Accommodating smooth surface curvature in corrugated kagome weaving

If a surface is triangulated by equilateral triangles (what is sometimes called an equilateral surface or deltahedral surface,) that surface can be woven with ready-made, frequency-1, corrugated kagome weavers. (As illustrated in a previous post, the ready-made pattern that is called for is a strip of squares having a "triangle-wave" of folds along selected diagonals of the squares.)

Just as pixelated curves in computer graphics are never actually smooth, equilateral surfaces are always crinkly. They have a quantized surface curvature that results in a crinkly surface texture no matter how closely they might approximate a smooth surface in their gross trajectory. A nearly smooth triangulation always has non-equilateral triangles.

The diagram below shows a non-equilateral triangulation of the plane that is representative of what happens when a slightly curved surface is triangulated with triangles as nearly equilateral as possible. Topological irregularities inevitably occur, such as 5 or 7 triangles around a vertex (a topological charge of +1 or -1) in a sea of 6-triangle vertices (topological charge = 0), yet the surface remains nearly flat. This results in some noticeably non-equilateral triangles, especially near charged vertices.

Often a 5-triangle vertex will neighbor a 7-triangle vertex—a topological dipole so to speak. Two of the  triangle strips in the triangulation will pass directly through such a dipole, and both will show a distinct curvature, bending toward the +1 vertex (5 triangles) and away from the -1 vertex (7 triangles.) In the diagram below the constructions follow the path of a triangle strip that passes through a dipole.

Any way we wish to corrugate such a strip, the tendency will be for the weaver, when laid out flat, to curve in the same direction as the corrugated strip viewed in projection. Whether we wish to get involved with custom bending, or custom cutting and bending, any curvature in the before-folding, laid-out-flat conformation is an expensive problem.

A few degrees of freedom are available in attempting to avoid this before-folding curvature. The nominally 6-way crossings can be left somewhat open (as they are in conventional kagome) and the 3-way crossings can be adjusted in 3-dimensional space somewhere above their corresponding triangles.

A starting point is to place each 3-way crossing directly above the center of the triangles inscribed circle. In this arrangement the distance (as measured on each weaver) between a 3-way crossing and the valley fold below it can be made the same everywhere. Laying the weaver out flat causes the local waviness to nearly disappear, but the overall curvature remains.

Path of corrugated weaver through a non-equilateral triangulation. 

A corrugated weaver with its 3-way crossings overlying with the incenters (centers inscribed circles) of the triangulation.

Laying the corrugated weaver out nearly flat causes the waviness of the weaver's projected path to nearly disappear, but the gross curvature of the path remains.

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.

Improved 2-phase unit weaver for korgome

This 2-phase weaver does not need an enantiomorphic version to mate with. It can also be origami folded from a square.

2-phase corrugated kagome unit-weaver

This 2-phase unit weaver has less overlap than the one in the previous post. The baskets it makes are 4 plies thick.

Corrugated kagome polyphase unit weaver

The image shows a paper model of a 3-phase corrugated unit weaver. Identical units overlap shingle-like to form a weaver that is everywhere 3 plies thick. Since every underlying triangle in (non-unit) corrugated kagome is overlaid by 3 weavers each covering 2/3 of the triangle, such a basket is 2 plies thick. Realized with 3-phase unit weavers, the same basket will be 6 plies thick.

Higher order corrugation of kagome weavers.

The triangle-base pyramids that make up the surfaces of both all-in and in-out korgome baskets, can be "popped in and out" as many times as desired to give a surface with less altitude variation. Here, illustrated with a paper model, are the first and second order corrugations of a kagome weaver. The latter is shown in both all-in and in-out folding.

The simplest in-out korgome basket on the sphere with non-zero volume

There are only a few small bipartite cubic graphs. They all direct in-out korgome baskets. The smallest on the sphere is the cube graph. (I'm not sure about K3,3, the 6-vertex graph that looks like a laced shoe. It embeds on the torus.)

Corrugated kagome weaving (korgome)

These images are some experiments with corrugated kagome weaving (both all-in and in-out) in thin aluminum sheet.

Korgome: all-in vs. in-out

If korgome is configured all-in there is no need for the underlying triangulation to be chess-colorable.

The all-out korgome weave of a tetrahedron is simply a cube with the tetrahedron visible as diagonals of the faces.

The all-in korgome weave of an octahedron has zero volume.

As shown in the previous post, the in-out korgome weave of the two-faced triangle (the dual map of the theta graph on the sphere) has zero volume. The same is true of the dual map of the cyclo-butadiene graph on the sphere.

The simplest corrugated kagome (korgome) basket

The simplest bipartite cubic map is the theta graph embedded in the sphere, so its dual, a triangle embedded in the sphere, is the simplest chess-colorable triangulation. The kagome weaving of this two-sided triangle yields the simplest corrugated kagome basket (or korgome for short.) The weave pattern is strictly kagome, but the weavers can be full-width and cross at 90-degrees because alternate triangles are folded inward. The corrugation also adds stiffness.

Marked up and pre-creased weaver for the simplest corrugated kagome basket.

Starting the weaving.

A tuck-through move.

Taping up the one splice. The basket is still convex at this point.

The convex face of the basket after "popping in."

The convex face of the simplest korgome basket.

Kumihimo tetrahelix

I haven't tried this yet, but these kumihimo moves should give a kagome weave (sparse triaxial weave) of a tetrahelix. The moves for a corrugated plain weave should be the same.