Wednesday, December 31, 2014

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.

Monday, December 29, 2014

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 .

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

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.

Wednesday, December 24, 2014

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.)

Friday, December 19, 2014

The deltahedra inside voxel-based solids

There's a tetrahedron inside every cube (and in two possible orientations.) (Image quoted from )
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 )

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 )

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.

Friday, December 12, 2014

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 .
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 )

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 ) 
 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.