If you’ve ever seen a mosaic made up of small, identical shapes that fit together perfectly without any gaps, then you’ve seen a tessellation. Tessellations are found in nature, in architecture, and in works of art dating back centuries. But what you may not know is that tessellations are also a mathematical concept with some pretty interesting applications. Read on to learn more about tessellations and the math behind this fascinating math art form, making it a great STEAM project for young kids.

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**What is a Tessellation?**

A tessellation is a repeating pattern of shapes that covers a surface without any gaps or overlaps. The word “tessellation” comes from the Latin word “tessella,” which means “small square.” And indeed, many tessellations are made up of small squares or regular polygons (shapes with straight sides, like triangles and hexagons). But tessellations can also be made from other shapes, like irregular polygons and stars!

Generally speaking, a regular tessellation is the one where all the shapes are uniform and regular, for example:

Whereas, a semi-regular tessellations consist of two regular shapes with identical vertex points (i.e. points where the two shapes meet). For example, octagons and squares can be arranged to form a semi regular pattern.

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**WHAT ARE THE THREE RULES OF TESSELLATIONS**

Despite the fact that tessellations can be created using a wide range of various forms, there are basic guidelines that all regular and semi-regular tessellation patterns must follow.

**Gaps and Overlapping**– There can be no overlaps or gaps between the shapes in the tessellation.**Regular Polygons**– Regular polygons are required for all regular tessellations. Polygons are geometric figures with connected straight sides. A regular polygon, such as a square or an equilateral triangle, is a shape with sides that meet to form equal angles. However, not all regular polygons can be used to form a tessellation since their sides do not line up evenly. A regular polygon that cannot be tessellated is a pentagon.**Symmerty**– A tessellation requires that each polygon have at least one line of symmetry. Often referred to as a mirror image, symmetry is defined as equal parts facing each other around an axis.**Common 36o degree Vertex**– To be used in a tessellation, all regular polygons that meet must share a common 360-degree vertex. A vertex is the place at which two sides meet to form an angle. For instance, two sides of an equilateral triangle meet to make a 60 degree angle. A vertex in a tessellation is the intersection of three or more shapes that add up to 360 degrees. For instance, three hexagons with interior angles of 120 degrees can be combined to make a vertex of 360 degrees, while a pentagon with interior angles of 108 degrees cannot.

When all of these criteria are met, we say that the shapes have tessellated the plane.

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**TYPES OF TESSELLATION TRANSFORMATIONS**

There are three types of tessellation transformations: translation, rotation, and reflection.

**Translation Tessellation**occurs when the shape is moved without turning it.**Rotation Tessellation**happens when the shape is turned around a point.**Reflection Tessellation**is when the shape is flipped over a line.

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**HOW TO MAKE A SIMPLE TESSELLATION PATTERN**

There are only three regular polygons that will tessellate by themselves: the triangle, the square, and the hexagon. To create a tessellation, you’ll need to start with one of these shapes and then use your creativity to divide it into unique pieces that fit together like a puzzle. The easiest way to do this is to use the ‘PART’ to ‘TRAP’ method.

1. Write the letters of the word “PART” on each of the four corners of a small square piece of paper as shown below.

2. On the paper, scribble a random line from left to right.

3. Next, scribble another random line from top to bottom, on the paper.

4. Cut out the shapes along the lines you drew.

5. Assemble the pieces together so that the corners of the square touch each other in the center and they form the word “TRAP” as shown. Tape the pieces together without any overlaps.

6. Your tessellation pattern is ready. Start tracing on a blank of sheet of paper with the pattern.

8. It should tessellate perfectly if done correctly. Now, go ahead and color it with alternate colors.

If you don’t feel like making your own tessellation pattern, use our tessellation coloring page to color while admiring the lovely patterns that are produced.

**EXAMPLES OF TESSELLATIONS IN REAL LIFE**

Tessellations are all around us! If you take a close look at the Hexagon quilt pattern, you’ll see that it’s made up of repeated hexagons. That’s a tessellation! Similarly, mosaic floors and tile patterns are also examples of tessellations.

In fact, Islamic architects were using tessellations centuries ago to create beautiful patterns in their buildings. Checkerboards are square tessellations. Each square meets edge to edge without gaps or overlaps. Bricks on a wall form a rectangular tessellation. Tessellations can also be found in nature, such as in honeycombs.

**Why Do Mathematicians Study Tessellations?**

Believe it or not, tessellations have some pretty important applications in the world of mathematics. For example, researchers have used tessellations to develop more efficient ways to pack objects together without any wasted space. This is especially important when packing items for shipping; by using tessellations, we can make sure that every inch of space inside a box or container is being used efficiently. Similarly, tessellations can be used to create patterns for floor tiles and wallpaper so that there is no wasted material during production.

Tessellations are a fun and easy way to explore geometry! In math, tessellations are often used to study symmetry and polyhedra. A polyhedron is a three-dimensional shape with flat faces, straight edges, and vertices (corners). The Platonic solids are a good example of polyhedra; they are the five perfect three-dimensional shapes: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Each of these shapes has faces that are made up of regular polygons (triangles, squares, hexagons), which means they will tessellate.

**TESSELLATION BOOKS AND STENCILS**

We enjoy incorporating reading and more hands-on activities related to the subject we are studying. So, here are a few ideas we have.

### 12 Roylco Tessellations Animal Templates

kids will have lot of fun tracing these tessellating animal stencils to make their artwork. Perfect for young kids.

**The Tessellation Coloring Book: 40 Mindless Geometric Patterns And Shapes**

This 40 page coloring book is perfect for older kids and adults as a relaxing activity.

### Toads and Tessellations (Charlesbridge Math Adventures)

It is a fantastic book for bridging reading and art. The story incorporates an art lessons on Escher/tessellations to explain mathematical concepts.

Next time you’re admiring a mosaic floor or tile pattern, see if you can spot the repeating shapes that make up the tessellation!

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