semi regular tessellation definition geometry

If you try tiling the plane with these units of tessellation you will find that they cannot be extended infinitely. Interestingly there are other combinations that seem like they should tile the plane because the arrangements of the regular polygons fill the space around a point. Here are the eight semi-regular tessellations: The arrangement of polygons at every vertex point is identical.A semiregular tessellation has two properties which are: You can also use a variety of regular polygons to make semi-regular tessellations. There are four polygons, and each has four sides.įor a tessellation of regular congruent hexagons, if you choose a vertex and count the sides of the polygons that touch it, you’ll see that there are three polygons and each has six sides, so this tessellation is called “ 6.6.6“:Ī tessellation of triangles has six polygons surrounding a vertex, and each of them has three sides: “ 3.3.3.3.3.3“. Now keep going around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with. Since it’s a square, it has four sides, and that’s where the first “4” comes from. Here’s how: choose a vertex, and then look at one of the polygons that touches that vertex. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.Ī tessellation of squares is named “ 4.4.4.4“. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. You can work out the interior measure of the angles for each of these polygons: Shape Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other – a slide (or a glide!) is involved. When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.We can’t show the entire plane, but imagine that these are pieces taken from planes that have been tiled. A regular tessellation means a tessellation made up of congruent regular polygons. The word “tessellate” is derived from the Ionic version of the Greek word “tesseres,” which in English means “four.” The first tilings were made from square tiles.Ī regular polygon has 3 or 4 or 5 or more sides and angles, all equal. The most famous pair of such tiles are the dart and the kite.Ĭlick here for the lesson plan of non-periodic Tessellations.A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.Īnother word for a tessellation is a tiling.Ī dictionary * will tell you that the word “tessellate” means to form or arrange small squares in a checkered or mosaic pattern. The pattern of shapes still goes infinitely in all directions, but the design never looks exactly the same. In the 1970s, the British mathematician and physicist Roger Penrose discovered non-periodic tessellations. Whatever direction you go, they will look the same everywhere. They consist of one pattern that is repeated again and again. It may be better to show a counter-example here to explain the monohedral tessellations.Īll the tessellations mentioned up to this point are Periodic tessellations. All regular tessellations are also monohedral. If you use only congruent shapes to make a tessellation, then it is called Monohedral Tessellation no matter the shape is. You can use Polypad to have a closer look to these 15 irregular pentagons and create tessellations with them. Among the irregular pentagons, it is proven that only 15 of them can tesselate. We can use any polygon, any shape, or any figure like the famous artist and mathematician Escher to create Irregular tessellationsĪmong the irregular polygons, we know that all triangle and quadrilateral types can tessellate.

The good news is, we do not need to use regular polygons all the time. If one is allowed to use more than one type of regular polygons to create a tiling, then it is called semi-regular tessellation.Ĭlick here for the lesson plan of Semi - Regular Tessellations. If you try regular polygons, you ll see that only equilateral triangles, squares, and regular hexagons can create regular tessellations.Ĭlick here for the lesson plan of Regular Tessellations. the most well-known ones are regular tessellations which made up of only one regular polygon. There are several types of tessellations.

Semi regular tessellation definition geometry