The regular hexagon also tessellates - bees, with their honeycombs, have been exploiting this for far longer than we have. For instance, the regular triangle has an internal angle of 60 degrees, and so it has six triangles that share one vertex, making the regular hexagon in this process. A vertex is a point where two or more lines meet, thus all of the corners of a polygon are vertices (which is the plural form of the vertex). The resultant integer shows how many polygons of that shape share one vertex. A regular polygon can tessellate if the internal angle of the shape can divide 360 degrees into a whole number. Other polygons, such as triangles and quadrilaterals, cooperate with mathematicians on this facet of nature. Fifteen different types of pentagon can tessellate, but the regular pentagon cannot. ‘Tiling the plane’ means that identical copies of a shape can be repeatedly used to fill a flat surface without any gaps or overlays. This discovery was of the 15th type of pentagon that can tile the plane. Last summer there was a discovery in the field of geometry. No, not the defence headquarters, but its eponym, the five-sided polygon that has been confusing mathematicians for over a century. This entry was posted in Geometry, Grades 9-12 and tagged proof only 3 polygons tessellate, regular tessellations, tessellation by Math Proofs. Therefore, they are the only polygons that can tessellate the plane. These are the representation of square, regular hexagon, and equilateral triangle respectively as we have stated above. Notice that the only possible ordered pair ( n, a) for this to be true are (4,4), (6,3) and (3,6). Subtracting 2 n from both sides and factoring the left hand side, we haveįactoring out $latex n-2$ we have $latex (n-2)(a-2) = 4$. That is $latex $latex an – 2a + 4 = 2n + 4$. Next, we add 4 to both sides to make it factorable. Multiplying both sides of the equation above by $latex n$, we have $latex 180a(n-2) = 360n$.ĭividing both sides by 180 results to $latex a(n-2) = 2n$ which simplifies to $latex an – 2a = 2n$. Now if we multiply this to $latex a$, the number or angles that meet at a point, the result must be 360 degrees for them not to have gaps or overlaps. The sum of the interior angles of a polygon with n sides $latex 180(n-2)$. Since the polygon is regular, the measure of each angle is equal to Theorem: The only regular polygons that tessellate are square, equilateral triangle, and regular hexagon. We will show below that these are the only possible regular polygons that will tessellate. Using this notation, we can describe the square as (4,4), the triangle as (3,6), and the hexagon as (6,3). Now, make the notation ( n, a), where n is the number of sides of the polygon and a be the number of angles that meet at a point. As you can see in the figure below, the sum of the interior angles meeting at a point is 360 degrees. Aside from these three, are there other regular polygons that can tessellate the plane? The answer is none.īefore we prove this theorem, let us first observe what make squares, equilateral triangles, and regular hexagons unique. In this post, we explore the properties of regular polygons such as the one shown in the first figure. Some polygons maybe combined with other polygons to do this. The tessellation below is composed of 12-sided polygons, squares, and triangles. The plane cannot always be tiled by a single shape.
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