Tessellations can be found in the hobby or art of origami. Paper is folded into triangles, hexagons, and squares to form many different patterns and shapes. This tessellation is called the honeycomb, another place to find tessellations in the real world. Another hobby of sorts are puzzles. There are puzzles on the market that you can manipulate different shapes and make your ow n tessellations out of those.
The art of M. Fathauer now promotes mathematical art at exhibitions and conferences. His products look excellent for any classroom teacher. Some people say that he is the expert in recreational math.
Roger Penrose , a professor of mathematics at the University of Oxford in England, pursues an active interest in recreational math which he shared with his father. While most of his work pertains to relativity theory and quantum physics, he is fascinated with a field of geometry known as tessellation, the covering of a surface with tiles of prescribed shapes.
Penrose received his Ph. While there, he began playing with geometric puzzles and tessellations. Penrose began to work on the problem of whether a set of shapes could be found which would tile a surface but without generating a repeating pattern known as quasi-symmetry.
After years of research and careful study, he successfully reduced the number to six and later down to an incredible two. Believe it or not, but the shapes he came up with are like the chemical substances that form crystals in a quasi-periodic manner.
Not only that, but these quasi-crystals make excellent non-scratch coating for frying pans. Penrose and Escher have been influences on each other.
Penrose saw some of Escher's work there and began playing with tessellations and came up with what he calls a tri-bar. A tri-bar is a triangle that looks like a three-dimensional object, but could not possibly be three-dimensional in real life. He published his work in the British Journal of Psychology. Escher read his article.
Says Penrose, "One was the tri-bar, used in his lithograph called Waterfall. Another was the impossible staircase , which my father had worked on and designed. Escher used it in Ascending and Descending, with monks going round and round the stairs. He did this, and then he wrote to me and asked me how it was done—what was it based on?
So I showed him a kind of bird shape that did this, and he incorporated it into what I believe is the last picture he ever produced, called Ghosts. Triangles, squares and hexagons are the only regular shapes which tessellate by themselves. If you put many of these shapes together side-by-side, they form a tessellation.
You can have other tessellations of regular shapes if you use more than one type of shape. You can even tessellate pentagons, but they won't be regular ones. The benefits of making connections to other subjects using art are well documented. Carol Goodrow, a first grade teacher saw improvement in math skills by making connections through other areas. For example, during year-end benchmark testing, her class completed sections on numeration more quickly, yet scored as well or better, than past classes.
In addition, Goodrow reported that many students demonstrated a better understanding of fractions. They represented the fractions by models, but they could also compute them in their heads. Canadian Math teacher Jill Britton uses Escher tessellations to help students learn the mathematics term, congruent.
While examining Escher's picture, Tessellation , she says, "When the students study a pegasus in its parent square, they discover how Escher modified the square to obtain his creature. Corresponding modifications are related by translation. The area of the parent square is maintained. Tessellation Artist Mike Wilson. Tessellation Artist Richard Hassell. Tessellation Artist Andrew Crompton. Tessellation Artist Emily Grosvenor. Tessellation Artist Bruce Bilney.
Tessellation Artist Seth Bareiss. Tessellation Artist Marjorie Rice. Tessellation Artist David B. Tessellation Artist Henk Wyniger. Tessellation Artist Yoshiaki Araki. Tessellation Artist David Bailey. Tessellation Artist Chris Watson. Tessellation Artist Chris Edwards. Each tile may contain non-tessellating decorative elements as well. Notice the faint vertical and diagonal guidelines used to align the tiles.
Cone mosaic pattern columns, ca. Tessellations are not the artistic stepchild of modern-day mathematics. They go back some five-thousand years to the Sumerian culture above of around BC located in modern-day Iraq. Since that time, they've been an element in virtually every civilization having developed an advanced decorative culture.
In the German mathematician, Johannes Kepler, made one of the earliest documented study of tessellations. He wrote about regular and semi-regular tessellations in his Harmonices Mundi.
He was the first to explore and explain the hexagonal structures of honeycombs and snowflakes. About two-hundred years later, in , the Russian scientist, Yevgraf Fyodorov, in studying the arrangement of atoms in the crystalline solids, proved that every periodic tiling features one of seventeen different groups of isometries reflections, rotations, and translations.
Fyodorov was the first to engage in a mathematical study of tessellations. An Alhambra tessellation as drawn by M. Escher,
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