Even-frequency PUCK weaving patterns via the map operation Meta

For orientable surfaces the map operation Bevel is a reliable source of bipartite cubic maps. The duals of these maps are chess-colorable triangulations, and therefore, the chess-colorable triangulations needed for even-frequency PUCK weaving can be generated directly through the map operation Meta ( since Mt(M) = Du(Be(M)). )

Here are some examples of Meta applied to maps that were already triangulations in the first place.

A chess-coloring of Mt(Octahedron)

A chess-coloring of Mt(Icosahedron).

A chess-coloring of Mt(Tetrahelix).

A chess-coloring of Mt(a high frequency triangulation of the Sphere).

A chess-coloring of Mt(a multi-resolution surface mesh.)

Using the map operation Meta to create a PUCK weaving pattern effectively replaces each edge rhomb with a 3-D module like this one, composed of four equilateral triangles: two "outtie" triangles and two "innie" triangles.

This tessellation is 3-regular and even-faced, but it is not the Bevel of anything.

This tessellation is 3-regular and even faced; it is the Bevel of the square grid.

This tessellation is 3-regular and even-faced; it is the Bevel of both the triangle and the hexagon grids.

Fold-less PUCK weavers

A fold-less PUCK weaver viewed perpendicularly to the plane of the apexes.

Replacing the folds in a 0-frequency PUCK weaver with triangles of conical curvature gives a PUCK weaver with points of infinite curvature but no actual folds. This may be the best weaver for materials that cannot be folded.

A fold-less PUCK weaver, top view.

Fold-less PUCK weaver above a 0-frequency (folded) PUCK weaver.

Fold-less and folded PUCK weavers compared.

PUCK weaver lengths

Brick-laid PUCK weavers are not guaranteed to come out commensurate with a loop unless they are either 1 or 2 squares in length. Thus the only universally applicable brick laid design is 2 squares in length with a phase shift of 1 (2/1,) which is probably too short an overlap between weavers.

Shingle-laid weavers can be any length, and thus can have any amount of overlap, but, again, to come out even around a loop, the only universally applicable phase shifts are either 1 or 2 squares. But a 1-square phase shift cannot give strapped-down edges. Also, odd lengths cannot give strapped-down edges.

Thus the practical choices for PUCK weavers are n/2 shingle-laid, where n is an even integer. Since 2/2 shingle-laid gives zero overlap, the only choices are shingle-laid:
4/2—basket is 4 layers thick,
6/2—basket is 6 layers thick,
8/2—basket is 8 layers thick, etc.

Even 6/2 probably makes it too difficult to complete a closed basket, and six layers thick is probably excessive. 4/2 shingle-laid may be the only practical option at this point.

4/2 shingle-laid weavers have simple splicing: the first 3-way crossing settles the splicing pattern in all three loops as these images show. That is, there is only one option for where to place new weavers in each direction so that the splice is strapped-down by the finished crossing.

Corrugation options for PUCK weaving

A length = 6, frequency = 1, PUCK weaver made from the wall of an aluminum soda can.

Every PUCK weaver has folds at 90 degrees to its sides that divide the weaver into squares. Optionally, there may be additional corrugations running at 45 degrees to the sides. PUCK weavers can be categorized by an integer, called the frequency, that gives the number of these 45-degree folds per square. (For example, the frequency is 0 when there are no 45-degree folds.)

If the folds can be assumed to be equally spaced and arrayed symmetrically about the parallel diagonal of the square, then the corrugation pattern is fully specified by its frequency.

Even frequencies (e.g., 0 or 2) require chess-colorable triangulations, which can be undesirable for smoothly curved surfaces because the smallest allowable deviations from a flat surface (6 triangles around a vertex) are 4 or 8 triangles around a vertex. With odd corrugation frequencies, 5 or 7 triangles around a vertex are also permissible.

Commercial corrugated steel sheets use a corrugation wavelength of 1.25" with 26-28-29 gauge steel and a wavelength of 2.5" with 18-20-22-24-26 gauge steel. Deeper corrugations are made in a wavelength of 2.67" with 18-20-22-24 gauge steel.

Steel sheet of these gauges are

#18 .048"
#20 .036"
#22 .030"
#24 .024"
#26 .018"
#28 .015"
#29 .014"

lambda/t ranges for different steel corrugation wavelengths

lambda   lambda/t    geometric mean
1.25"      69 to 89         78
2.50"      52 to 139       85
2.67"      56 to 111       79

Aluminum soda cans have about 100 micron (.004") thick walls. At lambda/t = 80, corrugation wavelength is 8mm = 0.315"

PUCK weavers with 0.6" squares (which can be cut as 6:1 weavers from can-height 3.75" strips) give

f      lambda     lambda/t
0          1.0"         250
1           0.5"        125
3           0.25"        63

At this scale, both f = 1 and f = 3 give acceptable corrugation wavelengths for aluminum can material.

Options for 4-square puck modules for unit origami

Since 4-square units are an even number of squares in length, shingling is out for hidden edges. The splicing must be brick-laid with an odd phase shift (1 is the only option since 3 is really the same thing.) Enantiomorphs are again required.

The good news is for the case with strapped-down edges. This 4-square unit can be simply shingled.

A 4-square puck module for unit origami.

Options for 3-square puck modules for unit origami

Options for puck origami units of three squares length.

None of these units require enantiomorphs when the phase shift is two squares. When shingled, the splice edges of the upper three designs can be hidden. The lower design, which has splice edges coinciding with the edges of the squares, cannot have all its splice edges strapped-down because the length of the unit is odd (see previous post.)

Options for 2-square Puck origami units

Options for a 2 square modular origami unit for Puck weaving.

The unit at the top of the figure is its own mirror image, thus only one type of unit is needed, but the splices cannot be kept internal. The other three designs need to work as left/right enantiomorphic pairs but splices can be hidden under other weavers.

Definition: Puck is an acronym for polyphase, unit-woven, corrugated kagome.

There are two ways weavers can be spliced. They can be shingled like books fallen-over on a shelf, or laid like two courses of bricks.

To hide splices:

Shingled:  the length of the shingle must be odd since the left edge of the shingle will show on one face of the weaving and the right edge on the other; the phase shift (the spacing between successive shingles) must be even so that the corresponding edge of the next shingle on the same face can be covered as well.

Laid: the length of the shingle must be even since both left and right edge show on the same face of the weaving; the phase shift between the two courses must be odd so that correspondig edges on the reverse face are covered by the weaving.

For example, the lower three unit designs, since they have even length, must rely on a laid configuration and an odd phase shift (1 unit is the only option) in order to hide the splice edges.

In the case where the splice edges coincide with square edges (e.g., the upper design), we must settle for having the splice edges strapped-down rather than hidden. In this case, the splice must be single-edged (i.e. a shingled pattern, not a laid pattern) and—just as required above—the phase shift must be even. Here the length will also be even since half a square is gained at each end by settling for an edge that is merely strapped-down rather than hidden. The smallest such design is a four-square weaver with a two-square phase shift.