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  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.

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 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.

Thursday, December 11, 2014

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.

Monday, December 8, 2014

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)

Friday, December 5, 2014

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.

Thursday, December 4, 2014

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.