Victoria Potter
Amy Donato
Renee Staudt
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Does there exist a non-periodic set of prototile’s
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.
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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 prototile’s that tile the plane
but never with translational symmetry.
Prototile’s
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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.
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In 1964, Robert Berger disproved Hao Wang’s
hypothesis and proved that set’s 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.
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Roger Penrose was the first to give a set of
variations in prototile’s.
First, he also found a set of 6 prototile’s that
cause non-periodicity in the plane.
He then soon lowered that number to four.
Then all of those led to Roger Penrose’s,
along with help from Robinson, John Conway
and Robert Ammann’s, discovery of just two
set’s of tile-types that were not squares that
cause non-periodicity.
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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.
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).
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
Emperor’s New Mind: Concerning Computers,
Minds, and the Laws of Physics (1989)
 Shadows of the Mind: A Search for the Missing
Science of Consciousness (1994).
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A Penrose tiling is a two tile-type, non-periodic
tiling in a plane generated by an aperiodic set
of prototile’s.
There are 3 types of Penrose tiling’s
 The original tiling
 The Kite and Dart tiling
 The Rhombus tiling
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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 tiling’s 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.
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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.
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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
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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 tiling’s 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.
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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 prototile’s.
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The thin rhombus has
angles 36°, 144°, 36°,
144°and can be bisected
along its short diagonal to
form a pair of triangles
that can be used for
substitution.
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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.
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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 nonperiodically.
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.
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Must be constructed so the slots and projections
on the edges fit together.
Rhombus tiling is the other most popular Penrose
tiling.
Repeated generations of deflation produce a tiling of the
original axiom shape with smaller and smaller shapes.
Original
Kite (half)
Dart (half)
Sun
Star
Generation 1
Generation 2
Generation 3
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
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The rhombus tiling is the most famous of all
Penrose tiles.
To construct a rhombus tiling you must follow
rules on construction of putting rhomb’s
together.
The point of a Penrose tile is that it is nonperiodic.
There are many ways to construct periodic
rhombus tiling’s that are obvious.
There is more to tiling then just the shape’s,
it has to do with using the rhomb’s and rules.
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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.
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.
Penrose tiling’s are known to be in 2D.
It seems that Penrose tiling’s could be
extended to a 3D case, but just the 2D case is
proven to agree with the structure and
algebraically.
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"Aperiodic Tiling's." . N.p.. Web. 19 Nov 2012.
http://www.math.cornell.edu/~mec/20082009/KathrynLindsey/PROJECT/Page5.htm.
Bartlomiej Kozakowski, Janusz Wolny. Faculty of Physics
and Applied Computer Science, AGH University of Science
and
Technologyhttp://www.uwgb.edu/dutchs/symmetry/penro
se.htm .
Eric G. Swedin, editor. Science in the Contemporary World:
An Encyclopedia. 2005.
http://rave.ohiolink.edu/ebooks/ebc/CSCIENE
"Penrose Kite-Dart." Tiling's Encyclopedia. N.p.. Web. 19
Nov 2012. <http://tilings.math.unibielefeld.de/substitution_rules/penrose_kite_dart>
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"Penrose Tiles and Aperiodic Tessellations'." . N.p.. Web. 19
Nov 2012. http://math.uchicago.edu/~mann/penrose.pdf
"Penrose Tiling." . N.p., 31 2008. Web. 19 Nov 2012.
http://www.cs.brandeis.edu/~storer/JimPuzzles/PACK/Czech
Farms/PenroseTilingWikipedia.pdf.
"Penrose Tiling." Science U. Geometry Technologies, 19 2005.
Web. 19 Nov 2012.
http://www.scienceu.com/geometry/articles/tiling/penrose.
html.
"Robinson Triangle." Tiling's Encyclopedia. N.p.. Web. 19 Nov
2012. http://tilings.math.unibielefeld.de/substitution_rules/robinson_triangle
Steven Dutch, Natural and Applied Sciences, University of
Wisconsin - Green Bay. Last Update 11 August 1999.
http://arxiv.org/ftp/cond-mat/papers/0503/0503464.pdf