Convex Polygons

Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane. In both cases, the angle sum of the shape plays a key role. Since triangles have angle sum 180° and quadrilaterals have angle sum 360°, copies of one tile can fill out the 360° surrounding a vertex of the tessellation.

The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon. The angle sum of any pentagon is 540°, because we can divide the pentagon into three triangles:


Unlike the triangle and quadrilateral case, the pentagon's angle sum of 540° is not helpful when trying to fit a bunch of pentagons around a vertex. In fact, there are pentagons which do not tessellate the plane.
some are not working:



some are working


Non-Euclidean Geometry:

design_fish(Tessellations)

I try to put the fish image which I made before by the Tessellations math.

design_aboriginal lin cloth(Tessellations)

Tessellations by Polygons

The picture works because all three corners (A, B, and C) of the triangle come together to make a 180° angle - a straight line. This property of triangles will be the foundation of our study of polygon tessellations

Since the angle sum of any triangle is 180°, and there are two triangles, the angle sum of the quadrilateral is 180° + 180° = 360°. Taking a little more care with the argument, we have:

Then

Math and the Art of MC Escher

I am reading the online book of Math and the Art of MC Escher.http://math.slu.edu/escher/index.php/Main_Page
there are some topic which help me a lot.(I will keep to post)
Tessellations:
A tessellation is a covering of the plane by shapes, called tiles, so that there are no empty spaces and no overlapped tiles. Tessellations are also called tilings.Some tessellations involve many types of tiles, but the most interesting tessellations use only one or a few different tiles to fill the plane. If the tiles are regular polygons (all sides are congruent and all angles are congruent) and vertices only meet vertices we say that the tessellation is a regular tessellation.An example of tessellations using only a few different tiles are the semi-regular tessellations. These tessellations are made up of two or more types of regular polygons, vertices only meet vertices and the configuration around every vertex is the same.



Escher lays out his methods for creating tessellations:

1.Begin with a tessellation by geometric shapes.
2."Mold the form". Here, he bends and manipulates the straight lines of the geometric scaffolding.
3.Look at the resulting pattern and attempt to recognize a figure.
4.Push the pattern towards the desired shape.

a interview from escher

just see a interview from M.C escher.
the idea of all the architecture is interesting.

cool effect

http://www.hypa.tv/tims/animations.html

there are some cool effect that might help me.