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A systematic method for parametrising periodic minimal surfaces: the F-RD surface A. Fogden
To cite this version: A. Fogden. A systematic method for parametrising periodic minimal surfaces: the F-RD surface. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.233-239. .
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J.
Phys.
J
France
(1992)
2
233-239
1992,
MARCH
PAGE
233
Classification
Physics
Abstracts
02.40
68.00
A systematic surfaces : the
parametrising
for surface
method F.RD
periodic
minimal
Fogden
A.
Department
(Received
Mathematics, University, PO
National
July 1991,
23
Abstract.
which
accepted
algorithm
An
developed by discovered by from
Applied
of
Australian
the Alan
all
for is
author
Schoen.
quantitative
in
the
exact
features
2
with
expression of
Physical Australia
December
1991)
of
construction
simply
exact
of
Canberra
final form
illustrated An
4,
2601,
School
Research Box
this
surface
the for
Sciences
triply periodic minimal parametrisation of
first the
may
F-RD be
Weierstrass
readily
and
Engineering,
surfaces the
function
recently
F-RD
surface
is
derived,
established.
Introduction.
key participant in all stages of the development of the physical microstructures increasing diversity of bicontinuous observed in condensed Iii provides a challenge to conventional geometric intuition. Recently, matter infinite periodic minimal surfaces (IPMS) have been adopted as the basis of a vocabulary for this specific class of this evolving language of It is an open question as to whether structure. periodic surfaces of the physical however context systems, possess any real significance in the their utility as a of classification is undeniable. The cubic TAMS of genus three the means D, P and G surfaces frequently invoked, in modelling the underlying a most structures are self-assemblies variety of molecular [2]. It has been postulated that these particular IPMS are favoured basis surface homogeneity uniform the of [3], that is, they represent the most on interracial scenario attainable globally. Bicontinuous phases related to higher genus TAMS with in the phase diagram increasing with also the region of their existence may occur, decreasing sample purity. This expectation has been bome out by recent experiments [4]. With the of sophisticated techniques, the principal limitation advent measurement to the observation of topologically possible and recognition complex remains the structures incomplete knowledge regarding accompanying IPMS descriptions. This problem has been partially redressed by the discovery of a multitude of new IPMS by Schoen [5] and, lately, by While
sciences,
geometry
has
explanation
been
a
of the
and Fischer [6]. These TAMS empirically, without the precise mathematical isolated were specification of the surface for quantitative comparison with observed systems. necessary Many of the Schoen examples have now been parametrised by Karcher [7]. A systematic of a general IPMS has recently been procedure for the parametrisation simplified developed by the author [8]. There the verified with greatly method a was description of the C(P) surface, then utilised in deriving the first rederivation of Neovius'
Koch
234
JOURNAL
PHYSIQUE
DE
I
N° 3
parametrisations of the hexagonal HS2 and orthorhombic surfaces PT of Koch and Fischer, together with those of the new orthorhombic IPMS and surface (denoted pentagonal new VAL and pCLP, respectively) arising from earlier work with Hyde [9, 10]. While the general formulation of the procedure is mathematically involved, the fundamentals simple and its are seeking application should accessible experimentalists classification of observed be to with With view this aim illustrate the method here the first parametrisation structures. to we a of the F-RD surface of Schoen. The choice of this example is physically motivated the Fsurface is a cubic IPMS of relatively simple topology (in fact, the lowest cubic IPMS RD genus of Schoen media. remaining undescribed) and thus may be expected to occur in bicontinuous The The
surface.
F-RD
topology
summarise lational
unit
absence two
by
the
of
(or
3 ~
m
m
).
and
RD).
The
in
the
Accordingly rotational
two-fold
partitioned by
Schoen's briefly is specified in report [5] we parametrisation here. The fundamental transfigure I, has a genus of six and the space group
surface
to
displayed
~
on-surface
labyrinths F
F
F-RD
relevant
IPMS,
the
of any
types
of the
symmetry information
Fm3m
symmetry
the
and the
the
interface
translational
unit
IPMS
the
axes
(as is the
are
geometrically
is
for
case
the
from
a
surface
I-WP
(and
distinct
constructed
planes
mirror
contains
basic
in II
the
])
so
are
characterised
surface
element
or
(identified in Fig. I ) delimited by four mirror planes of the surface. Propagation element by repeated reflection in the bounding mirror planes, until the full set of of this basic symmetries are exhausted, translational unit, thus comprising 48 such elements. generates the intersection surface is referred The of a mirror plane with the to as a plane line of curvature. Flhchenstfick
perpendicularly. The remaining defining the basic element intersect angles of w/3 and MN at points of 3-fold and 4-fold perpendicular respectively. These are degenerate points of the surface, rotational symmetry of the surface, is zero. called flat points, at which the Gaussian The degree of these flat points curvature surface of normal this point traced the number revolutions about the vector at out by the surface enclosing point is respectively. normal circuit the the 2 and 3, along any vector on Two
two
Fig.
pairs pairs
I.
of
such
curves
intersect
The
Parametrisation
at
fundamental
of
unit
of
F-RD
surface into the plane, such Mapping a minimal normal of of its surface point intersection the to in tum stereographically projected to a complex Weierstrass [12] to have the general form
(x,
y,
z
Re =
comprising
surface
48
of
the
surface
elements
shown.
surface.
F-RD
the
the
l~
(I
that
w'~, I(I
a
number
+
point (x, y, z) unit sphere
surface with
vector
w,
w'~),
the the
2
is
Gauss
centred
inverse
function
w') R (w')
dw'
was
mapped there, shown
and
by
(1)
N°
3
PARAMETRISATION
EXACT
OF
THE
235
SURFACE
F-RD
complex function R (w ). The image of the basic element of the F-RD surface in the some complex plane under this composite map is the shaded polygon in figure 2. For an IPMS the corresponding Weierstrass function R (w ) is multivalued, specifically, each generic and more surface image w is normal shared by the (finite) number of points vector same (x, y, z) on the translational unit. Gauss of gives the multiple Hence the this unit map covering of the unit sphere (and the complex plane on projection) comprising this number of sheets. This is a topological and is equal to the less for an number constant, genus one orientable TAMS [9]. For the genus six F-RD surface Gauss the translational unit map image from figure 2 then comprise five sheets. This is apparent just as successive reflections must of the basic in its bounding planes yields the set of 48 such surface element elements forming unit, in the Gauss map image repeated the translational reflection of the spherical geodesic polygon in its edges generates a set of 48 such polygons tessellating a five-sheeted spherical covering. for
Fig.
Projected
2.
Gauss
map
image
of
the
surface
F-RD
element.
surface normal images w of flat points, the number of points on the the special vector with the possessing this normal is only equal to the generic value if each is counted multiplicity of its flat point degree. Accordingly, above the image w on the multiple covering, identified numbering this degree, corresponding to the flat point are the set of points, as point, which is associated pinned at this equivalent. We regard the sheets set of as common referred branch point of order the degree less one. The projected Gauss map image to as a of the Weierstrass with imposed by this identification is the Riemann surface structure Riemann surface of the is function R (w ). For the F-RD surface the branch point structure generated by the propagation of the first and second order branch points (labelled in Fig. 2, and corresponding to the flat points of degree two and three, respectively, on the periphery of For
unit
the
basic
the
five As
and
The
the
is
surface
element)
at
the
2w/3
and
3 MN
vertices
of
the
polygon
in
tessellation
the
of
sheets. Riemann
hence
surface
specified by
Weierstrass
function
is
finite-sheeted,
polynomial
a
of
the
F-RD
Weierstrass
the
equation surface
of
is
function
degree equal then
the
to
solution
of the
of
an
IPMS
number the
is of
algebraic, sheets
[13].
quintic equation
a5(w)R~+a~(w)R~+a~(w)R~+a2(w)R~+aj(w)R+ao(w)
0 =
(2)
236
JOURNAL
particular completed specified.
for
As
the
cubic
on
Gauss
polygon
this
must
image of the surface triangulated by one
map be
symmetry
derived
are
images
reflection
polynomials a~(w ), combining equations (I)
of
from
geodesic
the
~~"'~)
*
/
((/
and
TAMS
an
figure tiling
the
of
the
sphere).
2.
For
[14]. The TAMS of sphere by the 48 w/3, w/2) (equivalent to unit
underlying
This
positioned
are
is
been
spherical covering,
a
orientation
this
surface have
tiles
angles (MN,
concentric
in
Schwarz
tiling
4
vertex
a
tessellates
basic
case
with
at
l) e'~"'~), respectively
I) e~"'~,
±
of
Schwarz
triangle
parametrisation of this these polynomials
once
fifteen
3
N°
The
(2)
and
element
the
I
,
of the
projection of the hexakis octalJedron onto tiling is shown in stereographic projection vertices of angle Mm, w/3 and w/2 in
~
ao(w ).
of six
set
a
then
PHYSIQUE
DE
(0,
(where
triangular
the 6, cc, m
is
8
12
and
e~"'~e~~"'~), integer).
an
of each coefficient polynomial a~(w) are necessarily symmetric with underlying Schwarz tiling. Thus they must comprise the 6, 8 and/or 12 tiling vertices and/or the 24 (respectively 48) images of a general edge (respectively face) point. The polynomials with roots at the Mm, w/3 and w/2 vertices are The
respect
set
of
to
the
roots
5
pi
fl
=
(w
w~)
=
w
=
w
(w~ +1)
(3a)
1=1
~
p2
fl
=
(w
w~)
(w
w~)
14 w~ +
~
(3b)
j=1 12
p~
fl
=
~~ =
+
w
33 w~
w~-
33
(3c)
,=1
(where the Mm vertex tessellated edges of the the degree of a~(w facts
These
where
n~,
ao
,
surface
4M
be
must
sufficient
equation (2)
particular
to
becomes
now
numbers,
real
are
complex plane is suppressed in pi). Further, for the IPMS, curvature to represent plane lines of on (again counting with the suppressed zero at infinity) [8]. specify the form of each coefficient polynomial in closed
in the
Riemann
then
are
infinity
at
and
the
indicate
braces
the
possibilities
two
for
the
cubic
term.
information
Additional known
that,
point
branch local
pinned image
to
a
the
at
the
wo,
results
demanding
from
Riemann
of the
structure
On
tessellation
the 2
three
unbranched.
Here
between utilise
we
the
equation (4) and the general observation
surface flat point of degree bo + I, on the bo + I sheets of the Riemann corresponding branch point (of order bo) above the flat point normal Weierstrass function branches asymptotic form [8] have the
R~°~ ~(w )
figure
consistency
surface.
of
sheets The
are
five
R~(w )
the
five
pinned branches
y~ (w
sheets, at
a
are
w~
wo)~~°
yo(w
then
each
over
second
order
given
)~~ (once),
Mm
w
-
vertex
point asymptotically branch
R(w )
surface vector
(5)
wo.
w~
of
with via
yj (twice)
:
the
underlying tiling
remaining equation (5) by the
w
-
w~
two
in
sheets
(6)
N°
PARAMETRISATION
EXACT
3
Substituting terms
be
applies.
requirement that the quintinc and quadratic form into equation (4), the (and equal) asymptotic order in (w w~) implies that the quartic term 0) and that the first of the two options for the cubic (that is, a~ term
present Hence
which,
terms
=
form
the
of
consistent
pinned
with
at
order
R~(w ) The
final
a~, a~, another
a~
and
may of
be
choice
a
for
units
w~ of the
vertices
and
of
loss
coordinate
the
~
vertex
wj of the
Schwarz
the
three
unbranched
sheets,
surface
Riemann
(~)
°
"0
w/3
R
yj (three
(w
w
scaling of
the
on
figure 2,
each
five
R be
to
is
coefficients
equation (7) while (w (amounting to specified. This is
unbranched
point
such
(8)
real
the of
out
parameters
two
now
w~
-
of
divided
suitable
a
structure
From
be
may
constant
is
tiling the pair of sheets display the asymptotics
times
determination
the
is
generality by axes), leaving
underlying tiling.
be
now
equation (6), the quadratic and required. Similarly, equation (7)
as
unknown
such
one
must
"2P2~~+ of
form
(once),
'
+
zero),
parametrisation
the
the
each
wj)~
Clearly ao. without chosen
by analysing
achieved
w/2
of
stage
second
point,
yj(w
"3P~~~
+
(namely
over
branch
~
the
order
that,
fact
the
first
a
of
equal leading
equation
polynomial
the
substitution
on
of
are
237
SURFACE
F-RD
leading
"5P~P2~~ for
THE
first
the
of
are
cannot
OF
sheets
covered
above
the
by four
once
w/2 vertex interior point of the polygons meeting with there, and four times as an common reflection latter tour coverings in the polygon. In the the related by two case are perpendicular underlying tiling edges intersecting at the point, and accordingly the four function For example, corresponding Weierstrass values at this point occur as a repeated pair. 2 and evaluating equation (7) at w (where, equations (3a) and (3b), pj(1) I from -12) exist real conjugate pair p~(1) there and and complex must constants a a a =
=
=
(c, n)
that
such
-48a~R~+4a~R~-12a~R~+ao= Equating
yields
coefficients
pair
the
16 a~ a~
removing
thus a~
5
and
ao
=
final
the
25 =
degrees
two
48, the
of
Weierstrass
a(R-a)(R-c)~(R-n)~.
homogeneous ao
135
a~,
of
relations
al
of
al
a~
the
say surface is
F-RD
=
p)p~R~-
(10)
=
Choosing,
freedom.
function
(9)
lsp)R~+5p~R~-48
a~= then
I,
a~=
-15,
given by (II)
0 =
where
and
pi
equations
(3a)
p~ and
are
the
Weierstrass
polynomials
of
the
I-WP
and
D
surfaces,
given
in
(3b).
parametrisation
attained by substituting the representation (I). Although the quintic polynomial equation cannot be solved analytically, all required surface information (such as local coordinates and the global properties of surface-to-volume ratio, surface area and integral squared Gaussian curvature) be extracted numerically in a simple and can economical The of the path integration in equation (I) may be restricted extent to manner. the polygonal region of the complex plane in figure 2, as the complete TAMS is then shaded reflection in the bounding planar generated by mirror surface of the basic element. The curves corresponding polygonal region of the Riemann surface (that is, the corresponding branch of the function) is Weierstrass ascertained by determining the five root of equation (I I) for a The
exact
Weierstrass
function
solutions
of
of
the
F-RD
equation
surface
(11)
into
is
the
thus
JOURNAL
238
PHYSIQUE
DE
N°
I
3
~ point
particular
comprise
real
a
the
on
w
real
number
edge
axis
and
0
w
