Hyperbolic Tiling on the Gyroid Surface in aPolymeric Alloy
Tomonari DOTERA
Department of Physics, Kinki University
Junichi MATSUZAWA
Department of Mathematics, Nara Women’s University

1Introduction

Biological amphiphilies (lipids) or synthetic surfactants in aqueous solutions
self-assemble to bilayers centered on the multiply connected surface dividing the space into two interpenetrating and nonintersecting “bicontinuous”
subspaces[1]. At high concentration of lipids or surfactants, these bilayers canorganize cubic phases based on mathematically well-characterized surfaces,
namely, triply periodic minimal surfaces: e.g., Schoen-Luzatti gyroid (G),Schwarz diamond (D), Schwarz primitive (P ), and Neovius surface (C(P ))[2].
Moreover, it is known that these surfaces abound in biological cells such as
the endoplasmic reticulum, the mitochondrion and the nucleus of certain
cells[3]. For high polymeric block copolymer systems[4], bicontinuous cubic
phases had attracted much attention as well[5, 6], and it is believed that only
the G phase has been established in most block copolymer systems[7, 8].
Recently, Hayashida et al. have found a gyroid surface related structure
in an ABC star polymer system. Interestingly, the gyroid surface is decorated and the structure can be regarded as a hyperbolic Archimedean tiling
structure[9]. The present paper discusses the mathematical aspects of thehyperbolic Archimedean tiling on the Gyroid surface in an ABC star block
copolymer system.

1

2

Self-organization of AB block copolymers

A block copolymer is a polymer consisting of at least two large connected
blocks of diﬀerent polymer types, say, A-B diblock copolymer:A − A − A − A − B − B − B − B.
For example, famous one is a styrene-isoprene diblock copolymer composedof plastic (polystyrene) and rubber (polyisoprene) polymer types. Supposethat A and B are immiscible with each other, molten copolymers can undergo
microphase separation exhibiting spatially periodic structures like lamellarstructures, bicontinuous double gyroid structures, hexagonal arrays of cylinders, body-center-cubic arrays of spheres depending upon the volume fraction of A and B as shown in Fig.1.[4] Such a morphological phase behavior
is common to surfactant or lipid systems.

Figure 1: Morphology of AB block copolymer systems. Light and dark graycorrespond to A and B components of AB block copolymers, respectively.
With decreasing the fraction of the A component, the shape of A component
becomes lamellae, bicontinuous double gyroid struts, cylinders, and spheres.
The lattice constants (periodic lengths) are the order of 10-100 nm.
The gyroid phase is called “bicontinuous phase” because domains of both
A and B components go to inﬁnity. It is also called “gyroid”, because thedark gray domain of the bicontinuous phase displayed in Fig.1 contains thegyroid minimal surface as a mid-surface. Notice that each polymer chain isvery ﬂexible and random in its chain shape in the domains; each domain is
made up of a random mixture of polymer chains of the same polymer type.

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3

Self-organization of ABC star block copolymers

Particular interests have focused on the possibility of creating new materials having new morphologies by controlling composition, molecular weight,and molecular architecture, such as multiblock copolymers and branched
or star block copolymers. For instance, ABC three-arm star-shaped[10, 11]copolymer systems (Figure 2(a)) have shown to have very fascinating morphologies. The ABC star block copolymer consists of three diﬀerent polymer
types connected at one junction. Because of this topological constraint, several polygonal cylindrical phases called “Archimedean tiling” phases havebeen obtained. See Fig.2.

Figure 2: (a) ABC star block copolymer consists of three diﬀerent polymers
linked at a junction point. Cylindrical phases called Archimedean tilingphases (b) are obtained when the fractions of A, B and C components are
not too diﬀerent.
For generic microphase-separated structures, there is a recent establishedscenario that bicontinuous phases exist in the boundary region between cylindrical and lamellar phases. In the case of a star polymer, the gyroid-typephase exsits between a lamellar phase and a cylindrical phase [9]. The problem here is the decoration on the gyroidal membrane.
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4

The gyroid surface and its symmetry via
hyperbolic geometry

The gyroid surface S is a triply periodic minimal surface given by the WeierstrassEnneper formula ([12]))(∫ ω
1 − w2√dw ,x = x0 + < exp(iθ)

8 − 14w 4 + 1

w0)
(∫ ω
i(1 + w2 )
√(1)
dw ,
y = y0 + < exp(iθ)
8 − 14w 4 + 1

w0


()
∫ ω
2w

√
 z = z0 + < exp(iθ)
dw .
w8 − 14w4 + 10
The angle θ is given by√
3K()2
θ = cot−1
1 ,K( 2 )where K(k) is the complete elliptic integral of the ﬁrst kind with modulus k.

Figure 3: The gyroid surface.
To study the symmetry of the gyroid surface, it is useful to regard S as4

a Riemann surface. Thus we recall a description of the gyroid surface as aRiemann surface.
The representation (1) of S enables us to regard S as a covering space ofa Riemann surface of genus 3 as follows.
√Let Σ be the Riemann surface of the function w8 − 14w4 + 1, which is
the two-sheeted covering surface of the Riemann sphere S 2√branched√ at 8±1±1
points which correspond to the zeros {±α , ±iα }(α = ( 3 − 1)/ 2) ofthe equation w8 − 14w4 + 1 = 0,
π : Σ −→ S 2 .
Then the 1-forms in (1)√1 − w2i(1 + w2 )
2wdw, √dw, √dw
w8 − 14w4 + 1
w8 − 14w4 + 1
w8 − 14w4 + 1

are holomorphic on Σ and the integrations of these holomorphic 1-forms
deﬁne the multiple-valued functions on Σ. The multiple-valuedness arises
from the integrations of the 1-forms around homotopically nontrivial loopson Σ: period integrals, which gives the translational symmetry (periodicity)of the gyroid surface S. Thus S is considered as a covering surface of Σ withcovering map p : S → Σ whose covering transformation group is just thetranslational symmetry group of S (Fig. 4).
Since the universal covering space of Σ is the hyperbolic plane, thereexists the covering map ϕ of the Poincaré disc D to the Gyroid surface S,
ϕ : D −→ S.
Let Γ be the symmetry group of S: the space group Ia3̄d ( No. 230, see [13]as for the notation). Since an isometry f of S are conformal automorphism,
there exists a conformal automorphism f˜ of D such that the projection ϕsatisfyϕ ◦ f˜ = f ◦ ϕ.
(2)
e to be the group of conformal automorphisms g of D suchThus we deﬁne Γthat there exists an element f of Γ satisfying ϕ ◦ g = f ◦ ϕ. Then ϕ is ane and Γ. Furthermore ϕequivariant mapping with respect to the actions of Γinduces a group homomorphisme −→ Γ,ϕ∗ : Γ5SBECOAgDpΣ

CB’Ỏ

B’’E’’πE’C’yC’’

A’A’’O’’xD’’

D’

Figure 4: The gyroid surface S and the Riemann surface Σ. The mapping
p is the covering map and g is the composition of the Gauss map of S to
S 2 and stereographic projection of S 2 to C. Bottom left is one sheet of theRiemann surface Σ. Bottom √right is the√ complex plane. The 8 intersection
points {±α±1 , ±iα±1 }(α = ( 3 − 1)/ 2) of 4 circles are the zeros of theequation w8 − 14w4 + 1 = 0, which are the branch points of π. The points
O0 , A0 , B 0 , C 0 , D0 of Σ are mapped to the points O00 , A00 , B 00 , C 00 , D00 by theprojection π.
whose kernel is isomorphic to the fundamental group π1 (S) of S.
6

The description of S as a covering surface of the Riemann surface Σ gives
e and Γ as follows.
the explicit representation of ΓThe representation (1) of the gyroid surface S shows that the Gauss mapof S to the unit sphere S 2 factors through a mapping of Σ to S 2 . Therefore
we can ﬁnd the triangle OAB on S with angles ( π6 , π6 , π2 ) such that OAB ismapped onto a triangle O0 A0 B 0 on Σ with same angles by p and the triangle
O00 A00 B 00 with angles ( π3 , π3 , π2 ) on C by the Gauss map, where O00 is the origin,
A00 , B 00 are the branch points of the covering π : Σ → C indicated in Fig. 4,and the covering π maps O0 A0 B 0 onto O00 A00 B 00 .
The quadrilateral A00 B 00 C 00 D00 on C (Fig. 4) is transformed into 6 copies
quadrilaterals in the Riemann sphere C ∪{∞} by successive reﬂections in thesides of the quadrilaterals. Thus the 12 copies of the quadrilateral A0 B 0 C 0 D0forms the Riemann surface Σ. This tiling is lifted to the Gyroid surface Sand gives a tessellation on S by the copies of OAB.
The triangle OAB is nothing but a fundamental domain of the symmetrygroup Γ of S: Γ is generated by the rotations w1 of order 4 around O and
w2 of order 2 around E
Γ = hw1 , w2 i,
where E corresponds to the midpoint on the arc A00 B 00 (Fig. 4) . The actionof Γ is easily seen from the view point of hyperbolic geometry. To do this weconsider the (2, 4, 6) tiling on the universal covering space of S: the Poincaré
disc D ([14], [15]) (Fig. 5).
The covering map ϕ maps the triangle P QR with angles ( π6 , π6 , π2 ) onto
e deﬁned above is the subgroup of index 2 in the Schwarz
OAB. The group Γe0 of type (2, 4, 6):triangle group Γ
e0 = hs1 , s2 , s3 s2 = s2 = s2 = (s1 s2 )2 = (s2 s3 )4 = (s3 s1 )6 = 1i,Γ123where s1 , s2 , s3 are the reﬂections in the sides of the triangle P QM withe consists of orientation-preserving elementsangles ( π2 , π4 , π6 ). The subgroup Γe0 and is generated by the rotations through angles 2π , 2π , 2π around theof Γ246
vertices of the triangle P QM .
e = hs1 s2 , s2 s3 iΓ' ha, b a2 = b4 = (ab)6 = 1i.
Then the mapping ϕ∗ is given by ϕ∗ (s1 s2 ) = w2 , ϕ∗ (s2 s3 ) = w1 .
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PRMQFigure 5: (2,4,6)tiling on the Poincaré disc D. Angles of the triangle P QRand P QM are ( π6 , π6 , π2 ) and ( π2 , π4 , π6 ).

5Hyperbolic Archimedean tiling

In the (2, 4, 6) tessellation on D given by the triangle with angles π2 , π4 , π6 , wecan ﬁnd a dodecagonal region indicated in Fig. 6. This region is mapped tothe domain in the gyroid surface S whose translations cover S ([14]).
If we put two adjacent triangles together to form a triangle ∆ with angles
e Let Γe0 be the( π4 , π4 , π3 ). This is a fundamental domain of the action of Γ.
e generated by s2 s3 and (s2 s3 s1 s2 )2subgroup of Γe0 = hs2 s3 , (s2 s3 s1 s2 )2 i,Γ
where s2 s3 and (s2 s3 s1 s2 )2 are both rotations through angle π2 around P ande0 ) is isomorphic to theangle 2πaround Q in Fig. 5. The image Γ0 = ϕ∗ (Γ3space group I 4̄3d (No. 220), which is the subgroup of index 2 in the spacegroup Γ = Ia3̄d.
e0 in Γe is 2, the action of Γe0 on D gives two colored
Since the index of Γtiling (Fig. 6).
The orbit Ω of a point in a fundamental domain (shaded region in Fig. 6
e0 deﬁnes a new tiling structure on D as follows (see Fig.
) by the action of Γ7 A).

8Figure 6: Two colored tiling on the Poincaré disc D.

Figure 7: A, Hyperbolic Archimedean tiling on the Poincaré disc D. The
circles on the shaded triangles form the tiling of vertex type (33 .4.3.4). B,ABC star-shaped block copolymer system. The I ( balls ) and the S ( membrane surrounding the balls ) components forming the gyroid membrane.
Remaining parts are ﬁlled with the P component.

9Every point P of Ω is surrounded by 6 faces: 4 triangles and 2 quadrilaterals. We denote its vertex type by (3.3.3.4.3.4) or (33 .4.3.4) which indicatesthe numbers of vertices of the consecutive faces around P . We call it hyperbolic Archimedean tiling, because it can be considered as a simple analogueof the Archimedean tiling of type (36 ) on a plane.
By the results of transmission electron microscopy and small-angle X-ray
scattering, we can ﬁnd the geometry of the gyroid surface in the geometricstructure of and ABC star-shaped block copolymer system, which is composed of polyisoprene (I), polystyrene (S), and poly (2-vinylpyridine) (P):the I and S components form a gyroid membrane and the complement of themembrane is ﬁlled with the P component (Fig. 7 B). The I component consists of isolated domains that form a new tiling structure on the gyroid surface
corresponding to the hyperbolic Archimedean tiling introduced above. Furthermore this tiling is characterized in three-dimensional Euclidean space asfollows.
Let P be a point on the gyroid surface S and ΩP the orbit of P undere0 . Then we havethe action of ΓTheorem There exists a unique point P in the triangle ϕ(∆) (a fundamental
domain of the action of Γ0 = I 4̄3d) on the gyroid surface S such that ΩPforms a hyperbolic Archimedean tiling (33 .4.3.4) in which the edges are all
the same length.
Outline of the proof. By the symmetry, points of ΩP are assembled in thesame way around each point. Thus when the faces surrounding one point ofΩP satisfy the conditions, all faces satisfy the conditions. Let P = (x, y, z) bea point of ΩP . Then the conditions that all edges connecting to P have thesame length are given by two quadratic equations. Let C be the intersectionof two quadric surfaces deﬁned by the quadratic equations. It is not hard toshow that the curve C and S intersect at one point in the triangle ϕ(∆).
We show the tiling of the theorem in Fig. 8.

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Figure 8: Hyperbolic Archimedean tiling of type (33 .4.3.4) in threedimensional Euclidean space whose vertices are on the gyroid surface.
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