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Tilings in Math, Art and
Science
Bob Culley
Tiling: universal, fundamental pattern making
Islamic Tiling Patterns
Perhaps the most extensive use and development of tiling patterns is found in Islamic Art.
These patterns spread with the rise of Islamic societies from Spain to China: the Umayyads,
Abbasids, Fatimids, Seljuqs, Ilkhanids, Timurids, Safavids, Ottomans, Mughals, 7th century on
Darb-i Imam Shrine, Isfahan, Iran 1453 Tash Hauli Palace, Khiva, Uzbekistan, 19th century
“Know, oh brother...that the study of sensible geometry leads
to skill in all the practical arts, while the study of intelligible
geometry leads to skill in the intellectual arts because this
science is one of the gates through which we move to the
knowledge of the essence of the soul, and that is the root
of all knowledge... “ from the Rasa’il of the Brethren of Purity,
10th century C.E., translated by S.H.Nasr, in “Islamic Patterns”,
Keith Critchlow
“The artist and the mathematician in Arab civilization have
become one. And I mean quite literally.” - Jacob Bronowski,
quoted in “Symmetries of Islamic Geometrical Patterns”,
Syed Jan Abas and Amer Shaker Salman
Connecting Art and Mathematics
Historical documents from the House of Wisdom:
On the Geometric Constructions Necessary for
the Artisan, by Abu’l-Wafa Buzjani (ca. 940–
998), anonymous work, On Interlocks of Similar
or Corresponding Figures (ca. 1300)
The Wallpaper (a.k.a. Plane Crystallographic) Groups
are 17 symmetries composed of translations, rotations,
reflections and glide reflections
Starting with Edith Müller’s thesis in 1944,
who found 12 of the groups, mathematicians
have debated whether all 17 occur in the
Alhambra.
Branko Grünbaum in 2006 questioned the
definition of the problem: does color count
in the symmetries or just shape? If colors
count, there are 17. He also wrote:
“Groups of symmetry had no significance to
the artists and artisans who decorated the
Alhambra”
Modern Mathematical View of Tiling
Some single (monohedral) shapes don’t tile.
Which do? What’s a good prototile?
Not a packing,
which can have
gaps
Not a covering,
which can have
overlaps
A plane tiling T is a countable family of closed sets which cover the
(Euclidean) plane without gaps or overlaps.
Each tile T is a closed topological disk, i.e. the tile boundaries are simple
closed curves. The union of the tiles is the plane, and the interiors of the
tiles are pairwise disjoint. (from Tilings and Patterns, Grünbaum and
Shephard)
Single Regular Polygon Tiling: 3, 4, 6, not 5
Many other kinds of tilings
Regular polygons, but not
edge-to-edge
Voderberg’s non-convex 9-gon
(ennagon) spiral tiling
One of Kepler’s tilings
including star polygons
Aperiodic Tilings
Periodic Tiling: lines up with itself after a plane translation. Aperiodic: can’t
1961: Hao Wang conjectures: If a finite set of tiles will tile the plane, it can do so periodically.
and that there should exist an algorithm to determine whether any given set of tiles will do this.
1966: Robert Berger showed (using an equivalence to the Halting Problem!) that no such
algorithm exists, and aperiodic sets of tiles exist. Berger came up with a set of 20,426 such tiles.
Then reduced them to 104. Don Knuth reduced them to 92. Karel Culik came up with these 13:
Aperiodic 2: Robinson to Penrose
1971: Raphael M. Robinson found an
aperiodic set of 6 modified rectangular tiles;
used projections and dents rather than
coloring matching
1974: Roger Penrose produces the dart
and kite aperiodic prototile set. John H.
Conway suggests a colored line based
matching rule.
Can a single prototile set
be aperiodic?
It is still an open
question.
“Everywhere there is
found...a silent swerving
from accuracy by an inch
that is the uncanny
element in everything…
a sort of secret treason
in the universe.”
G.K. Chesterton as
quoted by Martin Gardner
(in Penrose Tiles and
Trapdoor Ciphers)
Aperiodic patterns with Penrose tiles
X-Ray Crystallography
~1912 Max Von Laue developed X-Ray Crystallography (Physics Nobel 1914)
This becomes a standard
crystallography method:
over 400,000 solids were
characterized in 70 years. All fit
this “3,4,6 and not 5” model.
Three-fold, four-fold, and
six-fold symmetries. Five-fold
and higher than six-fold
symmetries are “proven”
impossible.
Connecting Aperiodicity to Crystals
1982: Alan Lindsay Mackay, a crystallographer, puts circles at the
intersections of a Penrose tiling, computes the diffraction
pattern: the result is a 10-fold symmetry
April 8 1982: Dr. Dan Shechtman, from the
Technion in Israel, doing metallurgy
experiments at NBS (now NIST) on AlMg
sees 10-fold symmetry in his electron
beam crystal diffraction patterns.
He tries unsuccessfully to publish his
results, has other crystallographers review
and check his results.
10 Fold ???
"There's no such thing as
quasicrystals, only quasi-
scientists." - Linus Pauling
1984: Physicists Paul Steinhardt and Dov
Levine connect the work of Mackay and
Shechtman, coin the term quasicrystal
in an article weeks after Shechtman’s work
is finally published.
Significant resistance to quasicrystals continued, but so did
experimental results.
In 1993, the International Union of Crystallographers
changed the definition of “crystal” to include
quasiperiodic crystals and many other structures.
2010: Mackay, Steinhardt and Levine get the Buckley Prize
in Physics
2011: Dan Shechtman receives the Nobel Prize in
Chemistry
Hundreds of quasicrystals have now been
found, including the natural quasicrystal
icosahedrite.
The Girih Tiles
In 2007, Peter Lu, a student of Paul
Steinhardt, came up with the Girih tiles
They match many Islamic
tiling patterns and they match the
Penrose dart/kites. Later Lu found
the Girih tiles match the Topkapi
scroll (~1500 c.e.), a Timurid set of
instructions for artisans
The Girih Tiles as a Penrose Tiling The Girih Tiles fit the tiling of the Darb-i
Imam Shrine (and many other tilings)
References and Resources:
The Wikipedia pages on tessellation, Wallpaper Group, Crystallographic Restriction Theorem,
aperiodic tiling, quasicrystal and each of the individuals named in these slides are pretty good.
YouTube has presentations of their work by Dan Shechtman, Paul Steinhardt, and Peter Yu.
Books:
Introduction to Tessellations by Dale Seymour and Jill Britton: very simple, graphic explanations, art
oriented, but includes the algebra shown in this talk
Tilings and Patterns by Branko Grünbaum and G.C. Sheppard: encyclopedic, definitive work on
mathematics of tilings
Penrose Tilings to Trapdoor Ciphers and the Return of Dr. Matrix by Martin Gardner: the first two
chapters give the history of aperiodic tilings up to the discovery of quasicrystals
Symmetries of Islamic Geometric Patterns by Syed Jan Abas and Amer Shaker Salman: discusses
some of the history of Islamic patterns (not just tilings) then catalogs many patterns according to
Wallpaper group symmetry
Appendix: Archimedean tilings
Archimedes (ca. 287-212 B.C.E) gave
us the Stomachion (see the Archimedes
Codex) - the oldest known geometric
puzzle - but not the Archimedean tilings
Semi-regular polygonal tilings
The general formula for the interior angle of an n-gon:
So for three regular polygons with sides n1, n2, n3 to fit together:
This simplifies to:
Extending this analysis to combinations of regular polygons, there are 21 combinations possible.
17 are distinct combinations: 4 are just different orderings of the same sets of polygons.
11 of these fit together to tile the plane, called the uniform or Archimedean tilings
Semi-regular Polygonal Tilings
Similarly for four, five or six polygons:
But six is a maximum, since 60 degrees is the smallest regular polygon angle
Each solution corresponds to a tiling pattern, e. g.

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Tilings in Art, Math and Science - Bob Culley

  • 1. Tilings in Math, Art and Science Bob Culley
  • 3. Islamic Tiling Patterns Perhaps the most extensive use and development of tiling patterns is found in Islamic Art. These patterns spread with the rise of Islamic societies from Spain to China: the Umayyads, Abbasids, Fatimids, Seljuqs, Ilkhanids, Timurids, Safavids, Ottomans, Mughals, 7th century on Darb-i Imam Shrine, Isfahan, Iran 1453 Tash Hauli Palace, Khiva, Uzbekistan, 19th century
  • 4. “Know, oh brother...that the study of sensible geometry leads to skill in all the practical arts, while the study of intelligible geometry leads to skill in the intellectual arts because this science is one of the gates through which we move to the knowledge of the essence of the soul, and that is the root of all knowledge... “ from the Rasa’il of the Brethren of Purity, 10th century C.E., translated by S.H.Nasr, in “Islamic Patterns”, Keith Critchlow “The artist and the mathematician in Arab civilization have become one. And I mean quite literally.” - Jacob Bronowski, quoted in “Symmetries of Islamic Geometrical Patterns”, Syed Jan Abas and Amer Shaker Salman
  • 5. Connecting Art and Mathematics Historical documents from the House of Wisdom: On the Geometric Constructions Necessary for the Artisan, by Abu’l-Wafa Buzjani (ca. 940– 998), anonymous work, On Interlocks of Similar or Corresponding Figures (ca. 1300)
  • 6. The Wallpaper (a.k.a. Plane Crystallographic) Groups are 17 symmetries composed of translations, rotations, reflections and glide reflections Starting with Edith Müller’s thesis in 1944, who found 12 of the groups, mathematicians have debated whether all 17 occur in the Alhambra. Branko Grünbaum in 2006 questioned the definition of the problem: does color count in the symmetries or just shape? If colors count, there are 17. He also wrote: “Groups of symmetry had no significance to the artists and artisans who decorated the Alhambra”
  • 7. Modern Mathematical View of Tiling Some single (monohedral) shapes don’t tile. Which do? What’s a good prototile? Not a packing, which can have gaps Not a covering, which can have overlaps A plane tiling T is a countable family of closed sets which cover the (Euclidean) plane without gaps or overlaps. Each tile T is a closed topological disk, i.e. the tile boundaries are simple closed curves. The union of the tiles is the plane, and the interiors of the tiles are pairwise disjoint. (from Tilings and Patterns, Grünbaum and Shephard)
  • 8. Single Regular Polygon Tiling: 3, 4, 6, not 5
  • 9. Many other kinds of tilings Regular polygons, but not edge-to-edge Voderberg’s non-convex 9-gon (ennagon) spiral tiling One of Kepler’s tilings including star polygons
  • 10. Aperiodic Tilings Periodic Tiling: lines up with itself after a plane translation. Aperiodic: can’t 1961: Hao Wang conjectures: If a finite set of tiles will tile the plane, it can do so periodically. and that there should exist an algorithm to determine whether any given set of tiles will do this. 1966: Robert Berger showed (using an equivalence to the Halting Problem!) that no such algorithm exists, and aperiodic sets of tiles exist. Berger came up with a set of 20,426 such tiles. Then reduced them to 104. Don Knuth reduced them to 92. Karel Culik came up with these 13:
  • 11. Aperiodic 2: Robinson to Penrose 1971: Raphael M. Robinson found an aperiodic set of 6 modified rectangular tiles; used projections and dents rather than coloring matching 1974: Roger Penrose produces the dart and kite aperiodic prototile set. John H. Conway suggests a colored line based matching rule. Can a single prototile set be aperiodic? It is still an open question.
  • 12. “Everywhere there is found...a silent swerving from accuracy by an inch that is the uncanny element in everything… a sort of secret treason in the universe.” G.K. Chesterton as quoted by Martin Gardner (in Penrose Tiles and Trapdoor Ciphers) Aperiodic patterns with Penrose tiles
  • 13. X-Ray Crystallography ~1912 Max Von Laue developed X-Ray Crystallography (Physics Nobel 1914)
  • 14. This becomes a standard crystallography method: over 400,000 solids were characterized in 70 years. All fit this “3,4,6 and not 5” model. Three-fold, four-fold, and six-fold symmetries. Five-fold and higher than six-fold symmetries are “proven” impossible.
  • 15. Connecting Aperiodicity to Crystals 1982: Alan Lindsay Mackay, a crystallographer, puts circles at the intersections of a Penrose tiling, computes the diffraction pattern: the result is a 10-fold symmetry April 8 1982: Dr. Dan Shechtman, from the Technion in Israel, doing metallurgy experiments at NBS (now NIST) on AlMg sees 10-fold symmetry in his electron beam crystal diffraction patterns. He tries unsuccessfully to publish his results, has other crystallographers review and check his results.
  • 17. "There's no such thing as quasicrystals, only quasi- scientists." - Linus Pauling 1984: Physicists Paul Steinhardt and Dov Levine connect the work of Mackay and Shechtman, coin the term quasicrystal in an article weeks after Shechtman’s work is finally published. Significant resistance to quasicrystals continued, but so did experimental results. In 1993, the International Union of Crystallographers changed the definition of “crystal” to include quasiperiodic crystals and many other structures. 2010: Mackay, Steinhardt and Levine get the Buckley Prize in Physics 2011: Dan Shechtman receives the Nobel Prize in Chemistry Hundreds of quasicrystals have now been found, including the natural quasicrystal icosahedrite.
  • 18. The Girih Tiles In 2007, Peter Lu, a student of Paul Steinhardt, came up with the Girih tiles They match many Islamic tiling patterns and they match the Penrose dart/kites. Later Lu found the Girih tiles match the Topkapi scroll (~1500 c.e.), a Timurid set of instructions for artisans
  • 19. The Girih Tiles as a Penrose Tiling The Girih Tiles fit the tiling of the Darb-i Imam Shrine (and many other tilings)
  • 20. References and Resources: The Wikipedia pages on tessellation, Wallpaper Group, Crystallographic Restriction Theorem, aperiodic tiling, quasicrystal and each of the individuals named in these slides are pretty good. YouTube has presentations of their work by Dan Shechtman, Paul Steinhardt, and Peter Yu. Books: Introduction to Tessellations by Dale Seymour and Jill Britton: very simple, graphic explanations, art oriented, but includes the algebra shown in this talk Tilings and Patterns by Branko Grünbaum and G.C. Sheppard: encyclopedic, definitive work on mathematics of tilings Penrose Tilings to Trapdoor Ciphers and the Return of Dr. Matrix by Martin Gardner: the first two chapters give the history of aperiodic tilings up to the discovery of quasicrystals Symmetries of Islamic Geometric Patterns by Syed Jan Abas and Amer Shaker Salman: discusses some of the history of Islamic patterns (not just tilings) then catalogs many patterns according to Wallpaper group symmetry
  • 21. Appendix: Archimedean tilings Archimedes (ca. 287-212 B.C.E) gave us the Stomachion (see the Archimedes Codex) - the oldest known geometric puzzle - but not the Archimedean tilings
  • 22. Semi-regular polygonal tilings The general formula for the interior angle of an n-gon: So for three regular polygons with sides n1, n2, n3 to fit together: This simplifies to: Extending this analysis to combinations of regular polygons, there are 21 combinations possible. 17 are distinct combinations: 4 are just different orderings of the same sets of polygons. 11 of these fit together to tile the plane, called the uniform or Archimedean tilings
  • 23. Semi-regular Polygonal Tilings Similarly for four, five or six polygons: But six is a maximum, since 60 degrees is the smallest regular polygon angle Each solution corresponds to a tiling pattern, e. g.

Editor's Notes

  1. Found all over the world history; Bricks: mass manufacturing, std. parts; natural question of possible patterns; pictures: modern mud brick palestine, road paving,Uruk (Sumerian ~3400 BC), Alhambra, Escher Lizards, graphene
  2. Elaborate on Abu’l-Wafa Buzjani (Astronomy, Spherical Trigonometry) or more generally House of Wisdom guys: Algebra, Algorithm Or Archimedes and Stomachicon.
  3. 3, 4 or 6 not 5 Do diagrams as click animation Show algebraic solution
  4. Confirmation bias congeals into dogma
  5. Shectman announced running for President in Israel, Jan. 2014! (explain Twining)
  6. Paradigm shift: A spectrum of organization from crystal to glass. Icosahedrite found
  7. 2007 Top 100 Science Stories Show Penrose dart/kite inside Girih tiles
  8. Actually, there are 21 solutions, but some are the same polygon combinations in different orders