Point substitution processes forgenerating icosahedral tilings
Nobuhisa Fujita
IMRAM, Tohoku University,
Sendai 980-8577, Japan

Outline
1.
2.

Part I.
Basic icosahedral tilings
Point substitution processes

3.
4.
5.

Part II.
Tilings constructed with PIRs
Canonical cell tilings
Toward icosahedral CCTs

Outline
Part I.

1.

Basic icosahedral tilings

2.

Point substitution processes

Part II.
3.
4.
5.

Tilings constructed with PIRs
Canonical cell tilings
Toward icosahedral CCTs

Icosahedral QCs
Icosahedral tilings

ED pattern along 5-foldaxis of an icosahedral
quasicrystalMn icosahedrab-c packing of icosahedralclusters (F-type) based on therhombohedral tiling(Ammann-Kramer tiling).
Model of i -(Al-Mn),
M. Audier and P. Guyot,
Phil. Mag. B 53, L43 (1986)

T *(P) tiling(Ammann-Kramertiling)

5-fold view

Basic tiles OR, AR(Ammann rhombohedra)

2-fold view

M. Duneau and A. Katz, Phys. Rev. Lett. 54, 2688 (1985).
P. Kramer and R. Neri, Acta Cryst. A 40, 580 (1984).

Icosahedral basis set(e1e2e3 e 4e6e3e1e4e5e2e5τ 0 1 1e6 ) =  1 τ 0 0
 0 1 τ −τ−τ10|| e1 ||= 2 + τ1+ 5
τ=2the golden mean0 −τ 1 

Icosahedral modules
M P := {n1e1 + n2e 2 + n3e 3 + n4e 4 + n5e5 + n6e 6 | ( n j ) ∈ Ζ 6 }M F := {n1e1 + n2e 2 + n3e3 + n4e 4 + n5e5 + n6e 6 |∑ n j = 0 mod 2, (n j ) ∈ Ζ6 }Ζ[τ 3 ]Ζ[τ ]jM I := {ν 1e1 +ν 2e 2 +ν 3e 3 +ν 4e 4 +ν 5e 5 +ν 6e 6 |(ν j )∈ Ζ ∪ Ζ + 1 (111111)}
266

Ζ[τ ]integer ring[1] T. Janssen, Acta Cryst. A 42 (1986) 261.
[2] D. S. Rokhsar et al., Phys. Rev. B 35 (1987) 5487.
[3] L.S. Levitov and J. Rhyner, J. Physique 49 (1988) 1835.

Scale invariance of the modules1
M I := M P ∪ M P + (111111) 2
11
 

= M F ∪ [M F + (100000)] ∪ M F + ( 1 11111)  ∪ M F + (111111) 22 

VeBeVoτ-scaling×Be××τVeττBo×τ

VoBoM P ×τ 3 = M P
M F ×τ = M F
M I ×τ = M I

T *(2F) tiling(Kramer et al.)P. Kramer et al., in Symmetries in Science V: Algebraic Structures,
their Representions, Realizations and Physical Applications, Ed. by B.
Gruber et al., Plenum Press, New York, 1991, pp. 395.

Outline
Part I.
1.

Basic icosahedral tilings

2.

Point substitution processes
Part II.

3.
4.
5.

Tilings constructed with PIRs
Canonical cell tilings
Toward icosahedral CCTs

Point substitution processes fordecagonal tilings

RPH
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)

RPH
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)(2)Place S atevery vertex

RP

H
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)(2)Place S atevery vertex

RP

H
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)(2)Place S atevery vertex(3)Eliminate
excessive points

RPH
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)(2)Place S atevery vertex(3)Eliminate
excessive points

RP

H
N. Fujita, Acta Cryst. A 65, 342 (2009)

Point substitution processes fordecagonal tilings(1)Expansion(σ =τ 2)(2)Place S atevery vertex(3)Eliminate
excessive points

RP

H
N. Fujita, Acta Cryst. A 65, 342 (2009)

Window

N. Fujita, Acta Cryst. A 65, 342 (2009)

Point Substitution Process(for constructing icosahedral quasiperiodic tilings)(1)Expansive similarity transformation:Ti σ Ti (Ti⊂M, σ =ρn, ρ=τ3(P), τ(F), τ(I))(2) Replicate the Ih-star at every vertex:
Ti’= σ Ti + S (S⊂M)(3) Decimation of points by local rules:
Ti’  Ti+1 (⊂Ti’ )N. Fujita, Acta Cryst. A 65, 342 (2009)

Step (3) is needed if there is redundancy in the points
generated through (1) and (2) (Point inflation rule)
K. Niizeki, J.Phys.A:Math.Theor.41,175208 (2008)

Outline
Part I.
1.
2.

Basic icosahedral tilings
Point substitution processes

3.

Part II.
Tilings constructed with PIRs

4.
5.

Canonical cell tilings
Toward icosahedral CCTs

Windows
T *(P)T *(2F)

P-typeF-typeτ || e1 |||| e1 ||Point density

WPΩP

WF τ 3 WP
=
ΩF
2 ΩP

Ih-starS

1 : (000000)
1
2 : (1 1 1 1 1 1 )
2
3 : (111000)

Point inflation rule(viewed in the external space)
1st iteration

seed×τ

2nd iteration×τ

Q∞(S,τ)Ih-star(mapped to the internal space)

1 : (000000)
12 : (1 1 1 1 1 1 )
23 : (111000)

TS

in the internal space|| e1 ||window
1 + τ −1 + τ − 2 + τ −3 + τ − 4 + τ −5 + L =

1 : (000000)
12=τ= 2.61803
1 − τ −1

12 : (1 1 1 1 1 1 )
23 : (111000)

5-fold direction∩MP
Q∞(S,τ)∩Inflation rule of the AmmannKramer tiling

Inflation rules by τ 3 scaling
T. Ogawa, J. Phys. Soc. Jpn. 54, 3205 (1985).

Outline
Part I.
1.
2.

Basic icosahedral tilings
Point substitution processesPart II.
3.

Tilings constructed with PIRs

4.

Canonical cell tilings

5.

Toward icosahedral CCTs

Canonical cell tiling‘Cell geometry for cluster-based quasicrystal models’,
C. L. Henley, Phys. Rev. B 43 (1991) 993.

4 polyhedra: A-, B-, C-, D-cells
A-cellC-cellB-cellc cbbccccbcc bbbcb

D-cellcbb cc bbcbc

There are 32 classes of nodes in a CCT(68)0(62)222222(67)333bbb

i-CdCanonicalCell Tiling(QC)
5.7Yb: Quasicrystal
ARRTH

H. Takakura &C.P. Gomez et al.,(2007).

M. Mihalkovic et al., Phys. Rev. B 53, 9002-9020 (1996).

M. Mihalkovic et al., Phys. Rev. B 53, 9002-9020 (1996).

Outline
Part I.
1.
2.

Basic icosahedral tilings
Point substitution processes

Part II.
3.
4.

Tilings constructed with PIRs
Canonical cell tilings

5.

Toward icosahedral CCTs

Is there an icosahedral CCT?
NO PROOF is given of the existenceof an ICOSAHEDRAL CCTMethods to construct approximant CCTs
(under periodic boundary conditions)
A Monte Carlo density optimization method:
M. Mihalkovic and P. Mrafko, Europhys. Lett. 21 (1993) 463.

A brute force algorithm:
M.E.J. Newman, C.L. Henley, and M. Oxborrow, Phil. Mag. B 71 (1995) 991.

Point sutstitution processes for
icosahedral CCTs

5-fold view

Ih-starthe magic starThe Ih-star is placed on every
vertex of the expanded CCT(scaling ratio=τ 3)

The Ih-star

Fix the center to be
the (68)0 type node.

A-packing(body centered cubic)

2 vertices

138 vertices

12 A-cells
0 B-cell
0 C-cell
0 D-cell

348 A-cells
136 B-cells
136 C-cells
24 D-cells

BC-packing(rhombohedral)

1 vertex

77 vertices

0 A-cell
2 B-cells
2 C-cells
0 D-cell

192 A-cells
76 B-cells
76 C-cells
14 D-cells

D-packing(simple hexagonal)(67)333

1 vertex
0 A-cells
0 B-cells
0 C-cells
2 D-cells

The center is missing!

D-packing(simple hexagonal)(67)333

The center is missing!

1 vertex

103 vertices

0 A-cells
0 B-cells
0 C-cells
2 D-cells

252 A-cells
102 B-cells
102 C-cells
20 D-cellsτ3×A-packing

τ3×D-packing

138 vertices

103 vertices

348 A-cells
136 B-cells
136 C-cells
24 D-cells

252 A-cells
102 B-cells
102 C-cells
20 D-cells

τ3×BC-packing

τ3×2/1 cubic-packing

77 vertices

584 vertices

192 A-cells
76 B-cells
76 C-cells
14 D-cells

1464 A-cells
576 B-cells
576 C-cells
104 D-cells

Conclusion
The present scheme has turned out to be useful for
constructing icosahedral tilings.
The magic star can generate all the vertices of an
inflated CCT except a point in the center of eachexpanded D-cell.
It is likely that there exist τ3-inflation rules forgenerating an icosahedral CCT, the proof ofwhich still needs to be worked out.