Hyperbolic Tiling on the Gyroid Surface in a Polymeric Alloy ...

1725 2011 80-91 80

**in**g**on** **the** **Gyroid** **Surface** **in** a

**Polymeric** **Alloy**

Tom**on**ari DOTERA

Department of Physics, K**in**ki University

Junichi MATSUZAWA

Department of Ma**the**matics, Nara Women’s University

1 Introducti**on**

Biological $aIIlp1_{1}ip1_{1}ilies$ (lipids) or surfactants ill aqueous $sy_{I1}t_{}1_{1}etic$ $s^{7}o1\iota 1tiollb$

self-assemble to bilayers centered 011 tlie multiply collIiected surface divid**in**g

**the** space **in**to two **in**terpenetrat**in**g $aIld1101$1 $i_{l1}tersecti_{Il}g$ “bic**on**t**in**uous“

subspaces[l]. At high c**on**centrati**on** of lipids or surfactants, **the**se bilayers can

organize cubic phases based 011 $mat1_{1}ematically$ well-characterized surfaces,

$11c1IIlt^{\backslash 1y}$ $(^{\backslash }.1^{\cdot}(((^{\mathfrak{l}}$ $\iota bl11^{\cdot}f_{C}\iota(0_{\backslash }\backslash ^{\backslash }:(.b^{r}\cdot,$ $S(1_{1}o(^{\backslash

, triply $I^{i1i}$

) m**in**imal }11$-Luzatti gyroid $(G)$ ,

$Sc1_{1}warz$ diam**on**d $(D)$ , Schwarz primitive $(P)$ , and Neovius surface $(C(P))[2]$ .

Moreover, it is known that surfaces abound **in** biological cells as

$t1_{1}ese$ $suc1_{1}$

**the** endoplasmic reticulum. **the** mitoch**on**dri**on** and **the** nucleus of certa**in**

cells[3]. For high polymeric block copolynier systems[4], bic**on**t**in**uous cubic

$1_{1_{C}’}\iota(1\dot{\subset}\iota t.\uparrow_{1_{C}\iota(}\cdot t,’ 1Ill1(^{\iota}1_{C}’\iota tt(^{\backslash }11f_{\mathfrak{l}}]_{()I1}$

$W(^{\backslash .11[^{\ulcorner}6],c1I1(1}\backslash J$

phases as , it is $|1i\backslash$ 1 $1[y$

**the** phase has beeli $G$ $est_{I}ablis1$ 1 $ed$ **in** most block copolylner systellls[7, 8].

$Recel1\{1y,$ et al. have found a gyroid surface related structure

$Haya_{k}s^{\urcorner}1_{1}ida$

**in** an ABC star polymer system. Interest**in**gly, **the** gyroid surface is decorated

a.lld tlie structure can be regarded as a hyperbolic Arch t,il**in**g

$i_{1}\iota 1edean$

structure[9]. The presol paper discusses **the** ma**the**matical aspects of **the**

$1t$

hyperbolic $Arc1_{1}i_{l1}1edeal1$ til**in**g **on** tlie **Gyroid** surface **in** an ABC star block

copolyll .

$1er\backslash ’yste,111$

81

2Self-organizati**on** of AB block copolymers

A block copolymer is a c**on**sist**in**g of at least two largc $poly_{I}rlc^{1}r$

c**on**nected

blocks of different polylner types, say. A-B diblock copolymer:

$A-A-A-A-B-B-B-$

$B$

For exaliiple. falnous **on**e is a styrene-isoprene diblock copolymer composed

of plastic (polystyrene) and rubber (polyisoprene) polylner types. Suppose

that A and are immiscible with $B$

each o**the**r, molten copolylners can undergo

microphase separati**on** exhibit**in**g spatially periodic structures like lamellar

structures, bic**on**t**in**uous double gyroid structures, hexag**on**al arrays of cyl**in**ders,

body-center-cubic arrays of spheres depend**in**g up**on** **the** volume fracti**on**

of A and as sliown Fig. 1.[4] Such a rnorpliological $B$ phase behavior

$iI1$

is commn**on** to surfactant or lipid systemls.

.

Figure 1: Morphology of AB block copolymer systems. Light and dark gray

corresp**on**d to A and $BCOlI$ 1$po$l $iel$ 1 of AB block copolymers, $ts$

respectively.

$Wit1_{1}$

decreas**in**g fracti**on** of tfic A $t1_{1}c$ $coirlp_{oIleI_{1}}t,$ shape of A $t1_{1}c^{Y}$ $c\cdot oInpoI$ lcllt

becomes lalllellae, $bico11ti_{1}1uous$ double gyroid struts, cyl**in**ders, and spheres.

The lattice c**on**stants (periodic lengths) are **the** order of 10-100 nm.

The gyroid phase is called“bic**on**t**in**uous phase“‘ because doma**in**s of both

A and $B$ comp**on**ell go to **in**f**in**ity. $ts$ It is also called $::_{gyroid’}$ , because **the**

dark gray doma**in** of thc bic**on**t**in**uous phase displayed **in** Fig.1 c**on**ta**in**s **the**

gyroid m**in**imal surface as a mid-surface. Notice that each polynler cha**in** is

very flexible and randoln **in** its **in** **the** doma**in**s; each dolna**in** is

$c1_{1_{C}t}ir1sI_{1}ape$

nlade up of a random niixture of polylner cha**in**s of **the** salne polymer type.

$\backslash ^{\}\backslash }\tilde{c}^{\backslash ^{\backslash }}.,\nu^{\iota}\vee$

$i’\iota|f\backslash \backslash$

$\sim_{s}s_{\wedge}$

3Self-organizati**on** of ABC star block copolymers

Particular **in**terests have focused **on** **the** possibility of creat**in**g ncw materials

hav**in**g new lllorpllologies by c**on**troll**in**g colnpositi**on**, molecular weight,

and lnolecular architecture, such as multiblock $copoly_{1}ners$ and branched

or star block copolyliiers. For **in**stance, ABC three-arm $star-s1_{1a}ped[10,11]$

$copoly_{1}ner$ systenls (Figure $2(a)$ ) have shown to very fasc**in**at**in**g morphologies.

The ABC star block copolymer c**on**sists of three different polymer

$1_{1}ave$

types c**on**nected at **on**e juncti**on**. Because of this topological c**on**stra**in**t, several

polyg**on**al cyl**in**drical phases called $(^{\text{}}Archi_{I}nedean$ til**in**g” phases have

been obta**in**ed. See Fig.2.

82

”,

$’=.J^{\backslash }\backslash \backslash 1||$

$”$

’

(a)

(b)

Figure 2: (a) ABC star block copolymer collsists of three different $poly_{1}ners$

l**in**ked at a juncti**on** po**in**t. Cyl**in**drical phases called Archimedean til**in**g

phases (b) are obta**in**ed when **the** fracti**on**s of A, B and comp**on**ents are

$C$

not too different.

For generic lnicrophase-separated structures, **the**re is a recent established

scenario that bic**on**t**in**uous phases exist **in** **the** boundary regi**on** between cyl**in**drical

and lalllellar phases. In **the** case of a star $poly_{1}1$ ler, **the** gyroid-type

phase exsit, $s$

between a $lam()1lar$ pha.se and a cyl**in**drical phase [9]. The problem

here is **the** decorati**on** **on** gyroidal membrane.

$t1_{1e}$

83

4 The gyroid surface and its symmetry via

hyperbolic geometry

The gyroid surface is a triply periodic m**in**imal surface given by **the** Weierstrass-

$S$

Enneper formula ([12])

$\{\beg**in**{array}{l}x=x_{0}+\Re(\exp(i\**the**ta)/0^{\omega}\frac{1-w^{2}}{\sqrt{w^{8}-14u1^{4}+1}}dw),y=y_{0}+\Re(\exp(i\**the**ta)\**in**t_{0}^{\omega}\frac{i(1+w^{2})}{\sqrt{w^{8}-14w^{4}+1}}dw),z=z_{0}+\Re(\exp(i\**the**ta)\**in**t_{0}^{\omega}\frac{2w}{\sqrt{w^{8}-14w^{4}+1}}dw).\end{array}$

(1)

The angle

$\**the**ta$

is given by

$\**the**ta=\cot^{-1}\frac{K(\frac{\sqrt{3}}{2})}{K(\frac{1}{2})}$

,

$w1_{1}e1^{\cdot}eK(k)$ is $t1_{1}e$ colnplete elliptic **in**tegral of **the** first k**in**d $witI_{1}$ **in**odulus $k$ .

Figure 3: The gyroid surface.

To study t,be sylnlnetry of $t1_{1}e$ gyroid surface, it is useful to regard $S$ as

$\tilde{I^{\urcorner}}$

of

$\tilde{\Gamma}$

is

a Riemann surface. Tlius we recall a descripti**on** of **the** gyroid surface as a

Rielnann surface.

The representati**on** (1) of enables us to regard as a cover**in**g space of

$S$ $S$

a Riemann surface of genus 3 as follows.

$\Sigma$

Let be **the** Riernann surface of t,he functi**on** $\sqrt{w^{8}-14w^{4}+1}$ , which is

**the** two-sheeted cover**in**g surface of **the** Riemann sphere branched at 8

$S^{2}$

po**in**ts which corresp**on**d

$\{\pm\alpha^{\pm 1}, \pm i\alpha^{\pm 1}\}(\alpha=(\sqrt{3}-1)/\sqrt{2})$

to **the** zeros of

**the** equati**on** $t\{)8_{-14w}4+1=0$ ,

$\pi:\Sigmaarrow S^{2}$ .

84

Then **the** $1$-forlns ill (1)

$\frac{1-w^{2}}{\sqrt{w^{8}-14w^{4}+1}}d\prime u),$ $\frac{i(1+w^{2})}{\sqrt{w^{8}-14w^{4}+1}}duf,$ $\frac{2w}{\sqrt{w^{8}-14w^{4}+1}}dv)$

are **on** $1_{1}01_{oIl1}orp1_{1}ic$ $\Sigma$

$t1_{1C^{\backslash }}i_{l1}t_{\}}c^{\backslash }g1^{\cdot}atioi_{1}s$

and of l-forms

$t1_{1(_{d^{\backslash }}}se1_{1}oloI_{I1}oI\cdot pllic$

def**in**e **the** -valued functi**on**s **on** $lil\iota 1ltiple$ $\Sigma$

. The lilultiple-valuedness arises

$\backslash b’$

from **the** illlegratioll of $1$-forlns around holnotopically n**on**trivial loops

$t1_{1}e$

$\Sigma$

**on** : period $i_{I1}tegrals$ , which gives **the** translati**on**al symmetry (periodicity)

of gyroid surface $t1_{1}e$ $S$ . Thus is c**on**sidered as a $S$ $coveri_{ll}g$

$\Sigma wit1_{1}$

surface of

$Sarrow\Sigma w1\iota osC^{\backslash }cov(_{\vee^{\backslash }}I^{\cdot}iIlg$

cover**in**g map $p$ : transforlnati**on** group is just **the**

translati**on**al $sym$11letry group of (Fig. 4).

$S$

$\Sigma$

S**in**ce **the** universal cover**in**g space of is tlle hyperbolic plalle. tllere

$\varphi$

exists **the** cover**in**g lnap **the** Po**in**car\’e disc $D$ to **the** **Gyroid** surface $S$ ,

$\varphi:Darrow S$ .

Let

$\Gamma$

be **the** symmetry group of : $S$ $Ia\overl**in**e{3}d$

**the** space group (No. 230, see [13]

as for **the** $11otati_{o1}1$ ). S**in**ce an isometry of are $co$llforlnal $f$ $S$ automorphisni,

**the**re exists a c**on**formal automorphisln $f$ of $D$ such that

$\varphi$

**the** projecti**on**

satisfy

$\varphi\circ f=f\circ\varphi$

. (2)

$b_{(}\backslash tt_{1(}\backslash$

Thus we def**in**e to group of c**on**formal $c\tau 11\uparrow(11orphi_{h_{t}}ms$ $g$ of $D$ such

$\Gamma$

that **the**re exists all $eleIiielltf$ of satisfyiiig $\varphi og=f\circ\varphi$ $\varphi$

. Then all

equivariallt lnapp**in**g with respect to acti**on**s of and $t1_{1}e$

$\Gamma$

. $Furt1_{1}erlnore\varphi$

**in**duces a group $11_{o1}1_{1O1}11orplli_{S1}\iota 1$ $\varphi_{*}:\tilde{\Gamma}arrow\Gamma$

,

$P\downarrow$

85

$\pi$

$arrow$

Figure 4: gyroid surface $S$ $\Sigma$

and **the** Riemann surface . Thc mapp**in**g

$r\Gamma h(\tau$

$p$ is tlie cover**in**g map and $g$ is **the** c**on**ipositi**on** of tlie Gauss lnap of to

$S$

$sf\backslash 1^{\cdot}c^{\backslash

and

}og$1 $ap1_{1}ic$ $S^{2}$

projecti**on** of to $C$ . Bottom left is **on**c $s1_{1}c^{1}et$ of $t1_{1}e$

$S^{2}$

$Riel\iota lall$

$\Sigma$

11 surface . Bott**on**l right is **the** complex plane. The 8 **in**tersecti**on**

po**in**ts $\{\pm\alpha^{\pm 1}, \pm i\alpha^{\pm 1}\}(\alpha=(\sqrt{3}-1)/\sqrt{2})$ of 4 circles are $t1_{1}e$ zeros of $t1_{1}e$

$e(1\iota lati**on**w^{8}-14w^{4}+1=0$ $\pi$

, which are **the** branch po**in**ts of . The po**in**ts

$\Sigma$

$O’,$ $A’,$ $B^{f},$ $C’,$ $D’$

of are mapped to $t1_{1}e$ po**in**ts $O”,$ $A”$

. $B”,$ $C”,$ $D”$ by **the**

$\pi$

projecti**on** .

whose kernel is isomorphic to **the** $ful$ 1$dan$ lelltal group

$\pi_{1}(S)$

of $S$ .

$\tilde{\Gamma}’$

$\tilde{\Gamma}’$

$\tilde{\Gamma}$

lnaps

$\tilde{\Gamma}$

$\tilde{\Gamma}$

$\frac{2\pi}{4},$ $\frac{2\tau}{6}$

around

The descripti**on** of as a cover**in**g surface of **the** surface $S$ $Riel\iota lanIl$ $\Sigma$

gives

**the** explicit representati**on** of

$\Gamma$

and as follows.

The representati**on** (1) of **the** gyroid surface $S$ shows that **the** Gauss map

of $S$ to **the** unit sphere $S^{2}$ $\Sigma$

factors through a mapp**in**g of to . $S^{2}$ Therefore

we can f**in**d **the** triangle $OAB$ **on** with angles $S$ $( \frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{2})$

such thaf $OAB$ is

$\Sigma$

mapped **on**to a $triangleO’A’B’$ **on** with same angles by $p$ and **the** $tI\cdot ial$ 1$gle$

$( \frac{\pi}{3}, \frac{\pi}{3}, \frac{\pi}{2})$

$O”A”B”$ witb angles **on** $C$ by **the** Gauss map. where $O$“ is orig**in**,

$t1_{1}e$

$A^{f/},$ $B$ “ $\pi$

are **the** branch po**in**ts of **the** cover**in**g

$\Sigmaarrow C$

: **in**dicated **in** Fig. 4,

and **the** cover**in**g $\pi 1\iota lapsO’A’B^{f}$ **on**to $O”A”B$“.

The $qat(ra1A”B”C^{ff}D$ “ **on** $C$ (Fig. 4) is transformed **in**to 6 copies

quadrilaterals **in** **the** Riemann sphere $C\cup\{\**in**fty\}$ by successive reflecti**on**s **in** **the**

sideb of **the** quadrilaterals. Tlius **the** 12 copies of **the** quadrilateral $A’B’C^{f}D’$

forms $tIle$ $\Sigma$

Riemann surface . This til**in**g is lifted to **the** **Gyroid** surface $S$

and gives a tessellati**on** **on** $S$ by **the** copies of $OAB$ .

Tho triangle $OAB$ is noth**in**g but a fundamental doma**in** of **the** symmetry

$\Gamma$

group of is generated by **the** rotati**on**s $S:\Gamma$ $w_{1}$ of order 4 around $O$ and

$w_{2}$ of order 2 around $E$ $\Gamma=\langle w_{1},$ $uf2\rangle$

,

86

where $E$ corresp**on**ds to **the** midpo**in**t **on** **the** arc $A”B$ “ (Fig. 4). The acti**on**

$\Gamma$

of is easily seen from **the** view poillt of hyperbolic geometry. To do this we

c**on**sider **the** (2, 4, 6) til**in**g **on** **the** universal cover**in**g space of : **the** Po**in**car\’e

$S$

disc $D$ ([14], [15]) (Fig. 5).

$T1_{1C^{\backslash }}$

$\varphi$

cover**in**g lnap tlie t,riangle **on**to

$PQR wit1_{1}Allglc^{\backslash }s(\frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{2})$

$t1_{1}c^{1}$ $t1_{1}c^{1}$

$OAB$ . Tlie group def**in**ed above is subgroup of **in**dex 2 **in** Scbwarz

triangle group of type (2, 4, 6):

$\tilde{\Gamma}’=\langle s_{1},$

$s_{2},$ $s_{3}|s_{1}^{2}=s_{2}^{2}=s_{3}^{2}=(s_{1}s_{2})^{2}=(s_{2}s_{3})^{4}=(s_{3}s_{1})^{6}=1\rangle$ ,

$t1_{1}c^{1}rc^{\backslash }flc^{\backslash }ctiorlSi_{I1}$

tbe

$T1_{1}e$

subgroup

are

.

where $s_{1},$ $s_{2},$ $s_{3}$

$t1_{1}c^{Y}$

sides of triangle $PQMwitl_{1}$

$( \frac{\pi}{2}, \frac{\pi}{4}, \frac{\pi}{6})$

angles c**on**sists of orientati**on**-preserv**in**g elements

of

$\frac{2\pi}{2},$

and is generated by **the** rotati**on**s through angles **the**

vertices of **the** triangle $PQM$ .

$\tilde{\Gamma}=\langle s_{1}s_{2)}s_{2}s_{3}\rangle$

$\simeq\langle a,$ $b|a^{2}=b^{4}=(ab)^{6}=1\rangle$ .

Then **the** mapp**in**g

$\varphi_{*}$

is

given by $\varphi_{*}(s_{1}s_{2})=w_{2},$ $\varphi_{*}(s_{2}s_{3})=w_{1}$ .

around

$\tilde{\Gamma}$

def**in**es

**in**

**on**

$\tilde{\Gamma}$

around

be

87

Figure 5: (2,4,6)til**in**g **on** **the** Po**in**care disc D. $A_{l1}gles$ of **the** triangle $PQR$

$( \frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{2})$ $( \frac{\pi}{2}, \frac{\pi}{4}, \frac{\pi}{6})$

and $PQM$ are and .

5 **in**g

$\frac{\pi}{4},$ $\frac{\pi}{6},$

$( \frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{3})$

In tlie (2, 4, 6) tessellati**on** 011 $D$ $t1_{1(}\backslash$

given by

$\frac{\pi}{2},$

triangle with angles

$w(^{1}$

can f**in**d a dodecag**on**al regi**on** **in**dicated **in** Fig. 6. This regi**on** is mapped to

fhe $dolnai\iota 1$ **in** **the** gyroid surface whose translati**on**s cover ([14]).

$S$ $S$

$\triangle$

If we put two adjacent triangles toge**the**r to form a triangle with angles

. This is a fundalilelltal $dolI$ 1 $a**in**$ of **the** acti**on** of .

$\tilde{\Gamma}_{0}$

Let $t1_{1}e$

subgroup of gellerated by $s_{2}s_{3}$ alld $(s_{2}s_{3}s_{1}s_{2})^{2}$

$\tilde{\Gamma}_{0}=\langle ss(S6SS_{2})^{2}\rangle$

,

$P$ and

where and are both rotati**on**s tIlrougli angle $s_{2}s_{3}$ $(s_{2}s_{3}s_{1}s_{2})^{2}$ $\frac{\pi}{2}$

$\frac{2_{7_{\mathfrak{l}}^{-}}}{3}$

angle **in** Fig. 5. The image $Q$ $\Gamma_{0}=\varphi_{*}(\tilde{\Gamma}_{0})$

is isomorphic to **the**

$I\overl**in**e{4}3d$

space group (No. 220). which is **the** subgroup of **in**dex 2 **in** **the** space

group $\Gamma=Ia\overl**in**e{3}d$ .

$\tilde{\Gamma}_{0}$

S**in**ce **the** **in**dex of fi

$\tilde{\Gamma}_{0}$

is 2. tlie acti**on** of $D$ gives two colored

til**in**g (Fig. 6).

$\Omega$

The orbit of a po**in**t **in** a fundamental doma**in** (shaded regi**on** **in** Fig. 6

$\tilde{\Gamma}_{0}$

$)$

by **the** acti**on** of a new structure **on** $tili_{Il}g$ $D$ as follows (see Fig.

7 A).

88

Figure 6: Two colored til**in**g **on** **the** Po**in**car\’e disc $D$ .

$I\{\urcorner ig_{l1}r(^{\backslash }7:A,$ $Hyp_{G1}\cdot boli(^{\rangle}$

Archimedean til**in**g **on** **the** Po**in**car6 disc $D$ . Tlie

circles **on** **the** sliaded triangles forlil tlie of $tili_{ll}g$ vertex type $(3^{3}.4.3.4)$ .

$B$ ,

ABC star-shaped block copolylner systelil. The I (balls) and **the** (lnembralle

surround**in**g **the** balls) colllpollellts forni**in**g tlle gyroid 111$eIllbra$lle.

$S$

Rema**in****in**g parts are filled with **the** comp**on**ent.

$P$

89

Every po**in**t of $P$ $\Omega$

is surrounded by 6 faces: 4 triangles and 2 quadrilaterals.

We denote its vertex type by (3.3.3.4.3.4) or $(3^{3}.4.3.4)$ which **in**dicates

**the** numbers of vertices of **the** c**on**secutive faces around $P$ . We call it hyperbolic

Archilnedean til**in**g, because it can be coilsidered as a $si_{1}nple$ analogue

of **the** Archimedean til**in**g of type (3) **on** a plane.

By **the** results of translnissi**on** electr**on** microscopy and small-angle X-ray

scatter**in**g, we can f**in**d **the** geometry of **the** gyroid surface **in** **the** geolnetric

structure of and ABC star-shaped block copolymer systenl, which is composed

of polyisoprene (I), polystyrene (S), and poly (2-v**in**ylpyrid**in**e) (P):

**the** I and comp**on**enfs form a gyroid membrane and **the** complement of **the**

$S$

membrane is filled with **the** comp**on**ent (Fig. 7 B). $P$ $T\iota_{1e}$

I $col11p_{o1}1ent$ c**on**sists

of isolated doma**in**s that forln a new til**in**g structure **on** **the** gyroid surface

corresp**on**d**in**g to **the** hyperbolic $Archi_{1}nedean$ til**in**g **in**troduced above. Furtliermore

tliis til**in**g is characterized **in** three-dimensi**on**al Euclidean space as

follows.

Let $P$ be a po**in**t **on** **the** gyroid surface and $S$ $\Omega_{P}$

**the**

$\tilde{\Gamma}_{0}$

**the** acti**on** of . Then we have

Theorem There exists a unique po**in**t $P$ $\varphi(\triangle)$

**in** **the** triangle (a

orbit of $P$ under

fundamental

doma**in** of **the** acti**on** of

$\Gamma_{0}=I\overl**in**e{4}3d$) **on** **the** gyroid surface such that $S$

$\Omega_{P}$

forms a hyperbolic Archimedean til**in**g $(3^{3}.4.3.4)$

**the** same length.

**in** which **the** edges are all

Outl**in**e of proof. By $t1_{1}e$ $t1_{1}esyInlrletl\cdot y$ , po**in**ts of tlie

$tl_{P}a\iota\cdot easse\iota flbledi_{I1}$

same way around eacb po**in**t. Thus when **the** faces surround**in**g **on**e po**in**t of

$\Omega_{P}$

satisfy

$s$

tlie collditio11 , all faces satisfy **the** collditi**on**s. Let $P=(x, y, z)$ be

$\Omega_{P}$

a po**in**t of . Then **the** c**on**diti**on**s that all edges c**on**nect**in**g to $P$ have **the**

same length are given by two quadratic equati**on**s. Let $C$ be **the** **in**tersecti**on**

$stl1^{\cdot}f\cdot a(^{\tau}c^{i}s$

of two quadric def**in**ed by **the** quadratic equati**on**s. It is not hard to

show that **the** curve $C$ and **in**tersect at **on**e po**in**t **in** tlie triangle $S$ $\varphi(\triangle)$

.

We show tlie til**in**g of **the** **the**oreln **in** Fig. 8.

References

[1] S. Hyde, S. Anderss**on**, K. Larssoll, Z. Blulil, T. Landh. S. Lid**in**, B.

W. N**in**ham, The language of Shape (Elsevier, Amsterdam 1997); W.

G\‘o

$\grave{z}d\grave{z}a1$ld R. Holyst, Macromol. Theory Simul. 5. 321 (1996) and a

90

Figure 8: **in**g of type $(3^{3}.4.3.4)$ **in** threedimensi**on**al

Euclidean space whose vertices are **on** **the** gyroid surface.

short of periodic lnal surfaces $1_{1}istoI\cdot y$ $t1_{1}e$ $1ni_{l1}i$ $t1_{1}eI^{\cdot}ei_{l1}$

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$P1_{1}ys$

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