What do tessellations have to do with transformations

Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry and certain other spaces.

These spaces are distinguished from two other types of spaces that belong to non-Euclidean geometry - hyperbolic and spherical. The essential difference between Euclidean geometry and these two other ones is the nature of parallel lines. While in Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it, in hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to the given line and in spherical geometry there are no such lines at all.

So is it possible to tessellate in non-Euclidean geometries? It is possible in the hyperbolic geometry. While in a Euclidean plane there are three regular tessellations, the number of regular ones in a hyperbolical plane is infinite.

When it comes to semi-regular ones, there are also much more combinations that in Euclidean plane. They are also possible in the three-dimensional hyperbolic space. Apart from the world of art and architecture, tessellation designs could be found in many types of tiling puzzles , from traditional jigsaw puzzles and tangram to more modern puzzles based on mathematics.

Combining art and math, the majority of tessellation puzzles can be assembled in several different ways, making them open-ended and encouraging creativity. When it comes to recreational mathematics, authors such as Henry Dudney and Martin Gardner have made many uses of tessellations in this field.

Famous for his puzzles and mathematical games, Dudney invented the hinged dissection puzzle - a geometric dissection in which all of the pieces are connected into a chain by 'hinged' points, such that the rearrangement from one figure to another can be carried out by swinging the chain continuously without severing any of the connections.

A mathematician and author in the magazine Scientific American , Gardner frequently wrote about rep-tiles , shapes that can be dissected into smaller copies of the same shape. His articles in the Scientific American have inspired many mathematicians interested in tessellations. One of them was Robert Reid who covered diverse areas of recreational mathematics including tessellation, geometry dissection, number theory, 'squares in squares' and more.

Another mathematician inspired by Gardner's articles was Marjorie Rice who despite having only a high-school education discovered four new tessellations with pentagons.

One of the most popular tessellation problems in recreational mathematics is the so-called 'squaring the square' , a tiling an integral square using only other integral squares. The task can go from an easy one to extremely complex one based on conditions and restrictions set. The connection between science, mathematics and art is an inevitable thing we need to mention if we wish to comment about various examples of tessellations in art , crafts, and architecture.

Mathematicians, scientists, and artists govern their practices to help us better understand the world around us and reflect upon questions that direct our existence. The division between the intuition and hard, cold logic that most of us tend to think about if we think about art and science is often blurred and many artists have focused their artistic production following certain scientific and mathematical rules.

The golden ratio in art is just one of various tools artists used to produce images that reflect the mysteries of the natural world or to produce the most pleasing image for the eye, and none of us can overlook the repetition , geometrical shapes , symmetry in some of the most celebrated works of art history or contemporary pieces today. The most renowned tessellation artist in history, presently still very much adored by mathematicians and the rest, whose production focused on the questions concerning the nature of visual perception, infinity, and patterns is M.

His use of tessellations in art pieces, the practice of using regular patterns that divide the plane, without his knowledge in mathematics and its principles beyond secondary school, fascinates still today. The 20th-century artist Maurits Cornelis Escher created some of the most memorable woodcuts, lithographs , mezzotints , and drawings inspired by the fictional investigation into the formal possibilities of perspective and the tiling of a plane using one or more geometrical shapes.

The defining moment that pushed the artist towards the creation of the art we most associate with his name was his trip to Granada, Spain and his visit to the Alhambra Palace. There, M. The exploration of patterns and the regular division of planes was the richest inspiration that the artist ever faced.

His most celebrated pieces following the principles of tessellation explored the basic patterns, but the artist elaborated further, distorting the shapes and rendered them into animals, birds, and other figures. His elaborate interlocking designs relayed heavily on his love for the natural world that helped him to construct hexagonal grids moving far beyond the human world and into more of a phantasmagoric world of strange creatures that resemble lizards, insects, fish, and birds.

In many of these images, the distinction between foreground and background is obliterated and the viewer can choose to see one or other set of shapes as foreground at will.

The distortions which the artist produced obeyed the three, four, or six-fold symmetry of the underlying pattern and the effects were both startling and beautiful and in keeping with the tessellation rules. The famous pattern designs that follow the principles of tessellation of one of the most decorative avant-garde movement Art Nouveau form the design production of Koloman Moser.

As one of the founders of the Vienna Secession , Moser, like Escher was a master of tessellation. Following with the love of spiral line, flowers, buds, leaves and geometrical shapes, his elaborate designs transcended onto a variety of works such as tapestry design, posters, furniture, tableware, jewelry, postage stamps etc. The play with reflection and repeat of certain motifs, along with the joining of different images like flowers, foliage, and birds, in different combinations across the reflection axes is known today as drop design.

The digital age and the computer dominance have greatly influenced the world of contemporary art. Moving away from graphic artworks by Escher, or prints by the leading Swiss Constructive artist Hans Hinterreiter that are also famous examples of the principle, today the computer generated tessellation patterns are easy to come by. The meshing of the images follows the principle that is also used to shape the images we see on our screens. The world of graphic art today is a vast area of beautiful examples of how art fuses with mathematics and science, and how a repetition of patterns is put to use.

This repetition of patterns also makes up a rich history of mosaic and mural paintings. From Ancient Greece to Roman Art, Byzantine and elaborate examples of Mexican mural and mosaic paintings, today we witness mosaics, murals, and graffiti art all over the world that are modern day examples of this old principle as well. Although the exactly repeating tilings are more than convenient for art and design at large, they are harder to find in nature.

They could also try shading triangular tiled tessellations so that there are not equal numbers of each colour of tile. Key ideas to reinforce in this include the conservation of area, length and orientation. As with the previous tessellation example the concept of area can be explored. Students should be describing the rotation s in the tessellation including angle of rotation and centre of rotation. They should also include the translation description as these two transformations combine to make the tessellation.

This session explores learning to create a base pattern for a tessellation using reflection. Rotations and translations are also used. Using reflections is more challenging than using rotations and translations.

A reflection is the flipping of points of the plane about a line, called a mirror line. The properties of size and shape remain invariant unchanged under the operation of reflection. In the figure below, the figure is reflected through line m. Use cut-outs on board to demonstrate how to alter a shape and then use reflection and rotation to create a tessellation. Original triangle Showing the nibble Cut and reflect the nibble Rotate the nibble.

It is useful to colour one side as stated previously. Students can then work on Copymaster 4 to practice and then create their own base templates. Ideas around creativity can be mentioned here. Key ideas to reinforce in this include the conservation of area and length. Students should be describing the reflections in the tessellation though not as straight forward.

They should also include the rotation and translation description as these three transformations combine to make the tessellation. In this session students are pulling on their previous learning experiences to show how tessellating shapes are developed.

Students choose a tile from the set in Copymaster 5. Students should trace or make several copies of the tile to see if the shape tessellates. If the shape tessellates students are to describe the transformations involved in the tessellation and to describe how the shape was created e. Students then choose one of these shapes or one of the previous shapes they have created or design a new shape and produce a tessellation onto an A3 size poster.

To add colour, students could trace their shapes onto square note pad paper and glue together to make the tessellation, or they could transfer their shape onto paper and colour using pencils, crayons or felts.

In this final session students can add their imagination to their design through colour, texture and imagination. Activity to show use of colour and texture: Hexagon to small rhombuses cut-out demonstrated on the board. It is such fun to do some tessellating! I've been using some Polydron plastic shapes which click together and making some rules before setting off: Rule one - we will use six equilateral triangles for a shape and that'll be our tessellating shape.

Rule two - we can flip the shape over and turn it around but the basic shape - the way the triangles are connected - must remain the same.

Well, here's one I started with: Then I tried to put a number of these together; I had to flip the brown one over to get the red one, rotate it to get the blue one, and so on. Having done this much of the tessellation, I could see that it could easily continue on for ever! Can you find out or see why? Then I tried another: This came about by just turning the brown one around to get the greens and blue.

Do you think this one could go on and on? Between and Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings. He adopted a highly mathematical approach with a systematic study using a notation which he invented himself. There are 4 ways of moving a motif to another position in the pattern.

These were described by Escher. A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place. The translation shows the geometric shape in the same alignment as the original; it does not turn or flip.

A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right over a "y" axis or flipped to the top or bottom over an "x" axis , reflections can also be done at an angle. If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical "mirror" images.