Tessellations in Boston

While we were in Boston for our D.C. trip, I noticed a sidewalk that contained tessellations. The shapes fit in perfectly with no gaps or spaces. The pattern would be able to continue indefinitely. The basic definition of a tessellation is a random shape repeated over and over covering a plane with out any gaps or overlaps. Another word is tiling.

   

















Some further research shows that there are two main types of tessellations: regular and semi-regular.

Regular tessellations are patterns made by repeating a regular polygon. In that case there are only three types: squares, triangles and hexagons. For regular tessellations, the pattern is always the same at the vertex.

A semi-regular tessellation consisted of two or more regular polygons. As with the regular tessellations, the patterns must be the same at each vertex. (~IR1)

Tessellations in a Table

I saw a table and noticed it had tessellations in it. First off, the shapes are hexagons. This can be proved using the formula (n-2)180 where n is the number of sides and the sum should be the sum of the angles. So (6-2)180= 720 degrees.

The hexagons fit together at the intersections because the two smaller angles are about 50 degrees and the two larger angles are about 130 degrees.  Altogether it is 360 degrees.  This means there are no gaps in the tessellation, and it fits together well.  (~NW5)

Soccer Tessellations

A soccer ball is one big tessellation, which is a repeating pattern of one or more shapes that covers the plane completely without overlap.  In this instance, however, the soccer ball is a curved surface filled with regular polygons (here pentagons and hexagons).  They are able to interlock because the ball is spherical, and so it works that each side of the pentagon is attached to a hexagon. This would not work on a flat surface because the sides would not fully connect. (~KM5)

Tessellations in Clothing

This is a picture of a Houndstooth skirt, which I realized was a tessellation. A tessellation is a pattern of shapes that fit perfect together. These tessellations are used a lot in geometry and show up everywhere. In this particular picture, the Houndstooth pattern is repeated over and over alternating black and white. As you can see there are no holes or gaps in this pattern, so it is a perfect tessellation. (~AD1)

Welcome to Geometry Hunters!

Welcome to the Geometry Hunters blog, a forum for students at Oakwood High School to share photographs, drawings, or observations of interesting geometric elements.

As a first example, consider this tiled pattern of a sidewalk, which I noticed while waiting for a friend to meet me for dinner in Palo Alto:

The tessellation includes squares and equiangular octagons, such that one right angle (90 degrees) and two interior angles from octagons (135 degrees each) meet at each vertex. Thus, the total angle sum at each vertex is 360 degrees, which enables a complete covering of the sidewalk without gaps.