Totally Tessellated: To Main Page
History and Culture of TessellationsEssentialsA Simple Type of TessellationM. C. Escher and His Unique Approach to TessellationsBeyond the Basics of Tessellations
Toolbar
Regular Tessellations (3/4)
 

1. Introduction
2. Trial and Error
3. Analysis of Results
4. Naming the Regular
Tessellations
5. The Regular
Tessellations
6. Hands-On Activities
Naming the Regular Tessellations
The naming convention for regular tessellations (as well as for semiregular and demiregular tessellations) requires knowledge of how to name polygons around vertices. The following subsection will explain this topic. Afterwards, we will finally give names to our three regular tessellations.

    How to Name an Arrangement of Regular Polygons Around a Vertex
    Since we have been concentrating on the arrangement of regular polygons around a vertex, it makes sense that the naming system is based on this idea.

    1. To name an arrangement of regular polygons around a vertex, first find the regular polygon with the least number of sides.
    2. Then find the longest consecutive run of this polygon, that is, two or more repetitions of this polygon around the vertex.
    3. Next, indicate the number of sides of this regular polygon. For example, to name a triangle with 3 sides, we name it 3 and follow it with a period (.). If you find more than one consecutive "run" of this polygon, then name it twice, i.e., 3.3.
    4. Proceeding in a clockwise or counterclockwise order, indicate the number the sides of each polygon as you see them in the arrangement.
    5. Do remember to start with the longest consecutive run of the regular polygon with the shortest number of sides.

    Examples of Polygon Arrangements

    Examples of how to identify the arrangement of polygons around a vertex

Consider the first arrangement shown in the figure. Since 3 is the lowest number of sides, we start from the triangle and proceed clockwise or counterclockwise around the vertex. Either way we go, we obtain a square, a hexagon, and another square. Putting the number of sides of each polygon in sequence, we obtain: 3.4.6.4.

Consider the second arrangement. Three is the lowest number of sides. There are two adjacent triangles in the arrangement, and there are no other longer consecutive runs of equilateral triangles. Thus, we start from the two adjacent triangles and go clockwise or counterclockwise around the vertex. Either way, we obtain: 3.3.4.3.4.

Consider the third arrangement. Even though there are two adjacent octagons, we must start with the polygon with the lowest number of sides: the square. Thus, proceeding clockwise or counterclockwise from the square, we obtain: 4.8.8.

 

Finally, let us now consider our regular tessellations:

Polygon Arrangments for the Regular Tessellations

The polygon arrangements for the regular tessellations

The polygon arrangement around each vertex in the regular tessellation of equilateral triangles is the same: 3.3.3.3.3.3. Similarly, the arrangment for the regular tessellation of squares is 4.4.4.4, and the arrangment for the regular tessellation of hexagons is 6.6.6.

Since this information is enough to distinguish between the regular tessellations and since it is also enough to reconstruct the original tessellation, we identify the regular tessellations as 3.3.3.3.3.3, 4.4.4.4, and 6.6.6.

Up

Real examples of regular tessellations:


TemplatesTo browse full-page templates of the regular tessellations that are ready to be printed, proceed to the templates page:

top of the page