#Escher tessellation software
#Escher tessellation free

In the first, Cycle, the running figures emerge from an orderly world to descend into a topsy-turvey chaos, but this chaos itself gives rise to the very order from which the figures emerge. The last two examples each use a hexagonal tesselation. To emphasize the nature of the underlying pattern, Escher allows us to trace the developing distortions of the tesselation that lead to the pattern at the center. The second example, Development I, uses a square tesselation. (To see an overlay of the triangle pattern, click on the thumbnail image to expand the large version, and then hover over it with the mouse pointer.) The first of the examples presented here, titled Regular Division of the Plane with Birds, uses a tesselation with triangles. The effect can be both startling and beautiful.

These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in order to preserve the tessellation. He also elaborated these patterns by distorting the basic shapes to render them into animals, birds, and other figures. (Many more irregular polygons tile the plane-in particular there are many tessellations using irregular pentagons.)Įscher exploited these basic patterns in his tessellations, applying what geometers would call reflections, glide reflections, translations, and rotations to obtain a greater variety of patterns. Whether or not this is fair to the mathematicians, it is true that they had shown that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation.

#Escher tessellation free

Escher, however, was fascinated by every kind of tessellation-regular and irregular-and took special delight in what he called “metamorphoses,” in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself. Typically, the shapes making up a tessellation are polygons or similar regular shapes, such as the square tiles often used on floors. Regular divisions of the plane, called tessellations, are arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps. Thus, for the student of mathematics, Escher’s work encompasses two broad areas: the geometry of space, and what we may call the logic of space. He was also fascinated with paradox and “impossible” figures, and used an idea of Roger Penrose’s to develop many intriguing works of art. He is of course also much imitated.Īs his work developed he drew great inspiration from the mathematical ideas he read about, often working directly from structures in plane and projective geometry, and eventually capturing the essence of non-Euclidean geometries, as we will see below.

Reproductions of his work remain in strong demand, and he has inspired thousands of other artists to pursue mathematical themes in their own work.

#Escher tessellation software

His work eventually appeared not only in printed form, but as commissioned or imitative sculptures on public buildings, as decorations on everything from neckties to mousepads, and in software written to automate the reproduction and manipulation of tesselations.

Zeno’s Paradox of the Tortoise and AchillesĮscher-like motif on a building in The Hague, Netherlands.