Why does every quadrilateral tessellate

A grid is now formed by parallelograms 4 times as big whose area equals that of four reference quadrilaterals. Below is a photograph of a outer house wall in New York City as may be observed from the City Highline. What is what? Simple Quadrilaterals Tessellate the Plane A shape is said to tessellate the plane if the plane can be covered without holes and no overlapping save for the boundary points with congruent copies of the shape. What if applet does not run?

References A. Hilton, D. Holton, J. Related material Read more Dancing Squares or a Hinged Plane Tessellation. Dancing Rectangles Model Auxetic Behavior. A Hinged Realization of a Plane Tessellation.

A Semi-regular Tessellation on Hinges A. A Semi-regular Tessellation on Hinges B. A Semi-regular Tessellation on Hinges C. Escher's Theorem. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Triangles are the easiest shape to tessellate, and the formlessness of ghosts makes tessellation easy.

To form into a mosaic pattern, as by using small squares of stone or glass. Tessellation Definition A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for a tessellation is a tiling. Inside was a red and white tessellated floor.

Normally, the whole of a floor is tessellated. Firstly, there are only three regular tessellations which are triangles, squares, and hexagons. To make a regular tessellation, the internal angle of the polygon has to be a diviser of Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane. This means every triangle, and every quadrilateral will tessellate. What regular polygons cannot tessellate? Triangles, squares and hexagons are the only regular polygons which tessellate with a single shape.

Other shapes may tessellate with more than one shape polygon. Yes, a trapezium tessellates. A tessellation is a tiling of the plane with two-dimensional shapes, such that there are no spaces or gaps between the…. Yes, absolutely. 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. Kershner [3] found three new types, and claimed a proof that the eight known types were the complete list. A article by Martin Gardner [4] in Scientific American popularized the topic, and led to a surprising turn of events. In fact Kershner's "proof" was incorrect. After reading the Scientific American article, a computer scientist, Richard James III, found a ninth type of convex pentagon that tessellates.

Not long after that, Marjorie Rice , a San Diego homemaker with only a high school mathematics background, discovered four more types, and then a German mathematics student, Rolf Stein, discovered a fourteenth type in As time passed and no new arrangements were discovered, many mathematicians again began to believe that the list was finally complete.

But in , math professor Casey Mann found a new 15th type. Recall that a regular polygon is a polygon whose sides are all the same length and whose angles all have the same measure. We have already seen that the regular pentagon does not tessellate. We conclude:.