Mesh resources of knotology weaving

Mesh resources of knotology weaving: the basket's folds and joins must lie along the edges of a mesh having up to four kinds of faces.

Knotology can realize a basket having a surface mesh composed of these four kinds of tiles: an isosceles right triangle, the squares of its side and hypotenuse, and an equilateral triangle with a side length matching that of the larger square.

The actual weavers are as wide as the smaller square and run parallel/perpendicular to the sides of the smaller square—and thus at 45° to the sides of the larger square. There is a fabric opening (hole) in the center of the larger square. Inside each equilateral triangle three weavers meet around a hole to form a cube corner that can be made either prominent (a peak) or recessed (a dell.)

More economically, the equilateral triangle is not needed, as the isosceles right triangle can build the capped triangles directly.

Still more economically, the right triangle alone suffices to build the smaller squares by joining hypotenuse to hypotenuse, and to build the larger squares by joining four together, side to side.

The difference between voxel weave and universal weave of a square grid

The difference between the voxel weave and the universal weave of a square grid—it's all a matter of where folds and cuts may be made.

The difference between the voxel weave of a square grid and its universal weave is simply a matter of where folds and cuts are permitted.

In the universal weave, the edges of the underlying graph are presumed to trace all possible locations of folds and cuts, just as they would in a mesh. This is so even though the underlying graph contains only black vertices (i.e., it locates fabric holes of only one parity.) Holes of the other parity will be interpolated by medialization, but these interpolated vertices are off-limits to folding and cutting.

In voxel weaving the same grid, the underlying graph is of course the same, but we are now also given geometric data on the white vertices (i.e., fabric holes of both parities are already located in 3D space.) In effect, we get to start with the radial of the underlying graph, and folds and cuts are permitted only along its edges which exclusively connect holes differing parity.

The 45° rotation between the two square grids of allowable folds and cuts, causes the modulus to be √2 larger in the universal weave. The fabric itself is identical.

In a plane of voxel weave, at a location where four diagonal creases can meet to form a square, a tube of square cross-section in universal weave can be woven in. This kind of join in effect permits the smaller right triangle in the knotology weaver to be used to transition between the two allowable-folds meshes.

Knotology weavers

Pattern for a knotology weaver.

A knotology weaver can weave mixes of tabby weave, corrugated kagome, and voxel weaving. Remarkably, all these weaves are woven with 100% fabric coverage using the same pattern of potential folds (not all potential folds necessarily being used in any given passage.)

The pattern of creases in a knotology weaver (see image above) can be described as a right-angled triangle wave with "altitudes" dropped from each apex to the sides of the strip. These two categories of creases are termed diagonal (solid lines in the image above) and perpendicular (dashed lines.)

Voxel weaving only ever uses the perpendicular creases.

Tabby weave, when done according to the universal method (i.e., with a hole in the center of each square and weavers running at 45°) does not use the perpendicular creases at all; it only uses the diagonal creases when passing over an edge (e.g., one of the 12 edges of a cubical basket.)

Corrugated kagome always uses the perpendicular creases (at fold angles of ±90°) to form its peak or dell caps, and uses the diagonal creases (at varying fold angles) when passing between two triangles of the same salience (i.e., peak-to-peak or dell-to-dell) or when crossing over an edge (e.g., one of the 6 edges of a tetrahedral basket.) When passing between triangles of opposite salience the diagonal crease is not folded.

The squares in voxel weaving have sides equal to the weaver width; the squares woven in universal tabby weave have sides equal to the length of the diagonal creases—larger by a factor of √2. Corrugated kagome must meet voxel weaving essentially at a truncated cube corner. Corrugated kagome must meet universal method tabby weave essentially at an octahedron equator.

Kagome (mutsume ami), tabby weave (yotsume ami), and knotology

There are three regular tessellations of the plane. Image quoted from Grunbaum and Shephard, "Tilings and Patterns an Introduction."

There are three regular tessellations of the plane (i.e., by triangles, by squares, and by hexagons) but only two regular ways to weave (a consequence of the fact that the tessellations by triangles and by hexagons are dual, thus not truly distinct.)

Tabby weave (yotsume ami.)

The first-discovered weave, based on the tessellation by squares, called tabby weave or yotsume ami in Japanese, was in use by 27,000 BP.

Kagome weave (mutsume ami.)

The weave based on tessellation by triangles/hexagons, called kagome weave or mutsume ami in Japanese, was in use by 7,000 BP.

A kagome weaving of wooden strips found at Higashimyo, Japan dating to 7,000 BP. Image quoted from http://archaeology.jp/sites/2007/higashimyo.htm .

In the recent era (19th century to present) has come the realization that all such strictly over-and-under weaves can be mathematically flattened to two-dimensionality by describing the openings of the fabric rather than the paths of the weaving elements.

Flattening an over-and-under weave: the checkerboard diagram of an alternating knot. Image quoted from P. G. Tait, "Some elementary properties of closed plane curves." (1876.)

A close look at any fabric that is woven over-and-under will show that the fabric openings have a resemblance to an impossible staircase.

A Penrose stairs or Escher staircase.

Proceeding around the fabric opening (clockwise or counter-clockwise) produces an apparent progression in the vertical direction (up or down) that ultimately proves illusory. By associating the direction of procession (clockwise or counter-clockwise) with curled fingers and the direction of progress (up or down) with the extended thumb, it is possible to categorize every opening in the fabric as either left-handed or right-handed. Importantly, we will categorize the opening the same way whether we view it from the front or the back: weaving a non-orientable surface does not encounter any contradictions in the weaving.

If we narrow the weaving elements until the basket surface becomes "all holes," and color each hole according to its left- or right-handedness, gives us a description of a basket as a surface colored like a chessboard or checkerboard. A every corner four tiles, two of each color, meet and no two tiles of the same color are ever found side-by-side.

A checkerboard graph describing a kagome basket.

Every graph can be converted into a checkerboard graph through a procedure called medialization, so all we need to weave a basket is a graph drawn on a surface. (A visual short-cut is to imagine growing the black dots that represent the graph's vertices out along their adjacent edges until they meet at mid-edge—what you will have left is a black and white checkerboard graph.)

From this graph theory perspective on weaving comes the realization that tabby weave and kagome are essentially the same thing. These two types of weave, one Paleolithic and the other Neolithic, can intermix as freely as triangles and quadrangles in a tessellation.

Triangles and quadrangles can mix fairly freely in a tessellation. After medialization, the result is a checkerboard graph describing a hybrid tabby/kagome weave.

Now we can weave any surface essentially any way we want. But there are problems. The weavers (weaving elements) will generally not be straight, and, except in tabby weave passages, coverage of the surface will not be 100%.

The perpendicular and oblique creases in knotology weavers allow them to weave any combination of (flat) tabby weave and (corrugated) kagome. Image quoted from www.origami-guide.com

Enter Heinz Strobl in the 1990's with his knotology technique. Knotology uses identically creased, straight weavers to weave any combination of tabby weave and kagome. The kagome passages are corrugated (which contributes stiffness to the basket wall) and coverage is 100%. The one constraint: the quadrangles must be squares and the triangles must be equilateral.

Things to korgome

There are many mathematical objects that can be made by corrugated weaving.

Deltahedra and equilateral surfaces. Obviously, any surface that is entirely faced by equilateral triangles is ready to be korgomed. (Since korgome is a variety of the universal weaving method, if the faces are suitable in shape, there is no need to worry that the weaving will produce an odd number of crossings.)

Equilateralized triangle meshes. Making all the triangles equilateral severely alters the shape of a mesh (but not completely.)

The deltahedra inside voxel-based solids

There's a tetrahedron inside every cube (and in two possible orientations.) (Image quoted from  http://burrtools.sourceforge.net/gui-doc/Spacegrids.html )

There is a tetrahedron inside every cube or voxel: its edges are six face diagonals. There are two possible orientations for the tetrahedron as there are two consistent sets of face diagonals. If every unit-cube voxel of Cartesian coordinates (x, y, z) is assigned the parity of x+y+z, and tetrahedron orientations are assigned according to this parity, neighboring tetrahedra always meet edge to edge.

In this way a vox solid (voxel-based solid) can be converted into a deltahedron (a solid faceted exclusively with equilateral triangles) that lies entirely within the original vox solid.

A structural problem arises with this technique. For example, a rigid stack of voxels becomes a flexible chain of tetrahedra.

The tetrahedra that lie inside a rigid stack of cubes form a flexible chain. (Image quoted from http://www.kaleidocycles.de )

A solution to this structural problem is to divide each voxel into eight sub-voxels, and replace those sub-voxels by tetrahedra. Here again there is a parity choice, one that makes these eight tetrahedra either a stella octangula or a "cumulated cuboctahedron."

A "cumulated cuboctahedron" (shown in red) inserted between the two halves of a stella octangula (gray.) (Image quoted from http://the-arc-ddeden.blogspot.com )

If stella octangula voxels are used, the corners are prominent. Using cumulated cuboctahedra (which can be imagined by slicing a half-voxel margin off this object) results in bevelled corners.

Whether the stella octangula or "cumulated cuboctahedra" units are chosen, weaving their surface triangles in corrugated kagome (a.k.a., knotology) restores the missing cube corners of the sub-cubes—the woven piece resembles the original vox solid externally, but its internal structure is a little different.

Weaving surfaces based on tetrahedral-octahedral voxels

Assembly of the tetrahedral-octahedral honeycomb. (Image quoted from TED-43, Wikipedia)

I've missed the obvious. The easiest way to obtain an all-equilateral-triangles surface approximating a given shape is to voxelize the shape in a hybrid voxel system containing tetrahedral and octahedral voxels: the tetrahedral-octahedral honeycomb.

The resulting equilateral surface can be woven directly in kagome with straight weavers:

Any equilateral surface (deltahedron) can be woven in kagome with straight weavers.

Or, each triangle can be decorated with a cube corner (either inward or outward facing) and the result can be woven in korgome (corrugated kagome) with straight weavers and nearly 100% coverage:

Any equilateral surface (deltahedron) can be woven in korgome (corrugated kagome) with straight weavers.

Images of corrugated kagome from www.origami-guide.com .

The coarse weaving of the cube is identical to the korgome weaving of the tetrahedron.

Adding cube corners to the faces of a tetrahedron yields a cube. 

Cubic and TO (tetrahedral-octahedral) voxels use the same grid. We can make a TO surface by performing local surgery on a cubic vox solid, but that will not get us anywhere if a korgome weave just resurrects the coarse weave of the cubic vox solid.

A squares and equilateral triangles tiling. (Image quoted from http://www.microscopy.ethz.ch/tate.htm )

Tilings of squares and equilateral triangles can easily be converted to all-equilateral-triangles by adding (or subtracting) octahedron caps at the squares.

A complex tiling of squares and equilateral triangles. (Image quoted from http://www.microscopy.ethz.ch/tate.htm ) 

 When octahedron caps are side-by-side, a tetrahedron can be added between them to form a ridge.

A quasiperiodic tiling of squares and triangles. (Image quoted from Michael Baake, "A Guide to Mathematical Quasicrystals.")

The korgome weave of an octagon cap can have any combination of in and out cube corners.

Korgome weaving of the octahedral caps leaves open the possibility of whether the quadrants will be in or out, and whether the overall shape protrudes or recedes from the surface. All-out and protruding turns an isolated square into a little toadstool in a field of cube corners. (When squares are adjacent they cannot both have this toadstool form.) All-in and protruding looks like a fold-up traffic barrier. Rows of squares (these are evidently quite common in tantalum telluride quasicrystals) when filled in with tetrahedra, become korgome ridges or grooves.

Weaving tetrakis vox surfaces

Another way to deal with the lack of surface corrugation on some vox solids is to corrugate each exposed voxel . When the corrugation takes the form of a square pyramid added to each exposed voxel face, the adjective tetrakis, usually applied to polyhedra, can be applied to the vox surface.

A parameter is the height of the pyramid, or, what is easier to measure, the altitude/base ratio, a/b, of its isosceles triangle faces.

When a/b = 0.5, the pyramid has zero height: we have a cube that has been decorated with crossing diagonals. The universal weave of a cube naturally produces this shape with its four straight weavers crossing on every face.

When a/b = √5 / 4 = 0.5590, all the dihedral angles are equal and we have a tetrakis hexahedron. The weave is the same as the universal weave of the cube, but the weavers are a little crooked. This is an appropriate a/b for folding.

When a/b = √2 / 2 = 0.7071, the original cube edges momentarily disappear and we have a rhomic dodecahedron. This is the tallest pyramid we can use without encountering mechanical interference at sharp internal corners. Again, the weave is the same as the universal weave of the cube except that the weavers are crooked and, in this case, some folds are missing, leaving rhombic rather than triangular facets.

When a/b = √3 / 2 = 0.8660, the facets are all equilateral triangles. The vox solid cannot have any internal corners—assuming it does not, the resulting  tetrakis surface is a deltahedron.

All the weaves described here are coarse weaves: the universal weaves of these shapes would require a fabric opening in the center of each triangular facet.

All of these tetrakis vox surfaces are in theory completely foldable (the rhombic dodecahedron would need to permit folding along its short diagonals.)

Weaving non-cubic voxels

Down-sampling the cubic grid yields two alternative, non-cubic shapes for voxels: the rhombic dodecahedron (down-sampled 2:1) and the truncated octahedron (down-sampled 4:1.)

Cubic voxels are easy, but they have problems. From a structural point of view, their big problem is anisotropy: depending on the orientation of the cubic grid, cubic voxels may alternately produce surfaces that are smooth (and therefore un-corrugated and flexible) and or quite rough (and therefore highly corrugated and stiff.) Down-sampling the cubic grid in the two most symmetrical ways (FCC and BCC) leads to voxels in two new shapes: the rhombic dodecahedron and the truncated octahedron. These non-cubic voxels are more isotropic than cubes.

Weaving of a surface based on voxels that are truncated-octahedral is complicated by the fact that the faces are not congruent: there are both square and hexagonal faces.

This weaver is both crooked and of varying width—and yet it fails to fully cover the hexagonal faces of the truncated octahedron.

Weaving surfaces that are faceted with truncated octahedra is complicated by the fact that the faces are not congruent, there are both squares and hexagons. The worst consequence of this is that the order of faces a weaving element crosses may change as voxels are added to or subtracted from the vox solid.

Weaving of a rhombic dodecahedron is simplified by the fact that the faces are all the same shape.

Weavers for the rhombic dodecahedron are crooked but they have constant width.

Weaving is simpler when the voxels are rhombic dodecahedra. All the faces of a rhomic dodecahedron are the same shape, and, though the weavers are crooked, their alternation "left-diamond, right-diamond" does not change as voxels are added or subtracted from the vox solid.

Basket math basics

Weaving is a matter of holes. (Image quoted from Kenneth Snelson, US Patent  6,739,937.

"Watch the donut, not the hole," is an old refrain, but it seems basket weaving works the other way. Kenneth Snelson has shown that a basket surface can be represented by its arrangement of fabric openings (a.k.a, holes) which are always to be found surrounded by openings of the opposite helical handedness.

The general method to weave a polyhedron or a map (a topological generalization of polyhedron) is to locate a hole of one helical handedness at each vertex, and a hole of the opposite helical handedness at each vertex of the dual polyhedron or map. When these two kinds of vertices are linked by edges that correspond to vertex-face adjacencies, the result is a quad-faced map called the radial of the original map (or, what is the same, the radial of its dual.) The radial is bipartite by construction, that is, its vertices fall into two classes such that no edge connects two vertices of the same class.

The radial of a map is always a bipartite, quad-faced map, the two vertex classes being the old (original) vertices and the new (inserted) vertices. (Image quoted from Extra Ways to See: An Artist's Guide to Map Operations.)

(Due to commonly invoked restrictions on what may constitute a quadrangulation, every bipartite quadrangulation is a bipartite quad-faced map, but not every bipartite quad-faced map is a bipartite quadrangulation. For example, the quads in a quadrangulation are not usually allowed to be self-adjacent, though this possibility is essential to combinatorial topology and a necessity in weaving the smallest possible baskets.)

While the radial contains all the information needed to do the weaving, its dual, the medial, is easier to read because the edges of the medial align with the weaving elements. Just as the vertices of the radial can be two-colored to reveal its bipartite nature, the faces of the medial can be two-colored in the manner of a chessboard. The same sort of chess-colored map, known as the checkerboard graph, is used in knot theory to represent alternating links.

The medial of a triangle, shown with its chess-coloring. Any triangulation (or triangle-faced map) can be woven as a basket after its triangles have been decorated in this way.

The color boundaries in the chess-coloring (here accentuated with narrow white edges) trace the paths of the weaving elements. A triangle-faced map yields what is called a kagome weave, a square-faced map yields tabby weave—and all sorts of hybrid weaves naturally crop up.

A triangulation decorated with a chess-coloring of the medial in preparation for weaving a kagome basket.

Any surface can be woven via the medial of a map. Non-orientability causes no particular difficulty for basket weaving (other than the inevitable requirement that the woven surface be able to cross through itself.)

Maps that are themselves bipartite, quad-faced maps (or are dual to such) are special because they can be woven directly without need for the refining subdivision implicit in taking the radial or medial.

An important example of bipartite quad-faced maps are the boundary surfaces of vox solids (bodies composed of unions voxels, or unit cubes) when they inherit the corners vertices and edges of the cubic network, what I will call vox surfaces.

Vox surfaces are quad-faced and bipartite. What then are they the radial of? They are the radial of a redaction of the cubic network where each cube is left represented by just four of its eight corner vertices, and the edges are a selected diagonal on each face (a total of six face diagonals) in place of the normal twelve edges of a cube. What this makes of the cubic network is an octahedron/cuboctahedron network. The octahedron/cuboctahedron network is perhaps most easily imagined as the edges in an open, point-on-point, packing of octahedra. (There is one octahedron surrounding each of the vertices redacted from the cubic network.) Carving a box solid out of this network leaves a boundary surface decorated with a single diagonal edge from each exposed cube face. Taking the radial of this embedded graph reconstitutes a bipartite and quad-faced vox surface. 

The centers of the cubic cells in a cubic network align with a second cubic network (that is to say, the cubic network is self-dual.) It is possible to pigment the cubic cells of this dual network in two colors, say red and black, such that no two cubes of the same color are face-adjacent. Carving a vox solid out of this two-colored space leaves every face of every voxel colored like four squares of a chessboard. This is exactly the chess-colored medial graph we need to weave the vox surface in the coarsest possible weave.

The somewhat finer universal weave can be carved out of two-colored space in a similar way. Each vertex in a cubic array of voxels can be enveloped by an octahedral cell in an octahedron/cuboctahedron network. Coloring each octahedral cell black and each cuboctahedral cell red, a vox solid carved out of this space will wear a chess-colored medial graph describing the universal weave of the vox surface.

A packing of octahedra and cuboctahedra. (Image quoted from http://robertlovespi.wordpress.com)

Pop goes the voxel

A collapsible voxel: the triangles have an altitude/base ratio of 0.55.

The same voxel reassembled with its pyramids "pushed" inward.

The voxel partially collapsed.

The voxel fully collapsed.

A collapsible voxel like this can be woven using weavers that are just slightly crooked.

Voxel-based surfaces are completely foldable

The boundary surface (vox surface) of any vox solid (i.e., a voxel-based solid) is a quadrangulation. Any such quadrangulation is bipartite because the edges and vertices in a three-dimensional packing of cubes form a bipartite graph and our quadrangulation is merely a subgraph of this.

Since the quadrangulation is already bipartite, we can convert it into a tripartite triangulation by placing a vertex of a third color in the center of each quadrangular (actually square) face, and connecting an edge to each of the four surrounding vertices.

Once triangulated in this way, a voxel-based surface is completely foldable because its tripartite.

Basket maps and duality

Definition: a map that is 4-valent and chess-colorable is termed a basket map.

A basket map can unambiguously direct the weaving of a basket when it has been chess-colored in two colors named "left-handed helicity" and "right-handed helicity." In weaving the basket, the four-valent vertices are crossings, the edges are the weaving elements, the faces are fabric openings, and the question,"Does this go over or under?" is settled by reference to the specified handedness of the fabric openings. 

More concisely, a map weaving strategy is: place a right-handed fabric opening at every vertex and place a left-handed fabric opening in the center of every face—or vice versa. 

An attribute more commonly discussed in the graph theory literature than chess-colorability is the closely related attribute of bipartite-ness. The dual of a basket map is a bipartite quad-faced map (BQFM) and vice versa. Given a suitably inclusive definition of quadrangulation, BQFM is synonymous with bipartite quadrangulation.

A map and its dual are just the two "parts" of a bipartite quad-faced map. We can generate the BQFM by applying the map operation radial to either the map or its dual. The dual pair can be recovered from the BQFM by applying inverse radial—a map operation that only operates on BQFM's. The action of inverse radial is to collapse quad faces into edges, conserving only one color of vertices in the process. The two possible choices of conserved color generate the dual pair.

Recall that the dual of any BQFM is a basket map. So, when we look at an arbitrary map we are looking at half of the holes in a basket!

Voxel unit-weaving with singly-folded weavers

Expanding the 'missing corner' pattern a little bit reduces the number of folds.

By expanding slightly the 'missing corner' pattern it is possible to reduce the number of folds in each weaver from three to one. The angular rotation needed to accomplish this is 22.5 degrees (pi/8 radians.)

When weavers need to be bent before weaving, having a single fold greatly reduces the parts inventory.

The 'missing center' pattern has no such solution: it always requires triply-folded weavers:

The 'missing center' pattern always needs triply-folded weavers.

Model of a cube (a single voxel) loop-woven with singly-bent loops.

There are only three bend states for singly-bent loop weavers: bent convex, bent concave, and unbent. Manufactured loops of those three types suffice for weaving any voxellated surface.

Unit-weaving voxel surfaces

A unit-weaving pattern for the enveloping surface of a union of voxels. The weavers are four squares long. The black squares are holes in the fabric. The dashed lines are voxel boundaries. (Alternatively, the corners of the voxels could be centered on the holes.) In either case, if the weave pattern is made up of unit squares, the modulus of the votels is the square root of 5. 

The over-and-under of weaving can play tricks on the imagination—for example, when weaving a non-orientable surface like a Klein bottle. The reliable way to picture an over-and-under weave is as a chess-coloring of a four-valent map. (A map is a generalization of a polyhedron we are no longer concerned with measurements of angle and distance, i.e., geometry, but only with the way the polygonal faces are mated with each other and the way the surface connects to itself; i.e., we are only concerned with topology.)

Starting with any map we can convert it to a chess-colorable, 4-valent map via the map operation medial. In this way we can discover a way to weave any tessellation of any surface, orientable or non-orientable. That's the universal method.

For special maps there may be alternative ways to do the weaving. For example, if the original map is already 4-valent and chess-colorable there is no need to take its medial, the original map is directly weavable. The dual of a 4-valent, chess-colorable map is a quad-faced, bipartite map. Thus, if we are given a quad-faced, bipartite map we do not need to take its medial—we can simply take its dual and weave that. As an example of the difference the choice makes, consider the map of a cube: it has 6 quad faces and its vertices can be colored black and white in such a way that no two adjacent vertices have the same color (in other words, the map of a cube is bipartite.) If we weave the map of a cube using the universal method we will end up with one weave crossing for every edge in the original map: a total of 12 weave crossings. If we use the special method (i.e., we directly weave the map's dual) we will end up with a crossing for each face in the original map: a total of 6 crossings. The special method yields a coarser and simpler way to weave the same surface.

A commonly encountered surface that is quad-faced and bipartite is the envelope of any union of voxels. Clearly such a surface is quad-faced. It is also bipartite because the entire 3-D, Cartesian mesh of unit squares is can be vertex 2-colored such that no two adjacent vertices have the same color. When we sculpt a union of voxels out of this space, the vertices can keep their colors. Left in view is a surface that is clearly bipartite.

Alternate ways to unit-weave a voxel whose modulus is the square root of five times the weaver width.

Weaving strips of aluminum flashing into a Penrose tiling

Straight strips of aluminum flashing folded and woven into a portion of a Penrose tiling.

The weaving above was difficult to do because the folds were not bent to the precise angle and the springy strips resist being bent in the weaving. The strips had to be trimmed from a nominal 1.5" width down to about 1.38" to avoid interferences.

Weaving a Penrose tiling with straight weavers: the concave face.