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 leadsto skill in all the practical arts, while the study of intelligiblegeometry leads to skill in the intellectual arts because thisscience is one of the gates through which we move to theknowledge of the essence of the soul, and that is the rootof 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Historical documents from the House of Wisdom:On the Geometric Constructions Necessary for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 havegaps

Not a covering,which can haveoverlaps

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’t1961: 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 inchthat is the uncanny element in everything…a sort of secret treasonin the universe.” G.K. Chesterton asquoted by Martin Gardner(in Penrose Tiles andTrapdoor 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, andsix-fold symmetries. Five-foldand 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 diffractionpattern: 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 quasicrystalin 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 beenfound, including the natural quasicrystalicosahedrite.

The Girih TilesIn 2007, Peter Lu, a student of Paul Steinhardt, came up with the Girih tiles

They match many Islamictiling patterns and they match thePenrose 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 tilingsPenrose 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 quasicrystalsSymmetries 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 TilingsSimilarly 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.