Tessellations

A shape will tessellate if its vertices can have a sum of 360˚ . In an equilateral triangle, each vertex is 60˚ . Thus, 6 triangles can come together at every point because 6×60˚=360˚ . This also explains why squares and hexagons tessellate, but other polygons like pentagons won’t.

Equilateral triangles, squares and regular hexagons are the only regular polygons that will tessellate. Therefore, there are only three regular tessellations.

There a three types of tessellations: Translation, Rotation, and Reflection.

• TRANSLATION – A Tessellation which the shape repeats by moving or sliding.
• ROTATION – A Tessellation which the shape repeats by rotating or turning.
• REFLECTION – A Tessellation which the shape repeats by reflecting or flipping.

Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane.

• RULE #1: The tessellation must tile a floor (that goes on forever) with no overlapping or gaps.
• RULE #2: The tiles must be regular polygons – and all the same.
• RULE #3: Each vertex must look the same.

There are shapes that are unable to tessellate by themselves. Circles or ovals, for example, cannot tessellate. Not only do they not have angles, but you can clearly see that it is impossible to put a series of circles next to each other without a gap.

There are only three shapes that can form such regular tessellations: the equilateral triangle, square, and regular hexagon. Any one of these three shapes can be duplicated infinitely to fill a plane with no gaps.

No, A regular heptagon (7 sides) has angles that measure (n-2)(180)/n, in this case (5)(180)/7 = 900/7 = 128.57. A polygon will tessellate if the angles are a divisor of 360. 128.57 is not a divisor of 360.

Skipping a vertex Are there other regular polygons that now tessellate? We can see from this that the pentagon, hexagon, octagon, and dodecagon tesselate with one skipped vertex. The corresponding holes are shaped decagon, hexagon, square, and triangle.

No, a nonagon cannot tessellate the plane. A nonagon is a nine-sided polygon.

A regular decagon does not tessellate. A regular polygon is a two-dimensional shape with straight sides that all have equal length.

No other regular polygon can tessellate because of the angles of the corners of the polygons. In order to tessellate a plane, an integer number of faces have to be able to meet at a point. For regular polygons, that means that the angle of the corners of the polygon has to divide 360 degrees.

There a three types of tessellations: Translation, Rotation, and Reflection.

A Simple Method For Creating Tessellations From Rectangles

1. Cut out a rectangle out of an index card or poster board.
2. Draw a line from one side to the opposite side.
3. Cut along the line you drew and interchange the pieces.
4. Draw another line on the resulting figure in a perpendicular direction to the first line.
5. Cut along the line you just drew and interchange the pieces.

1-Step Cutting Tessellation

1. Take one square piece of paper and cut a weird shape out of one side of the square.
2. Line your oddly-shaped cut-out on top of a second square of paper, lining up the long edges.
3. Repeat for each of the remaining three squares.
4. Take one of your squares and cut out your tracing.

1a : mosaic. b : a covering of an infinite geometric plane without gaps or overlaps by congruent plane figures of one type or a few types. 2 : an act of tessellating : the state of being tessellated.

Discuss the three basic attributes of tessellations:

1. First, they are repeated patterns. Ask students to find examples of repeated patterns in the room.
2. Second, tessellations do not have gaps or overlaps.
3. Third, tessellations can continue on a plane forever.

Tiles used in tessellations can be used for measuring distances. Tiles that are arranged so there are no holes or gaps can be used to teach students that area is a measure of covering. Students could trace around a small set of tiles in a pattern and find that by counting the tiles they can tell how big the shape is .

In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons.