Triply Periodic Minimal Surfaces

A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece. Soap films are minimal surfaces. Minimal surfaces necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic. Many triply periodic minimal surfaces are known, some of which are pictured on this page.

These images were made with the Surface Evolver. The Evolver datafiles are the *.fe files linked to below. The surfaces are generally made by defining and evolving the fundamental region of the surface, which is usually very simple due to the high symmetry, and then displaying many copies of it, suitably transformed. The fundamental regions usually are one of Coxeter's kaleidoscopic cells. Surfaces with "adj" in the file name were made by evolving the adjoint surface. These surfaces need the Evolver file adjoint.cmd.txt along with the datafile.

NOTE: Many of these files also need the file cube_transforms.inc, which contains the transformations for showing multiple copies of the fundamental region. (The downloaded file is actually named cube_transforms.inc.txt, since our university web server will only permit certain extensions on downloaded files. Remove the ".txt" extension after downloading.)

Many of these surfaces were originally discovered by Alan Schoen in his famous 1970 NASA technical report Infinite Periodic Minimal Surfaces without Self-Intersection.

I have made plastic models of several of these using 3D printing via Shapeways.com. These models are available for public purchase from Shapeways; see my list.

Some of the images below made the cover of AMS Notices in December, 2000.

Edo Timmermans makes triply periodic minimal surfaces out of magnetic balls. See his videos of the P-surface, D-surface, H-surface, and more.


Contents:


pcell start pcell end pcell cube four pcells pcell cube pcell cube
Quadrilateral
before evolving.
Quadrilateral
after evolving.
Unit cubic cell
showing mirror planes.
Four unit cubes. Unit cubic cell
showing embedded
straight C2 axes.
A unit made
of a catenoid
between squares.

Schwarz' P Surface

The fundamental region, as bounded by mirror planes, is a quadrilateral in a tetrahedron which is 1/48 of a cube. The left two images show the fundamental region before and after evolution. The third image is one cubical unit cell, and the fourth is four unit cells. The fundamental quadrilateral has a C2 axis (straight line with 180 degree rotation symmetry), and the fifth image shows all the embedded C2 axes. The surface divides space into two congruent labyrinths, as do many of the other surfaces on this page.
Evolver file: pcell.fe
Buy from Shapeways.com

D surface cube D surface rhomb

Schwarz' D Surface

The left image shows a cubical unit cell. Note that the translation period of the labyrinths is twice that of the surface itself; i.e. a unit translation of the surface switches the sides of the surface. The right shows the surface in a rhombic dodecahedron. The skeletons of the labyrinths have a diamond lattice pattern, which gives the surface its name. Small metal sculptures of the D surface are available from Bathsheba Sculpture.
Evolver file: dcell.fe
Buy from Shapeways.com

gyroid

Schoen's Gyroid Surface

The only triply periodic surface known with no embedded straight lines or mirror symmetries. It is locally isometric to both the P and D surfaces, being an intermediate in the Bonnet rotation family of the P and D surfaces. More on my gyroid page.

complementary D surface

Schoen's Complementary D Surface

This surface has the same space symmetry group and same straight lines as Schwarz's D surface. If you look carefully, you will see that the piece shown does not have full 6-fold rotational symmetry; a 1/6 rotation requires a flip. For more on this and related surfaces, go here.
Evolver file: cd.fe
Buy from Shapeways.com (rhombic dodecahedron unit cell, fine mesh)
Buy from Shapeways.com (rhombic dodecahedron unit cell, coarse mesh)

Neovius surface

Neovius' Surface

A cubical unit cell, basically a central chamber with necks out to the middle of each edge of the cube.
Evolver file: neovius.fe
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Neovius genus 14 surface Neovius genus 14 surface

N14 Surface

One version of the cubical unit cell basically a central chamber with necks out to the middle of tubues running along each edge of the cube. Another cubical unit cell, offset from the first by half an edge in each direction, looks like a Neovius surface with extra tunnels. Genus 14. For lack of a better name, I am calling this "N14" for "Neovius genus 14". Proposed by Alan Schoen from sketches he dug up from 1969.
Evolver file: N14.fe

Neovius genus 26 surface Neovius genus 26 surface

N26 Surface

This is the N14 surface with some extra holes I punched in it to make it genus 26. Again, two different unit cells are shown.
Evolver file: N26.fe

Neovius genus 38 surface Neovius genus 38 surface

N38 Surface

This is the N14 surface with some extra holes i punched in it a different way to make it genus 38. Again, two different unit cells are shown.
Evolver file: N38.fe

Schoen C15(P) surface

Complementary P Surface Family Page

A family of surfaces generalizing the Neovius surface.

batwing pair batwing cube

Batwing Family Page

Surfaces having a quadrirectangular tetrahedron as kaleidoscopic cell.

Starfish family member

The Starfish Family Page

This is a two-parameter family, members of which are shown in tables of kaleidoscopic cells, cubelets, and rhombic dodecahedra, labeled by their genus. The family may be parameterized by (p,q), where p is the number of holes along the cube edge, and q along the cube diagonal. Pictured at left is (3,3). Putative members of the family may not actually exist; attempts to get all the edges of the fundamental region to match up properly (called "period killing" by the cognoscenti) may leave a gap. Starfish 4-2 fails to period kill by only 0.005 (so far).

disphenoid p=31

Disphenoid Families Page

Several families of surfaces whose kaleidoscopic cell is the tetragonal disphenoid. There are two C2 axes, of unequal length.

P3a disphenoid P3a line frame P3a rhombic

P3a surface of Lord and Mackay

A disphenoid surface, using two C2 axes of equal length. This surface has the same skeleton of embedded straight lines as the P surface and Neovius' surface (middle). At right is a rhombic cell.

S'-S''

Hybrids

Some combination surfaces.

H surface unit H surface four

Schwarz' H Surface

The left image shows a equilateral triangular prism cell, which is actually half of a lattice unit cell. The right shows four cells connected.
Evolver file: hcell.fe

RII unit

Schoen's RII Surface

The image shows an isosceles right triangular prism cell, which again is half of a lattice unit cell.
Evolver file: RII.fe

RIII unit

Schoen's RIII Surface

The image shows a 30-60-90 triangular prism cell, once more half of a lattice unit cell.
Evolver file: RIII.fe

I-6 unit

Schoen's I-6 Surface

Surface between plane square grids.
Evolver file: I-6.fe

I-8 unit

Schoen's I-8 Surface

Surface between plane square grids with diagonals.
Evolver file: I-8.fe

I-9 unit

Schoen's I-9 Surface

Another surface between plane square grids with diagonals.
Evolver file: I-9.fe

CLP surface

Schwarz' CLP Surface

This surface can be viewed as two orthogonal sets of parallel planes with alternating tunnels through the intersections to remove the lines of intersection.
Evolver file: clp.fe

F-RD surface

Schoen's F-RD Surface

Unit cell with tetrahedral symmetry. It may be viewed as a central chamber with tubes to alternating corners of the cube. This is actually only an eighth of a lattice cell; to get a lattice cell, reflect in the cube faces.
Evolver file: FRDadj.fe
Buy from Shapeways.com (unit cell)
Buy from Shapeways.com (1/8 unit cell)

Hybrid-1[P,F-RD] surface

Schoen's Hybrid-1[P,F-RD] Surface

Unit cell with tetrahedral symmetry. It may be viewed as a central chamber with tubes to alternating corners of the cube and to the faces of the cube. This is actually only an eighth of a lattice cell; to get a lattice cell, reflect in the cube faces.
Evolver file: hybrid-1adj.fe

GW surface, parameter 0.1 GW surface, parameter 0.1 GW surface, parameter 0.1 GW surface, parameter 0.1 GW surface, parameter 0.1 GW surface, parameter 0.1

Schoen's GW Surface

One labyrinth of this surface has the graphite hexagonal sheet structure, and the other labyrinth has the wurtzite structure, hence the name GW. As a parameter is varied from small (left) to large (right), the surface goes from horizontal parallel sheets with catenoid connections to pairs of vertical sheets in a hexagonal layout with cross-tunnels at the junctions.
Evolver file: GW5adj.fe

I-WP surface

Schoen's I-WP Surface

The unit cell may be viewed as a central chamber with tubes to the corners of the cube.
Evolver file: IWP.fe
Buy from Shapeways.com (1/8 unit cell)
Buy from Shapeways.com(full unit cell)

O,C-TO surface

Schoen's O,C-TO Surface

The unit cell may be viewed as a central chamber with tubes to the corners and faces of the cube, thus being a hybrid of the P surface and the I-WP surface.
Evolver file: octoadj.fe

Schoen-12 surface

Schoen's F-RD(r) Surface (formerly unnamed Surface 12)

The unit cell may be viewed as a chamber in a box with tubes running to the centers of four edges. The unit cell has 8-fold symmetry (there being horizontal C2 axes as well as the mirror lines shown). The box is square based, and the height is variable. The cell shown is actually only half of a lattice cell; reflect in the top or bottom cube face to get a full lattice cell. This image was actually constructed by evolving the adjoint surface (which is in the datafile), adjusting a parameter to kill one period, and then doing an adjoint transformation.
Evolver file: s12adj.fe

Schoen-14 surface

Schoen's I-WP(r) Surface (formerly unnamed Surface 14)

The unit cell may be viewed as a chamber in a box with tubes running to the centers of eight edges. The unit cell has 16-fold symmetry. The box is square based, and the height is variable. This image was actually constructed by evolving the adjoint surface (which is in the datafile), adjusting a parameter to kill one period, and then doing an adjoint transformation.
Evolver file: s14adj.fe

manta pair manta part manta cube manta octahedron

Schoen's Manta Surface of Genus 19

The fundamental region is a tetrahedron which is 1/96 of a cube. The left image shows two fundamental regions, whose appearance is the source of the name "manta". The second image shows 12 fundamental regions in a cube. The third image is the full cubical unit cell. The fourth image shows the surface as a chamber with tubes in a slightly flattened octahedron.
Evolver file: mantaadj.fe
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manta35 pair manta35 cube manta35 cell manta35 octahedron

Schoen's Manta Surface of Genus 35

The second member in the Manta series.
Evolver file: manta35adj.fe

manta51 pair manta51 cube manta51 cell manta51 octahedron

Schoen's Manta Surface of Genus 51

The third member in the Manta series.
Evolver file: manta51adj.fe

0 tunnel triplane 1 tunnel triplane 2 tunnel triplane 3 tunnel triplane 4 tunnel triplane 5 tunnel triplane
A sequence of surfaces converging to three sets of orthogonal planes. One intuitive way to construct a lot of minimal surfaces is to take a set of minimal surfaces and resolve their intersections to smooth surfaces by drilling tunnels crosswise. This sequence, starting with the D surface, shows lattice cells of the surface. Note the labyrinth periods are twice the surface periods. The surfaces are of genus 3, 9, 15, 21, 27, and 33.
Evolver files: triplane0adj.fe, triplane1adj.fe, triplane2adj.fe, triplane3adj.fe, triplane4adj.fe, triplane5adj.fe

F-RD surface 1 tunnel hexplane 2 tunnel hexplane 3 tunnel hexplane 4 tunnel hexplane
A sequence of surfaces converging to three sets of orthogonal planes, beginning with the F-RD surface, which intercalates the sequence above. Each cube pictured here is actually 1/8 of the lattice cell. The surfaces are of genus 6, 12, 18, 24, and 30.
Evolver files: hexplane1adj.fe, hexplane2adj.fe, hexplane3adj.fe, hexplane4adj.fe, hexplane5adj.fe

Fischer-Koch S surface S surface unit cell

Fischer-Koch S Surface

Left: Piece of the Fischer-Koch S surface. The piece is bounded by straight lines, which are axes of 180 degree rotation symmetry in the directions of the diagonals of the faces of a cube. Right: a cubic unit cell. Note that this surface has no mirror symmetry, so the sides of the cube are not mirror planes. There is a C3 axis down one diagonal of the cube
Evolver file: Scell.fe

Fischer-Koch C(S) surface

Fischer-Koch C(S) Surface

Piece of the Fischer-Koch C(S) surface which they later realized was the same as the P surface. The piece is bounded by straight lines, which are axes of 180 degree rotation symmetry.
Evolver file: CScell.fe

Y piece Fischer-Koch Y surface

Fischer-Koch Y Surface

At left is a piece of the Fischer-Koch Y surface, which they later realized is the same as the D surface. The piece is bounded by straight lines, which are axes of 180 degree rotation symmetry. At right is a bunch of pieces together, producing two labyrinths of tunnels.
Evolver file: ycell.fe

Fischer-Koch C(Y) surface C(Y) unit cell

Fischer-Koch C(Y) Surface

Piece of the Fischer-Koch C(Y) surface. The piece is bounded by straight lines, which are axes of 180 degree rotation symmetry.
Evolver file: CYcell.fe


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