orchid

math art

Escher spent many days sketching these tilings, and later claimed that this “was the richest source of inspiration that I have ever tapped.” In 1957 he wrote an essay on tessellations, in which he remarked:

In mathematical quarters, the regular division of the plane has been considered theoretically . . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature thay are more interested in the way in which the gate is opened than in the garden lying behind it.

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. (Many more irregular polygons tile the plane – in particular there are many tessellations using irregular pentagons.) Escher 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. He also elaborated these patterns by “distorting” the basic shapes to render them into animals, birds, and other figures. These distortions had to obey the three, four, or six-fold symmetry of the underlying pattern in order to preserve the tessellation. The effect can be both startling and beautiful.


In Reptiles the tessellating creatures playfully escape from the prison of two dimensions and go snorting about the destop, only to collapse back into the pattern again. Escher used this reptile pattern in many hexagonal tessellations. In Development 1, it is possible to trace the developing distortions of the square tessellation that lead to the final pattern at the center.

3D Game

Object Locative Environment Coordinate System

levelHead: A 3D Spatial Memory Game

inspired

I am trying to design some shape which is related to the triangle trick.

how to draw an impossible triangle

Method 1
1.Sketch an equilateral triangle. This will be the center of your triangle.


2.Lightly sketch two parallel lines outside one side of the triangle. The lines should be equally spaced. Take caution that your lines are drawn straight.
3.Do this for each of the other two sides. Your sketch should look like three triangles nested together.





4.Choose one side of the "center" triangle. Extend one end of that straight line until it reaches the "middle" triangle.
5.Find the same side of the "middle" triangle. Extend one end of that straight line, in the same direction as before, until it reaches the "outside" triangle.

6.Repeat steps for the other two sides of the triangle.

7.Erase short segments so that the triangle begins to look three-dimensional rather than flat. Each edge of this "3-D" shape should look like a reverse "L".

8.Add short segments at an angle in the corners. These short segments will finish off the outside points.

9.Cleanup your drawing by erasing the points outside of the short segments drawn in the previous step.
10.Add shading if desired.

Method 2
Draw a triangle and extend the ends of the lines past where they join.
Draw lines from these tips, extending them beyond the corners of the inner triangle.
Draw in the 'corners'.
Draw in the final long lines to connect the corners.

flying fish

Escherization

the Dutch graphic artist M.C. Escher created over a hundred ingenious tesselations in the plane. Some were simple and geometric, used as prototypes for more complex endeavors. But in most the tiles were recognizable animal forms such as birds, fish and reptiles. The definitive reference on Escher's divisions of the plane is Doris Schattschneider's Visions of Symmetry, now in a second edition.

It is based on three large components:

• A parameterized space of tilings. We develop a parameterization of the space of isohedral tilings, a family of tilings that can express the designs Escher created. Every possible tile shape boils down to a sequence of real numbers. An interactive tool lets us modify these numbers in a natural way (corresponding to Escher manipulating drawings by hand). More importantly, we can direct the computer to search through possible sequences of numbers, trying to find good tilings by brute force.
• A comparison function for shapes. We can dial up a collection of tile shapes using the parameterization above. We need to know how well each of these tiles approximates the goal shape provided by the user. We use a metric from the computer vision literature that efficiently finds the L² distance between two polygons. Our objective then becomes to locate the particular tile shape that compares most favourably to the goal shape.
• An optimization algorithm. We need a meaningful strategy for sorting through the large shape of possible tile shapes for the best one (the one that's closest to the goal shape). We use a simulated annealing algorithm from Numerical Recipes. When suitably tuned, it tends to find good tile shapes without getting stuck in objectionable local minima.

We might enjoy playing with the edges of shapes:

movie inspired by escher

15-year-old Sarah accidentally wishes her baby half-brother, Toby, away to the Goblin King Jareth who will keep Toby if Sarah does not complete his Labyrinth in 13 hours.

there are a lot of scenes that was inspired by escher
escher
escher labyrinth

escher