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Mathematical Mosaics

In your home, down a street, or even in ordinary objects, mosaics make their mark on our daily lives.  A mathematical mosaic is a group of shapes (with sides of equal length and measurement) arranged together to form a repetitious pattern in which they share sides at each corner point.  Each corner point contains the same neighbors throughout the pattern. (Mathematics: A Human Endeavor, 3rd Edition)

For example, everyday I walk over this.

At first glance it looks like ordinary tile.  However, if you look closer it is mathematical mosaic composed of rectangles and squares.  To make this arrangement you need four rectangles and one square.  The four rectangles create the point all having interior right angles.  The equation for the angles would be 90°+90°+90°+90° = 360°.   The rectangles have rotational symmetry creating a square in the pattern.

Everyday I cuddle up to this seemingly innocent lap quilt.

But in reality it is a mathematical mosaic.  The pattern is composed of eight isosceles triangles. The triangles have one right angle and two forty-five degree angles. The eight triangles around the point each have an interior angle of 45.  Thus the equation would be 45°+45°+45°+45°+45°+45°+45°+45°= 360°

Incorporating activities like finding mathematical mosaics in everyday life into your lessons give the student not only an understanding of how mosaics influence our world but also a real world application of math.  It helps the student understand that math can be applied to our everyday environment and lives not just in the classroom.

Here are some questions you might consider using in a lesson on mathematical mosaics.

How do you know if a mosaic is really a mathematical mosaic?

If you were given triangles and squares how many different math mosaics could you create in a 6×6 square?

If you have a mathematical mosaic made up of octagons and dodecagons, what other polygons could you use to complete the mosaic?  Give reasons why you chose each polygon.

Do you think the number of sides of a polygon has any correlation between the corresponding interior angles of that polygon?  Can you prove your answer?


3 responses to “Mathematical Mosaics

  1. Love your pictures and your questions!! The question about the sides and angles corresponding helps students to understand an important mathematical relationship!
    Great Blog!

  2. Jennifer,

    I spent 29 years looking at those same tiles in my school! I wonder if they were sold at a discount….. The real world connections that you have modelled for kids can be very powerful motivators in the classroom. They can take a confusing term and make sense of it for our students. Nicey done!


  3. Jennifer,

    That is a beautiful quilt! I asked a similar question to your second one, but I didn’t give a dimension for a guideline. I think that is much better. I will adjust my question! Thanks for the tip;)


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