Example 7įigure 9 shows an example of irregular tessellation, in which all the polygons are regular and congruent. Irregular tessellations are those that are formed by irregular polygons, or by regular polygons but that do not meet the criterion that a node is a vertex of at least three polygons. It is a tessellation consisting of triangles, squares and hexagons, in the configuration 3.4.6.4, which is shown in figure 8. Example 6: rhombi-tri-hexagonal tessellation Figure 7 clearly illustrates this type of tessellation. Like the tessellation in the previous example, this one also consists of triangles and hexagons, but their distribution around a node is 3.3.3.3.6. It is the one that is composed of equilateral triangles and regular hexagons in the 3.6.3.6 structure, which means that a node of the tessellation is surrounded (until completing one turn) by a triangle, a hexagon, a triangle and a hexagon. Some examples of semi-regular tessellations are shown below. 4.6.12 (truncated tri-hexagonal tessellation).3.12.12 (truncated hexagonal tessellation).3.4.6.4 (rhombi-tri-hexagonal tessellation).3.3.3.4.4 (elongated triangular tessellation).3.3.3.3.6 (blunt hexagonal tessellation).There are eight semi-regular tessellations: Each node is surrounded by the types of polygons that make up the tessellation, always in the same order, and the edge condition is completely shared with the neighbor. Semi-regular or Archimedean tessellations consist of two or more types of regular polygons. The nomenclature for a regular hexagonal tessellation is 6.6.6 or alternatively 6 3. In a hexagonal tessellation each node is surrounded by three regular hexagons as shown in figure 5. The notation that is applied to this type of square tessellation is: 4.4.4.4 or alternatively 4 4 Example 3: Hexagonal tessellation It should be noted that each node in the tessellation is surrounded by four congruent squares. In figure 1 we have a good example.įigure 4 shows a regular tessellation composed only of squares. The most common type of tessellation is that formed by rectangular and particularly square mosaics. In the case that a single type of mosaic formed by a regular polygon is used, then a regular tessellation, but if two or more types of regular polygons are used then it is a semi-regular tessellation.įinally, when the polygons that form the tessellation are not regular, then it is a irregular tessellation. In this way, there are no spaces left uncovered and the tiles or mosaics do not overlap. Tiles or tiles are flat pieces, generally polygons with congruent or isometric copies, which are placed following a regular pattern. They are everywhere: in streets and buildings of all kinds. The tessellated are surfaces covered by one or more figures called tiles. Example 12: tessellation in video games.Example 6: rhombi-tri-hexagonal tessellation.Example 5: Blunt hexagonal tessellation.The most famous pair of such tiles are the dart and the kite.Ĭlick here for the lesson plan of non-periodic Tessellations.Video: 12.1 Tessellations of Regular and Irregular Polygons Content 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.
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