If these were tiles on the floor they would fit together with no gaps, and be flat on the floor no bumps. Triangles tesselate on a flat plane. Similarly, we can do the same with squares. A pentagon can be divided into three triangles. Three regular pentagons is too small, four regular pentagons too large. There is no Goldilocks integer number of regular pentagons to make a perfect tessellation.
For hexagons, these tesselate. Just like pentagon example above, a regular n-sided polygon can be broken up into triangles. For an n-sided shape, it can be broken up into n-2 triangles. For a shape to tessllate, the internal angle has to be a factor of Only the triangle, square, and hexagon fit this criterion. Here's a more formal proof. Before continuing, try the Quadrilateral Tessellation Exploration.
Taking a little more care with the argument, we have:. The point of all the letters is that the angles of the triangles make the angles of the quadrilateral, which would not work if the quadrilateral was divided as shown on the right. Begin with an arbitrary quadrilateral ABCD. The angles around each vertex are exactly the four angles of the original quadrilateral.
Recall from Fundamental Concepts that a convex shape has no dents. All triangles are convex, but there are non-convex quadrilaterals. The technique for tessellating with quadrilaterals works just as well for non-convex quadrilaterals:.
It is worth noting that the general quadrilateral tessellation results in a wallpaper pattern with p2 symmetry group. Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane. In both cases, the angle sum of the shape plays a key role. The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon. Rather than repeat the angle sum calculation for every possible number of sides, we look for a pattern.
In fact, there are pentagons which do not tessellate the plane. Attempting to fit regular polygons together leads to one of the two pictures below:. Both situations have wedge shaped gaps that are too narrow to fit another regular pentagon. Thus, not every pentagon tessellates. On the other hand, some pentagons do tessellate, for example this house shaped pentagon:. The house pentagon has two right angles. Thus, some pentagons tessellate and some do not.
The situation is the same for hexagons, but for polygons with more than six sides there is the following:. This remarkable fact is difficult to prove, but just within the scope of this book. However, the proof must wait until we develop a counting formula called the Euler characteristic, which will arise in our chapter on Non-Euclidean Geometry. Nobody has seriously attempted to classify non-convex polygons which tessellate, because the list is quite likely to be too long and messy to describe by hand.
However, there has been quite a lot of work towards classifying convex polygons which tessellate. Because we understand triangles and quadrilaterals, and know that above six sides there is no hope, the classification of convex polygons which tessellate comes down to two questions:.
Question 2 was completely answered in by K. Reinhardt also addressed Question 1 and gave five types of pentagon which tessellate. In , R. A regular pentagon is a geometric object that is a plane figure with straight lines. It is a regular polygon with five sides. Shapes tessellate to fit around an interior angle.
They also tessellate because they are regular polygons; non-regular polygons cannot tessellate. All triangles and quadrilaterals will tessellate, whether regular or irregular. Contrary to the above answer, a regular pentagon will not tessellate but there are 14 different irregular pentagons which will tessellate the last was discovered in Three convex hexagons will do so as well.
No polygon of 7 or more sides will tessellate - whether they are regular contrary to the above answer or irregular. A regular octagon will not tessellate but an irregular one can. Log in. Study now. See Answer. Best Answer. Study guides. Science 20 cards. Is glucose solution a homogenous mixture.
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