Surface Evolver Documentation - Torus Example

# Surface Evolver Documentation

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# Example: Torus partitioned into two cells (Kelvin's foam)

This example has a flat 3-torus (i.e. periodic boundary conditions) divided into two bodies. The unit cell is a unit cube, and the surface has the topology of Kelvin's partitioning of space into tetrakaidecahedra [TW], which was the least area partitioning of space into equal volumes known until recently [WP].

The datafile handles the wrapping of edges around the torus by specifying for each direction whether an edge wraps positively (+), negatively (-), or not at all (*).

Note the use of the keyword `TORUS_FILLED` in the datafile. This informs Evolver that one of the volume constraints is redundant, preventing a singular matrix when the time comes to enforce volume constraints. One could use just `TORUS` and only put on one volume constraint.

The display of a surface in a torus can be done several ways. The `connected` command (my favorite) makes each body show as a single unit. The `clipped` command shows the surface clipped to the fundamental parallelpiped. The `raw_cells` command shows the unedited surface.

The Weaire-Phelan structure [WP]. is in the datafile `phelanc.fe`. It has area 0.3% less than Kelvin's.

| The initial two-cell Kelvin shape. Note that due to periodidity, a single vertex or edge may appear multiple times in the image.

    // twointor.fe
    // Two Kelvin tetrakaidecahedra in a torus.

    TORUS_FILLED   // signals that domain is a torus and bodies fill it.

    periods
    1.000000 0.000000 0.000000
    0.000000 1.000000 0.000000
    0.000000 0.000000 1.000000

    vertices // values from another program
    1  0.499733 0.015302 0.792314
    2  0.270081 0.015548 0.500199
    3  0.026251 0.264043 0.500458
    4  0.755123 0.015258 0.499302
    5  0.026509 0.499036 0.794636
    6  0.500631 0.015486 0.293622
    7  0.025918 0.750639 0.499952
    8  0.499627 0.251759 0.087858
    9  0.256701 0.499113 0.087842
    10 0.026281 0.500286 0.292918
    11 0.500693 0.765009 0.086526
    12 0.770240 0.499837 0.087382

    edges // with wraps in axis directions
    1    1 2  * * *
    2    2 3  * * *
    3    1 4  * * *
    4    3 5  * * *
    5    2 6  * * *
    6    2 7  * - *
    7    1 8  * * +
    8    4 6  * * *
    9    5 9  * * +
    10   3 10 * * *
    11   3 4  - * *
    12   6 8  * * *
    13   6 11 * - *
    14   7 4  - + *
    15   8 12 * * *
    16   9 8  * * *
    17   9 11 * * *
    18   10 7 * * *
    19   11 1 * + -
    20   12 5 + * -
    21   5 7  * * *
    22  11 12 * * *
    23  10 12 - * *
    24   9 10 * * *

    faces
    1    1 2 4 9 16 -7
    2    -2 5 12 -16 24 -10
    3    -4 10 18 -21
    4    7 15 20 -4 11 -3
    5    -1 3 8 -5
    6    6 14 -11 -2
    7    5 13 -17 24 18 -6
    8    -12 13 19 7
    9    -16 17 22 -15
    10   -10 11 8 12 15 -23
    11   -21 9 17 19 1 6
    12   -14 -18 23 -22 -13 -8
    13   -24 -9 -20 -23
    14   -19 22 20 21 14 -3

    bodies
    1    -1 -2 -3 -4 -5 9 7 11 -9 10 12 5 14 3 volume 0.500
    2     2 -6 -7 8 -10 -12 -11 -13 1 13 -14 6 4 -8 volume 0.500

Doing some refining and iterating will show that the optimal shape is curved a bit.

Next: Spinning ring example.
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