Victoria Potter Amy Donato Renee Staudt Slide 2 Does there exist
a non-periodic set of prototiles to create an aperiodic set in the
plane? Tiling problems have been studied for years by computer
scientists and exist in discrete and computational geometry. There
are many rules when it comes to tiling problems and much of them
involve symmetry restrictions (whether tiles can be flipped or
rotated) according to certain rules. For many years it had been
belief that no such set exists. Slide 3 Tiling-when you fit
individual tiles together with no gaps or overlaps to fill a flat
space (plane). Prototile-is a (finite or infinite) set of tiles
that represent all the tiles of a tiling, or of a class of
tiling's. Periodic-has translational symmetry, but must be in at
least two non-parallel directions. Non-periodic means it lacks
translational symmetry, a shifted copy will never match the
original exactly. Aperiodic-a set of prototiles that tile the plane
but never with translational symmetry. Slide 4 Prototiles Slide 5
Slide 6 Slide 7 This problem was first posed in 1961 by Hao Wang,
when he used 4-way dominoes, now known as Wang tiles, to
hypothesize that these dominoes can tile the plane if and only if
it can do so periodically (in a pattern that repeats in two
different non-parallel directions). Wang tried to find a method for
deciding if any set of dominoes will tile non-periodically by
putting the same colored sides against each other. Rotations and
reflections not being allowed. Wang concluded that any set of tiles
that can tile the plane will do so periodically, so no set of
non-periodic tiles could possibly exist. Slide 8 Slide 9 In 1964,
Robert Berger disproved Hao Wangs hypothesis and proved that sets
of tiles can tile the plane non-periodically. He found an aperiodic
set with 20,426 dominoes and later reduced that number to 104.
Donald Knuth reduced the number to 92 dominoes. Raphael Robinson
gave a set of just 6 tiles by putting projections and slots into
the edges to get jigsaw pieces that work the same with the colors
and created a non-periodic prototile. Slide 10 Slide 11 Roger
Penrose was the first to give a set of variations in prototiles.
First, he also found a set of 6 prototiles that cause
non-periodicity in the plane. He then soon lowered that number to
four. Then all of those led to Roger Penroses, along with help from
Robinson, John Conway and Robert Ammanns, discovery of just two
sets of tile-types that were not squares that cause
non-periodicity. Slide 12 Roger Penrose was born in 1931 in
Colchester, Essex, England, Father Medical Geneticist Mother
Medical Doctor Penrose earned his Ph.D. in mathematics at Cambridge
University in 1957 and married in 1959He argued that the human
brain can carry out processes that no computer can do, running
counter to the general tendency among other researchers in the
field of artificial intelligence. Slide 13 Visited various
universities in Brittan and the United States before settling down
as a professor of applied mathematics at Birbeck College, London,
in 1967. Penrose became the Ball Professor of Mathematics at Oxford
University in 1973. Penrose retired from Oxford in 1998 Professor
of Geometry at Gresham College in London Penrose and his father
gained some popular recognition when they devised some geometrical
figures later used by the Dutch surrealist artist M. C. Escher
(1898 1972). Slide 14 Penrose and Hawking shared the 1988 Wolf
Prize in Physics for their work on black holes and relativity.
Penrose became a fellow of the Royal Society in 1972, received the
Royal Society Royal Medal in 1985, and was knighted in 1994. His
interests turned to computers and artificial intelligence, and he
published the best-selling The Emperors New Mind: Concerning
Computers, Minds, and the Laws of Physics (1989) Shadows of the
Mind: A Search for the Missing Science of Consciousness (1994).
Slide 15 A Penrose tiling is a two tile-type, non-periodic tiling
in a plane generated by an aperiodic set of prototiles. There are 3
types of Penrose tilings The original tiling The Kite and Dart
tiling The Rhombus tiling Slide 16 Proposed in 1974 in a paper
titled, Role of aesthetics in pure and applied research. Penrose
got his inspiration from Johannes Kepler. Kepler in his book,
Harmonices Mundi, discusses tilings built around pentagons. Penrose
used this to discover that it could be expanded into a Penrose
tile. When tiling the plane with regular pentagons it leaves gaps
between the tiles, which breaks the rule of what tiling is. Slide
17 Penrose formed one specific tiling that could be filled with
three different shapes. A star A boat A diamond (or thin rhombus)
Earlier ideas of this had been traced back, but Penrose was the
first to prove this idea. Slide 18 Slide 19 The Kite is a
quadrilateral whose interior angles are 72, 72, 72, and 144 The
Dart is a non-convex quadrilateral whose interior angles are 36,
72, 36, 216 Kite Dart Slide 20 The green and red arcs in the Kite
and Dart restrict the placement of the tiles. When two tiles share
an edge in a tiling, the patterns must match at these edges. First
of the tilings that contained only two distinct tiles. The kite and
dart can be bisected to form pairs of triangles that can be used
for substitution tiling. Slide 21 The kite and dart are one of the
most well known and popular tiling patterns. These are two of the
most used kite and dart prototiles. Slide 22 Slide 23 The thin
rhombus has angles 36, 144, 36, 144and can be bisected along its
short diagonal to form a pair of triangles that can be used for
substitution. The thick rhombus has angles 72, 108, 72, 108 and can
be bisected along its long diagonal to form a pair of triangles
that can be used for substitution. Slide 24 Ordinary rhombus-shaped
tiles can be used to tile the plane, but they tile periodically.
There are specific rules that you must follow when placing rhombus
tiles so they tile non- periodically. Two tiles cannot be put
together to form a single parallelogram. Only particular sides can
be put together with one another. They must be assembled so that
the curves match in color and position. Must be constructed so the
slots and projections on the edges fit together. Rhombus tiling is
the other most popular Penrose tiling. Slide 25 Slide 26 Repeated
generations of deflation produce a tiling of the original axiom
shape with smaller and smaller shapes. Original Generation 1
Generation 2 Generation 3 Kite (half) Dart (half) Sun Star Slide 27
Drop City artist Clark Richert used Penrose rhombi in artwork in
1970. In more recent times Computer artist Jos Leys has produced
numerous variations on the Penrose theme Art historian Martin Kemp
has commented a contemporary decoration which used Penrose tiles
and observed that Albrecht Durer has sketched similar motifs of a
rhombus tiling Slide 28 The rhombus tiling is the most famous of
all Penrose tiles. To construct a rhombus tiling you must follow
rules on construction of putting rhombs together. The point of a
Penrose tile is that it is non- periodic. There are many ways to
construct periodic rhombus tilings that are obvious. There is more
to tiling then just the shapes, it has to do with using the rhombs
and rules. Slide 29 When constructing a Penrose tiling, two
adjacent vertices must both be blank or must both be black. If two
edges lie next to each other they must both be blank, or both have
an arrow. If the two adjacent edges have arrows, both arrows must
point in the same direction. Slide 30 The inner section in purple
is surrounded blue sections making a decagon. The outer section is
made up of two parts. The ten blue spokes and ten yellow sectors.
Slide 31 Penrose tilings are known to be in 2D. It seems that
Penrose tilings could be extended to a 3D case, but just the 2D
case is proven to agree with the structure and algebraically. Slide
32 "Aperiodic Tiling's.". N.p.. Web. 19 Nov 2012.
2009/KathrynLindsey/PROJECT/Page5.htm Bartlomiej Kozakowski, Janusz
Wolny. Faculty of Physics and Applied Computer Science, AGH
University of Science and
se.htm.http://www.uwgb.edu/dutchs/symmetry/penro se.htm Eric G.
Swedin, editor. Science in the Contemporary World: An Encyclopedia.
http://rave.ohiolink.edu/ebooks/ebc/CSCIENE "Penrose Kite-Dart."
Tiling's Encyclopedia. N.p.. Web. 19 Nov 2012. Slide 33 "Penrose
Tiles and Aperiodic Tessellations'.". N.p.. Web. 19 Nov 2012.
"Penrose Tiling.". N.p., 31 2008. Web. 19 Nov 2012.
Farms/PenroseTilingWikipedia.pdf "Penrose Tiling." Science U.
Geometry Technologies, 19 2005. Web. 19 Nov 2012.
"Robinson Triangle." Tiling's Encyclopedia. N.p.. Web. 19 Nov 2012.
bielefeld.de/substitution_rules/robinson_triangle Steven Dutch,
Natural and Applied Sciences, University of Wisconsin - Green Bay.
Last Update 11 August 1999.