Thus, this statement is false.Į) If a polygon has more than six sides, then it cannot tessellate the plane. Consequently, we can conclude that we actually need six angles to tessellate the equilateral triangle. However, this would only give us 240 degrees, leaving us 130 degrees short. 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. 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. In order for the triangle to tessellate, the sum of the angles at a vertex must equal to 360 degrees. This can be proven when looking at an equilateral triangle. Therefore, any triangle will tesselate the plane as well.ĭ) If a triangle is used to tessellate the plane, then exactly 4 angles will fit around each vertex. As mentioned above, a quadrilateral will tessellate a plane, sine 360 degrees is the total number needed around each vertex point of a tesselation. 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. This falls along the same lines as the statement above. This will account for the 360 degrees.Ĭ) Any triangle will tessellate the plane. The quadrilateral will need to be rotated so that a different interior angle is touching each vertex point of the tessellation. Therefore, any quadrilateral can tessellate the plane, using its four interior angles. In order for a figure to tesselate the plane, there needs to be figures that fit to account for 360 degrees around each vertex. Since a quadrilateral contains two triangles, we can conclude that any quadrilateral possess an interior angle sum of 360 degrees. Within a triangle, the total amount of degrees of the interior angles is 180 degrees. 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. 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.ī) Any quadrilateral tessellates a plane. These would be two straight lines and therefore cannot tessellate. However, this cannot happen because the angles would have to be 180 degrees each in order to produce the 360 degrees around the vertex. The last option we can have to tessellate would be two tiles at a vertex. We can see that six works, however, since the number of tiles that are needed to tessellate are 3. 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 3 and 4 sides work because they produce whole numbers. This means that the total number of tiles we can fit around a vertex will keep getting smaller. We can see that as the number of sides increase so do the sums of the interior angles. # of sides Measure of angles (MOA) # of tiles at a vertex (360/ MOA) 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. Therefore, we must first find regular polygons whose interior angles can fit at a vertex perfectly in order to form 360 degrees. A) There are exactly 5 regular tessellations FALSEĪ regular tessellation is comprised of a single type of tile in which the tiles fit perfectly around a vertex.
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