Ideas in Geometry/Instructive examples/Section 2.3

Problem # 6

a) There are exactly 5 regular tessellations FALSE

A regular tessellation is comprised of a single type of tile in which the tiles fit perfectly around a vertex.

Therefore, we must first find regular polygons whose interior angles can fit at a vertex perfectly in order to form 360 degrees. Through examining several types of polygons (figure 1 below), we can find the number of tiles needed for each polygon by dividing 360 degrees by the sum of each polygons interior angles.

 # of sides	         Measure of angles (MOA)		# of tiles at a vertex (360/ MOA)
      3			       60 degrees				        6
      4			       90 degrees		                         4
      5			      540/5 degrees = 108	 	             3.333333
      6			      120 degrees				        3

We can see that as the number of sides increase so do the sums of the interior angles. This means that the total number of tiles we can fit around a vertex will keep getting smaller. We can see that 3 and 4 sides work because they produce whole numbers. However a polygon with five sides will not work, since it would require a fraction of an amount of a tile to completely tessellate. We can see that six works, however, since the number of tiles that are needed to tessellate are 3.

The last option we can have to tessellate would be two tiles at a vertex. However, this cannot happen because the angles would have to be 180 degrees each in order to produce the 360 degrees around the vertex. These would be two straight lines and therefore cannot tessellate.

Therefore, since 3, 4, and 6 are the only numbers for sides of polygons that will tessellate perfectly, there are only 3 regular tessellations, not five.

b) Any quadrilateral tessellates a plane. TRUE

Since a quadrilateral is a figure that contains exactly four enclosed sides, we know that if lines were drawn from either of its diagonals it would create two trianlges. Within a triangle, the total amount of degrees of the interior angles is 180 degrees. Since a quadrilateral contains two triangles, we can conclude that any quadrilateral possess an interior angle sum of 360 degrees. In order for a figure to tesselate the plane, there needs to be figures that fit to account for 360 degrees around each vertex. Therefore, any quadrilateral can tessellate the plane, using its four interior angles. The quadrilateral will need to be rotated so that a different interior angle is touching each vertex point of the tessellation. This will account for the 360 degrees.

c) Any triangle will tessellate the plane. TRUE

This falls along the same lines as the statement above. Since the interior angle sum of a triangel is exactly 180 degrees, two triangles can be rotated so that they fit together to form a quadrilateral with a total interior angle sum of 360 degrees. As mentioned above, a quadrilateral will tessellate a plane, sine 360 degrees is the total number needed around each vertex point of a tesselation. Therefore, any triangle will tesselate the plane as well.

d) If a triangle is used to tessellate the plane, then exactly 4 angles will fit around each vertex. FALSE

This can be proven when looking at an equilateral triangle. In order for the triangle to tessellate, the sum of the angles at a vertex must equal to 360 degrees. When we examine the interior angles of an equilateral triangle, we know that since there are 180 degrees in a triangle, each angle must be an equal portion. Therefore, each angle is 60 degrees. If the statement above were to be true, then that would mean that only 4 of the angles of the equilateral triangle would be needed to form the 360 degrees around a vertex. However, this would only give us 240 degrees, leaving us 130 degrees short. Consequently, we can conclude that we actually need six angles to tessellate the equilateral triangle. Thus, this statement is false.

e) If a polygon has more than six sides, then it cannot tessellate the plane. FALSE

Although we learned from part a) that the highest number of sides a polygon can have in order for it to tessellate the plane completely is six, A polygon can semi-tessillate a plane with more than six sides. Within a semi-tessellation, a polygon with more than 6 sides can tessellate along with other polygons inbetween. These other polygons are needed in order to account for the 360 degrees that must lie around a single vertice. Using an octagon for example, we can see that each vertice of an octagon has an angle of 135 degrees. Thus three octagons placed together would equal 405 degrees and would therefore not tessellate. However, If we place two octagons so that they share a common vertice, we are now left with (2 x 135) = 270 degrees. In order to get up to 360 degrees we must add another polygon with an interior angle of 90 degrees. Since a square has an angle of 90 degrees, we can use 2 octagons and 1 square at a single vertice to create the 360 degrees required to tessellate the plane.