4.4 Soothing Symmetry and Spinning PinwheelsFriday, February 20, 2009

Symmetry

Mirror Images Beauty? Rigid Symmetry – motion of the plane that

preserves the pattern and does not shrink, stretch, or otherwise distort the plane

Shift, rotation, flip or combination of these

Types of Symmetry

Line Symmetry

Rotational Symmetry

Symmetry of Scale

Also known as scalable If the tiles that make up the pattern can be

grouped into super-tiles that still cover the plane and, if scaled down, can be rigidly moved to coincide with the original pattern

Checkerboard

Tessellations

Tiling the plane

Regular tessellation - means a tessellation made up of congruent regular polygons

Semi-regular Tessellations

Name Some More

Demi-regular Tessellations

And Another

Patterns in Nature

Chaotic Patterns Penrose patterns – no rigid symmetries

that use only two tile shapes, kites and darts

Penrose Patterns

More about Penrose Patterns

Every tile occurs in one of 10 possible orientations in the plane

Penrose Tiling

December 2003: Sir Roger Penrose, the eminent British mathematician, came face to face with his own copyrighted polygon pattern in Kleenex quilted toilet paper. When his wife returned from the market with the embossed rolls, Penrose expressed "astonishment and dismay" upon seeing the use to which his discovery had been put.

Penrose devised the nonrepeating five-fold symmetrical pattern in the 1970s by using two kinds of diamond shapes—fat and thin—to create what is now called Penrose tiling. The pattern, which was thought not to exist in nature before Penrose's discovery, has subsequently been found in many physical and biological phenomena.

Pinwheel Pattern 1994 – John Conway of Princeton and Charles

Radin of the University of Texas-Austin Uses one single triangular tile Symmetry of scale, but no rigid symmetry Tiles occur in infinitely many orientations Group by 5 to form super-tiles

Pinwheel Properties

Uniqueness of Scaling – there is only one way to group the Pinwheel Triangles into super-tiles to create a Pinwheel super-pattern in the plane

No rigid symmetries

MC Escher

M.C. Escher was a Dutch graphic artist, most recognized for spatial illusions, impossible buildings, repeating geometric patterns (tessellations), and his incredible techniques in woodcutting and lithography.

M.C. Escher was born June 1898 and died March 1972.

Escher’s Works

More Escher

Problem of the Day

You're a cook in a restaurant in a quaint country where clocks are outlawed. You have a four minute hourglass, a seven minute hourglass, and a pot of boiling water. A regular customer orders a nine-minute egg, and you know this person to be extremely picky and will not like it if you overcook or undercook the egg, even by a few seconds. What is the least amount of time it will take to prepare the egg, and how will you do it?