Transcript 4.4 Soothing Symmetry and Spinning Pinwheels

4.4 Soothing
Symmetry and
Spinning Pinwheels
Friday, 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 superpattern 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?