellate and others do not, the students need to have examined the interior angles of polygons. The activity on page 5 covers this. If they understand this, they will be able to explain why the three regular tessellations equilateral triangles, squares, and hexagons work, whereas pentagons do not tessellate.

The students should use similar reasoning to explain why some combinations of shapes tessellate and others do not. Drawings will vary, but all sets of pattern blocks can be made into tessellations. In thinking about which patterns and shapes cover more of the plane, we have started to reason about area. We will continue this work, and to learn how to use mathematical tools strategically to help us do mathematics. Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

For example, the area of region A is 8 square units. A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon.

Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere. Which square—large, medium, or small—covers more of the plane? Explain your reasoning.

The area of this shape is 24 square units. Which of these statements is true about the area? Select all that apply. By "the plane," we mean the 2-dimensional Euclidean plane -- i.

From this point on, we will use the word "tiling" to refer to a tiling of the plane. In mathematics, a tiling of the plane is a collection of subsets of the plane, i. There is one more detail to add to this definition — we want a tile to consist of a single connected "piece" without "holes" or "lines" for example, we don't want to think of two disconnected pieces as being a single tile.

Thus, we require that each tile be a topological disk this is math lingo for "consists of a single connected piece without holes or lines". We can't draw an entire tiling of the plane it is of infinite size! Two tiles are said to be congruent if one can be transformed into the other by a rigid motion of the plane, i.

Similarly, two tilings are said to be congruent if one can be transformed into the other by a rigid motion of the plane. Two tilings are said to be equal or the same if one can be changed in scale magnified or contracted equally throughout the plane so as to be congruent to the other.