Non-balanced minimal surfaces. IWP (Schoen 1970) Structural unit in a unit cell (a mirror cube)....
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![Page 1: Non-balanced minimal surfaces. IWP (Schoen 1970) Structural unit in a unit cell (a mirror cube). Eight catenoidal tunnels in [111] directions Im3m A =](https://reader036.fdocuments.net/reader036/viewer/2022062421/56649d825503460f94a68703/html5/thumbnails/1.jpg)
Non-balanced minimal surfaces
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IWP(Schoen 1970)
Structural unit in a unit cell
(a mirror cube). Eight catenoidal tunnels in
[111] directions
Im3m
A = 23
= 3.4641
= –12
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IWPTranslational subunit in a truncated
octahedron (Voronoi
region of bcc lattice)
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IWP8 cubic
unit cells
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IWPunit inscribed in
a rhombic dodecahedron
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IWP
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IWP-1(IWP with one
catenoidal tunnel missing)
Pm3m
area =
3.241 x 2 =
6.482
= –82
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IWP-1view of unit cell along 3-fold axis
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IWP-2IWP with 2 catenoids removed
A =
3.0174 x 2 =
6.0348
= –68
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IWP/ Neovius
hybridPm3m
A = 4.4562
= –30
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Modulation
(polytype)
of C(P) /C(P)a
[The octagonal panels can be of four different types, 0,1,2 or 4 catenoids and can be combined in various ways.]
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FRD(Schoen)
A = 4.7758
= –40
CaF2
type
structure
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FRD from a structural unit in a
rhombic dodecahedron!
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A minimal surface in a
regular dodecahedr
onStructural unit for a minimal surface in S3 rather than for a triply periodic surface in E3. 120 units in the polytope {5,3,3}.
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Minimal surface in
an icosahedro
nA structural unit for a minimal surface in S3 rather than for a triply periodic surface in E3. 600 tetrahedral units in the polytope {3,3,5}.
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OCTO(Schoen)
P/IWP hybrid
This structural unit of OCTO derived from
IWP by insertion of six tunnels through the cube faces
Pm3m
A = 3.67612
= –18
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A modification of IWP
Only 3 of the 6 possible tunnels through faces of
the cubic unit cell are included. Eight units make a unit cell of the
new surface
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Another modified IWP
Only two of the six tunnels are
retained
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Another structural
unit derived
from IWP
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Double P
Pm3m
A = 3.972
= –22
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Pm3m
A = 4.0548
= –28
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Pm3m
A = 4.0548
= –28
Same surface, different choice of origin!
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Triple PPm3m
A = 4.5835
= –46
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P/T2 hybrid
(Mackay)
Pm3 = –30
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P/T2 hybrid
1/8 unit cell
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ALM
P/T2 hybrid
A=5.3878
Pm3
= –30
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P,FRDHybrid P/T4
(unit cell)
Fm3m
A = 5.7875
= –44
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P,FRD Hybrid P,T4
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Hexplane2
(Brakke)
1/8 unit cell
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Hexplane2
Unit cell
A = 5.3742
= – 88
Fm3m
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betaWPm3n
A = 4.3419
= –20
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OCTONOCTO/Neovius
hybrid
Pm3m
A = 4.497
= –36
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OCTONHybrid
OCTO/C(P)
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NPinNeovius with P tunnels inwards
Gozdz & Holyst’s
CPDA = 4.3441
= –26
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NPin
(CPD)1/8 unit cell
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CPCPB(Neovius with tunnels out;
hybrid C(P)/C(P)b
Pm3m
A = 3.55836
= –22
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CPCPBalternative
origin
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Hybrid P/T1
Structural unit,
1/8 unit cell
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Hybrid P/T1Unit cell
Pm3
A = 5.0745
= –28
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Orthorhombic variants of the P surface
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= –46
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= –52
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= –56
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= –56
(Same as previous surface. Different origin!)
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Batwing 1/8 unit cell
Pseudobatwing unit cell
Patches of two surfaces that are almost identical. This leads to interesting possibilities for hybrid surfaces, a 3D analogue of Truchet
tilings
3D “Truchet tiling”
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Batwing
Im3m/Pm3m
unit cell
Pseudobatwing
R3m/R3m
unit cell
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The four orientations of the 3D
Truchet tile
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Batwing3 Truchet-type
tessellation
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