will need to use some of the pattern blocks located in our classroom to
build some of the examples on this page.
patterns with some pattern blocks, making certain that you leave no
fit together easily?
shapes don't seem to fit with the others?
patterns could be repeated over and over again in the plane?
shapes fit together making a pattern using only one type of block?
shapes fit together making a pattern using two blocks that are
shapes fit together making a pattern using three blocks that are
should also visit the Tessellation Tool at: http://www.boxermath.com/plp/modules/online/workshop/toolbox/mosaictool.html?offer_id=PMTHF
which will allow you to build tessellations and other designs by
attaching the corners of various shapes.
regular polygon has 3 or 4 or 5 or more sides and angles, all equal.
A regular tessellation means a tessellation made up of
congruent regular polygons.
Note: Regular means that the sides of the polygon are
all the same length.
Congruent means that
the polygons are all the same size and shape.
A polygon is a many-sided
shape. A regular polygon is one in which all of the sides
and angles are equal. Some examples are shown below.
Name the regular polygons shown above.
A vertex is a point at which three or more tiles in a tessellation
meet. Two tiles
cannot meet in a point, but have to meet in line.
Using pattern blocks, try tessellating with hexagons as shown below.
How many hexagons meet at each vertex?
Next, try tessellating with squares.
How many square tiles meet at each
Finally, using the pattern
blocks, try tessellating with triangles.
How many regular triangles meet at each
is a regular tessellation?
is an example of pentagons.
Are the pentagons tessellating?
Explain your reasoning.
When you look at the three examples above,
you can easily see that the
squares are lined up with each other while the triangles and hexagons
are not. Also, if you look at 6 triangles at a time, they form a
hexagon, so the tiling of triangles and the tiling of hexagons are
similar and they cannot be formed by directly lining shapes up under
each other - a slide is involved.
Complete the following
|Measure of each
by the measurement
of the interior angle
|Will the polygon
tessellate using only
60 = 6
Since the regular polygons in a
tessellation must fill the
plane at each vertex, the interior angle must be an exact
divisor of 360 degrees. This works for the triangle, square,
and hexagon, and you can show working tessellations for these
figures. For all the others, the interior angles are not
exact divisors of 360 degrees, and therefore those figures
cannot tile the plane.
In a tessellation
the polygons used will fit together with their angles
arranged around a point with no gaps or overlaps. When using just one
polygon (for example, only equilateral triangles), the interior measure
of each angle will need to be a factor of _____ degrees (meaning that
____ degrees can be divided evenly by that angle measure). The only
regular polygons that qualify are the __________________,
___________________, and ___________________.
any polygon other than a triangle, square, or hexagon.
Illustrate and explain why it will not tessellate regularly in the Euclidean
Visit Hyperbolic Tessellations to
help you find out: http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
sure to try out the Java Applet to create your own hyperbolic