decagonal tilings in medieval islamic architecturepborwein/pborwein_resources/laferriere.pdf ·...

Decagonal Tilings in Medieval Islamic Decagonal Tilings in Medieval Islamic ArchitectureArchitecture

OutlineOutline

OutlineOutline

► IntroductionIntroduction

OutlineOutline

► IntroductionIntroduction Plane TilingsPlane Tilings Regular TilingsRegular Tilings Periodic TilingsPeriodic Tilings

OutlineOutline

► IntroductionIntroduction Plane TilingsPlane Tilings Regular TilingsRegular Tilings Periodic TilingsPeriodic Tilings

►Aperiodic TilingsAperiodic Tilings

OutlineOutline

► IntroductionIntroduction Plane TilingsPlane Tilings Regular TilingsRegular Tilings Periodic TilingsPeriodic Tilings

►Aperiodic TilingsAperiodic Tilings Penrose TilingsPenrose Tilings Girih TilingsGirih Tilings

Plane TilingsPlane Tilings

Plane TilingsPlane Tilings

►Collection of plane Collection of plane figuresfigures

Plane TilingsPlane Tilings

►Collection of plane Collection of plane figuresfigures

►Covers plane entirelyCovers plane entirely

Plane TilingsPlane Tilings

►Collection of plane Collection of plane figuresfigures

►Covers plane entirelyCovers plane entirely►No gaps or overlapNo gaps or overlap

Plane TilingsPlane Tilings

►Collection of plane Collection of plane figuresfigures

►Covers plane entirelyCovers plane entirely►No gaps or overlapNo gaps or overlap

Regular TilingsRegular Tilings

Regular TilingsRegular Tilings

►Plane tiling of congruent Plane tiling of congruent regular polygonsregular polygons

Regular TilingsRegular Tilings

►Plane tiling of congruent Plane tiling of congruent regular polygonsregular polygons

►Only three existOnly three exist

Regular TilingsRegular Tilings

►Plane tiling of congruent Plane tiling of congruent regular polygonsregular polygons

►Only three existOnly three exist

Regular TilingsRegular Tilings

►Plane tiling of congruent Plane tiling of congruent regular polygonsregular polygons

►Only three existOnly three exist

Regular TilingsRegular Tilings

►Plane tiling of congruent Plane tiling of congruent regular polygonsregular polygons

►Only three existOnly three exist

Periodic TilingsPeriodic Tilings

Periodic TilingsPeriodic Tilings

►Tiling of the plane with translational Tiling of the plane with translational symmetrysymmetry

Periodic TilingsPeriodic Tilings

►Tiling of the plane with translational Tiling of the plane with translational symmetrysymmetry

►Only 2-fold, 3-fold, 4-fold and 6-fold rotational Only 2-fold, 3-fold, 4-fold and 6-fold rotational symmetry is allowedsymmetry is allowed

Periodic TilingsPeriodic Tilings

►Tiling of the plane with translational Tiling of the plane with translational symmetrysymmetry

►Only 2-fold, 3-fold, 4-fold and 6-fold rotational Only 2-fold, 3-fold, 4-fold and 6-fold rotational symmetry is allowedsymmetry is allowed

Periodic TilingsPeriodic Tilings

►Tiling of the plane with translational Tiling of the plane with translational symmetrysymmetry

►Only 2-fold, 3-fold, 4-fold and 6-fold rotational Only 2-fold, 3-fold, 4-fold and 6-fold rotational symmetry is allowedsymmetry is allowed

Penrose TilingPenrose Tiling

Penrose TilingPenrose Tiling

►Penrose tiling was developed by Roger Penrose tiling was developed by Roger Penrose in 1973Penrose in 1973

Penrose TilingPenrose Tiling

►Penrose tiling was developed by Roger Penrose tiling was developed by Roger Penrose in 1973Penrose in 1973

►Has no translational symmetry and is Has no translational symmetry and is therefore aperiodictherefore aperiodic

Penrose TilingPenrose Tiling

►Penrose tiling was developed by Roger Penrose tiling was developed by Roger Penrose in 1973Penrose in 1973

►Has no translational symmetry and is Has no translational symmetry and is therefore aperiodictherefore aperiodic

► Is quasi-periodic (in physics this property is Is quasi-periodic (in physics this property is called quasi-crystalline)called quasi-crystalline)

Penrose TilingPenrose Tiling

►Made up of two rhombiMade up of two rhombi

Penrose TilingPenrose Tiling

►Made up of two rhombiMade up of two rhombi {72, 108, 72, 108} degrees{72, 108, 72, 108} degrees

Penrose TilingPenrose Tiling

►Made up of two rhombiMade up of two rhombi {72, 108, 72, 108} degrees{72, 108, 72, 108} degrees {36, 144, 36, 144} degrees{36, 144, 36, 144} degrees

Penrose TilingPenrose Tiling

►Made up of two rhombiMade up of two rhombi {72, 108, 72, 108} degrees{72, 108, 72, 108} degrees {36, 144, 36, 144} degrees{36, 144, 36, 144} degrees

►Only one rule: no two adjacent tiles can form Only one rule: no two adjacent tiles can form a parallelograma parallelogram

Penrose TilingPenrose Tiling

►Made up of two rhombiMade up of two rhombi {72, 108, 72, 108} degrees{72, 108, 72, 108} degrees {36, 144, 36, 144} degrees{36, 144, 36, 144} degrees

►Only one rule: no two adjacent tiles can form Only one rule: no two adjacent tiles can form a parallelograma parallelogram

►Ratio between number of each tiles is Ratio between number of each tiles is golden ratiogolden ratio

Penrose TilingPenrose Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

►Very complex patterns that appear Very complex patterns that appear throughout Islamic art and architecturethroughout Islamic art and architecture

Girih TilingGirih Tiling

►Very complex patterns that appear Very complex patterns that appear throughout Islamic art and architecturethroughout Islamic art and architecture

►Locally display 10-fold rotational symmetry, Locally display 10-fold rotational symmetry, and therefore cannot be periodicand therefore cannot be periodic

Girih TilingGirih Tiling

►Very complex patterns that appear Very complex patterns that appear throughout Islamic art and architecturethroughout Islamic art and architecture

►Locally display 10-fold rotational symmetry, Locally display 10-fold rotational symmetry, and therefore cannot be periodicand therefore cannot be periodic

► Initially thought to have been created using Initially thought to have been created using compass and straight edge methodcompass and straight edge method

Girih TilingGirih Tiling

►Peter J. Lu at Harvard and Paul J. Steinhardt Peter J. Lu at Harvard and Paul J. Steinhardt at Princeton found that Girih Tilings exhibit at Princeton found that Girih Tilings exhibit advanced decagonal quasicrystal geometry advanced decagonal quasicrystal geometry like that of Penrose Tilingslike that of Penrose Tilings

Girih TilingGirih Tiling

►Peter J. Lu at Harvard and Paul J. Steinhardt Peter J. Lu at Harvard and Paul J. Steinhardt at Princeton found that Girih Tilings exhibit at Princeton found that Girih Tilings exhibit advanced decagonal quasicrystal geometry advanced decagonal quasicrystal geometry like that of Penrose Tilingslike that of Penrose Tilings

►Girih Tilings used five different tilesGirih Tilings used five different tiles

Girih TilingGirih Tiling

►Peter J. Lu at Harvard and Paul J. Steinhardt Peter J. Lu at Harvard and Paul J. Steinhardt at Princeton found that Girih Tilings exhibit at Princeton found that Girih Tilings exhibit advanced decagonal quasicrystal geometry advanced decagonal quasicrystal geometry like that of Penrose Tilingslike that of Penrose Tilings

►Girih Tilings used five different tilesGirih Tilings used five different tiles►Can be mapped to Penrose TilingsCan be mapped to Penrose Tilings

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

Girih TilingGirih Tiling

ExamplesExamples

http://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1chttp://www.sciencemag.org/cgi/content/full/315/5815/1106/DC1c

ReferencesReferences

1.1. Decagonal and Quasi-Crystalline Tilings in Medieval Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture. Islamic Architecture. Peter J. Lu, et al. Peter J. Lu, et al. ScienceScience 315315, 1106 , 1106 (2007).(2007).

2. 2. Category: Periodic Tiling Image.Category: Periodic Tiling Image. Wikipedia.org, Wikipedia.org, http://en.wikipedia.org/wiki/Category:Periodic_tiling_imagehttp://en.wikipedia.org/wiki/Category:Periodic_tiling_image

3. 3. Penrose Tilings. Penrose Tilings. Science U, Science U, http://www.scienceu.com/geometry/articles/tiling/penrose.htmlhttp://www.scienceu.com/geometry/articles/tiling/penrose.html