Cristallography and Art
-
Upload
robert-mercier -
Category
Documents
-
view
25 -
download
0
Transcript of Cristallography and Art
![Page 1: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/1.jpg)
Crystallography and Art
Bernd SouvignierRadboud University Nijmegen
3rd de Brun Workshop: Algebra, Algorithms, Applications
Galway, December 7-10, 2009
![Page 2: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/2.jpg)
Overview
• M.C. Escher
• Penrose patterns
• Quasicrystals
• Islamic Art
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 3: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/3.jpg)
M.C. Escher
Dutch artist, 1898 - 1972
Famous e.g. for: regular plane tilings, impossible figures
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 4: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/4.jpg)
Theoretical work
Escher developed his own classification of regular plane tilings, partiallycoinciding with the crystallographic approach (e.g. based on lattice types),partially contrasting it.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 5: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/5.jpg)
Relation to the crystallographic community
Escher was invited to give a plenary talk at the International Congress ofCrystallography in Cambridge in 1960.
In his abstract, he states:
From the beginning of my investigations, the use of contrasting colours orshades was both self-evident and necessary in order to distinguish visuallybetween neighbouring figures. I was therefore surprised to learn that thenotion of antisymmetry has only recently been introduced into and acceptedby crystallography. My own applications are unthinkable without the use ofcolour contrast.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 6: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/6.jpg)
Illusion of space
While creating regular plane tilings, Escher enjoyed playing with the illusionof representing a 3-dimensional reality.
Lattice-equal subgroup of index 2.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 7: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/7.jpg)
Construction of a plane filling tile
Often, the construction of the figures is highly ingenious.
The plane groups of Escher’s patterns almost never allow proper reflecti-ons, since that would require the figures to have straight edges.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 8: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/8.jpg)
Other geometries
Inspired by an illustration in a paper of H.S.M. Coxeter, giving a regular tilingof a hyperbolic plane, Escher also worked in planes with curvature 6= 0.
Plane tilings with triangles of angle sum 56π (hyperbolic), π (Euclidean) and
76π (elliptic). Note that the same motif is used in all three cases.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 9: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/9.jpg)
Ghosts
In 1962, R. Penrose challenged Escher with a set of congruent jigsaw pie-ces which would tile the plane in a unique way. Based on these tiles, Escherdesigned his last print Ghosts (1971).
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 10: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/10.jpg)
Penrose tiles (kites and darts)
Developed out of the subdivision of a pentagon into smaller pentagons,filling the gaps with other tiles which allow iteration of the subdivision.
Self-similarity with scaling factor τ = 1+√
52 ≈ 1.618
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 11: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/11.jpg)
Non-periodic tiling via subdivision and inflation
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 12: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/12.jpg)
Diffraction from Penrose tilings
Despite its non-periodicity, the diffration pattern of a Penrose tiling (with thevertices of the tiles as diffraction gratings) displays sharp Bragg peaks.The decagonal symmetry of the diffraction pattern is incompatible with a2-dimensional lattice.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 13: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/13.jpg)
Quasicrystals
In 1984, Shechtman published an article with diffraction patterns of anAl Mn-alloy, displaying 2-fold, 3-fold and 5-fold rotational axes and in facticosahedral symmetry.
Conflict: Sharp Bragg peaks ⇒ crystal5-fold rotation axis ⇒ non-periodic
}⇒ quasicrystal
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 14: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/14.jpg)
Cut-and-projection method
Quasicrystals can be described as certain projections of periodic structuresin higher-dimensional spaces to the real space (usually R3 or R2).
Let Rn = VP ⊥ VI be an orthogonal decomposition, where VP is calledthe physical or external space and VI is called the internal space.
A (periodic) structure S ⊂ Rn is projected to the physical space VP bymapping v = vp + vi to vp if vi lies in some compact subset C of Rn, i.e. ifv is close to VP .
A common choice for the window C is the Voronoı cell of S.
In order to preserve certain symmetry elements, the external space VP
is typically chosen as an R-irreducible submodule of a subgroup of Π(G)(where G is the space group of S).
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 15: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/15.jpg)
Fibonacci quasicrystal
x = τy
τ := 1+√
52
@@@@
@@
@
@@
@
@@
@@
@@@u eu u
e u eu u
(5,3)@
@@I
e u ue u
(8,5)@
@@I
eu u
e u eu u
(13,8)@
@@I
e u ue u
l s l s l l s l s l l s l l s l s l l s l s l l s l
��
�
L
��
�
S L
��
�
L S
��
�
L S
��
�
L
��
�
L S
��
�
L
��
�
L S
��
�
L S
��
�
L
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
""
"
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 16: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/16.jpg)
Self-similarity
A scaling of the window S in the Fibonacci quasicrystal by τ induces theself-similarity transformation
S → L, L→ S L.
for the Fibonacci quasicrystal.
The self-similarity property holds generally for the cut-and-project method.
Together with the fact, that a periodic structure in the higher-dimensionalspace is projected, this explains the sharp Bragg peaks in the diffractionpattern.
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 17: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/17.jpg)
The Penrose pattern is cubic
The Penrose pattern (with thick and thin rhombs) can be obtained by thecut-and-project method from the 5-dimensional cubic lattice Z5.
This lattice admits a 5-fold rotation permuting the vertices adjacent to onevertex. One such axis is along the vector v = (1,1,1,1,1)tr.
The 4-dimensional space v⊥ splits into two 2-dimensional real subspacesinvariant under the 5-fold rotation. One of these is chosen as physicalspace.
Thus, the Penrose pattern is designed to have 5-fold rotation symmetry(even dihedral), whereas the symmetry elements of Z5 that do not fix thephysical space are lost (in particular all translations).
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 18: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/18.jpg)
Pattern no. 28 from the Topkapi scroll (∼ 1500)
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 19: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/19.jpg)
Girih (line) pattern
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 20: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/20.jpg)
Girih pattern with underlying tiles
The tiles are indicated by red-dotted lines in the Topkapi scroll
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 21: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/21.jpg)
Girih tiles from pattern no. 28
except for upper right ‘bat’
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 22: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/22.jpg)
Large-scale tiles
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 23: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/23.jpg)
Subdivision of girih tiles
Scaling factor 2τ + 2 ≈ 5.236
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 24: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/24.jpg)
Further subdivision of pattern no. 28
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 25: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/25.jpg)
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 26: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/26.jpg)
Pattern no. 28 with third level of tiles
The white stars are the defects in the subdivision of the pentagons
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 27: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/27.jpg)
Pattern no. 28 with third level of line ornament
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 28: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/28.jpg)
Darb-i Imam shrine in Isfahan (1453)
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 29: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/29.jpg)
Girih pattern on two scaling levels
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 30: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/30.jpg)
Girih pattern and underlying tiles on two scaling levels
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 31: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/31.jpg)
Girih tiles on two scaling levels
no large-scale hexagon occursB. Souvignier de Brun Workshop Galway 10-12-2009
![Page 32: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/32.jpg)
Subdivision of girih tiles
Scaling factor 4τ + 2 ≈ 8.472
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 33: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/33.jpg)
Three levels of tiles
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 34: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/34.jpg)
Girih pattern on three scaling levels
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 35: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/35.jpg)
Smaller scaling factor −→ more scaling levels
Scaling factor τ + 1 ≈ 2.618
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 36: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/36.jpg)
Subdivision of tiles
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 37: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/37.jpg)
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 38: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/38.jpg)
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 39: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/39.jpg)
Self-similar girih patterns on up to four scaling levels
B. Souvignier de Brun Workshop Galway 10-12-2009
![Page 40: Cristallography and Art](https://reader033.fdocuments.in/reader033/viewer/2022051411/547fa257b4af9fc9158b5b12/html5/thumbnails/40.jpg)
B. Souvignier de Brun Workshop Galway 10-12-2009