Presentation on theme: "4.4 Soothing Symmetry and Spinning Pinwheels"— Presentation transcript:
14.4 Soothing Symmetry and Spinning Pinwheels Friday, February 20, 2009
2Symmetry Mirror Images Beauty? Rigid Symmetry – motion of the plane that preserves the pattern and does not shrink, stretch, or otherwise distort the planeShift, rotation, flip or combination of these
3Types of SymmetryLine SymmetryRotational Symmetry
4Symmetry 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 patternCheckerboard
5Tessellations Tiling the plane Regular tessellation - means a tessellation made up of congruent regular polygons
13More about Penrose Patterns Every tile occurs in one of 10 possible orientations in the plane
14Penrose TilingDecember 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.
15Pinwheel Pattern1994 – John Conway of Princeton and Charles Radin of the University of Texas-AustinUses one single triangular tileSymmetry of scale, but no rigid symmetryTiles occur in infinitely many orientationsGroup by 5 to form super-tiles
18Pinwheel PropertiesUniqueness of Scaling – there is only one way to group the Pinwheel Triangles into super-tiles to create a Pinwheel super-pattern in the planeNo rigid symmetries
19MC EscherM.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.
22Problem of the DayYou'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?