Tessellations are also used in computer graphics where objects to be shown on screen are broken up like tessellations so that the computer can easily draw it on the monitor screen. Each of these has many fascinating properties which mathematicians are continuing to study even today. There are many other types of tessellations, like edge-to-edge tessellation (where the only condition is that adjacent tiles should share sides fully, not partially), and Penrose tilings. 3.3.3.4.4 (elongated triangular tessellation) 3.3.4.3.4 (blunt square tessellation) 3.4.6.4 (rhombi-tri-hexagonal tessellation) 4.8.8 (truncated square tessellation) 3.12.12 (truncated hexagonal tessellation) 4.6.12 (truncated tri-hexagonal tessellation) Some examples of semi-regular tessellations are shown below. There are eight such tessellations possible All the other rules are still the same.įor example, you can use a combination of triangles and hexagons as follows to create a semi-regular tessellation. If you look at the rules above, only rule 2 changes slightly for semi-regular tessellations. If you use a combination of more than one regular polygon to tile the plane, then it's called a 'semi-regular' tessellation. The mathematics to explain this is a little complicated, so we won't look at it here So what's unique to those 3 shapes (triangle, square and hexagon)? As it turns out, the key here is that the internal angles of each of these three is an exact divisor of 360 (internal angle of triangle is 60, that of square is 90, and for a hexagon is 120). You can see that there is a gap and that's not allowed. Let's try with pentagons and see what shape we come up with. You may wonder why other shapes won't work. Let me show you examples of these two here. What are the other two? They are triangles and hexagons. Of course, you would have guessed that one is a square. Each vertex (the points where the corners of the tiles meet) should look the same.All the tiles must be the same shape and size and must be regular polygons (that means all sides are the same length).The tessellation must cover a plane (or an infinite floor) without any gaps or any overlaps.There are only three rules to be followed when doing a 'regular tessellation' of a plane If you use only one kind of polygon to tile the entire plane - that's called a 'Regular Tessellation'Īs it turns out, there are only three possible polygons that can be used here.

There are different kinds of tessellations – the ones of most interest are tessellations created using polygons. The word “Tiling” is also commonly used to refer to 'tessellations'. Of course, when we are talking about floors, the shapes used to cover it are mostly rectangles or squares (in fact, the word ' tessellation' comes from the Latin word tessella - which means ' small square'). The one difference here is that technically a plane is infinite in length and width so it's like a floor that goes on forever. That is a good example of a 'tessellation'. And you'll notice that the floor is covered with some tiles or marbles of different shapes. That is a flat surface - called a 'plane' in mathematical terms. To explain it in simpler terms – consider the floor of your house. 'Demiregular Tessellation.'įrom MathWorld-A Wolfram Web Resource.A tessellation is simply is a set of figures that can cover a flat surface leaving no gaps. Referenced on Wolfram|Alpha Demiregular Tessellation Cite this as: Geometrical Foundation of Natural Structure: A Source Book of Design.

'Die homogenen Mosaike -ter Ordnung inĭer euklidischen Ebene. There are 20 such tessellations, illustrated above, as first enumerated by Krötenheerdt (1969 Grünbaum and Shephard 1986, pp. 65-67).

Caution is therefore needed in attempting to determine what is meant by 'demiregularĪ more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65). However, not all sources apparently give the sameġ4. (which leads to an infinite number of possible tilings). Tessellations (which is not precise enough to draw any conclusions from), while othersĭefined them as a tessellation having more than one transitivity class of vertices Some authors define them as orderly compositions of the three regular and eight semiregular A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical.

Types of tessellation