Twofold orthogonal weavings on cuboids

Abstract

Some closed polyhedral surfaces can be completely covered by two-way, twofold (rectangular) weaving of strands of constant width. In this paper, a construction for producing all possible geometries for such weavable cuboids is proposed: a theorem on spherical octahedra is proven first that all further theory is based on. The construction method of weavable cuboids itself relies on successive truncations of an initial tetrahedron and is also extended for cases of degenerate (unbounded) polyhedra. Arguments are mainly based on the plane geometry of the development of the respective polyhedra, in connection with some of three-dimensional projective properties of the same.

1. Introduction

Rectangular twofold, two-way weavings [1] on cubes [2] raised interest in investigation of topology of weavings on a broader set of polyhedral surfaces. Terms used in the introductory sentence are explained as follows: a weaving pattern of straight strands over a flat surface is called twofold if almost all points of the surface are covered exactly by two strands; two-way means that strands can only be oriented in any of two fixed directions along the surface; rectangular corresponds to the special choice of such directions being orthogonal to each other. This can also be generalized to polyhedral surfaces, which are composed of some flat woven regions (see also figure 1 for illustration). Research activity was also encouraged by some applications of weavings applied in architecture (the most famous example is the Beijing Olympic Stadium) as well as in manufacturing of fibre coverings [3]. In parallel, there has been persisting a purely mathematical interest on woven surfaces that dates back to [4,1], focusing primarily on the study of symmetric objects. Later on, a graph-theoretical approach also appeared: octahedrite as well as i-hedrite graphs described in [5] were proven to be dual of tilings in polyhedral weaving patterns [6]. Specific research was done on the topology of strands in weavings on a cube [7,8], where several weaving patterns (for a complete double covering) exist owing to symmetry.

Figure 1.

Weavings on closed eight-vertex polyhedra: overlapping strands always meet at an angle of π/2 (‘two-way rectangular’) and produce a double cover (‘twofold’). Three-dimensional image and development of a woven cube (a,d), a cuboid with less symmetry (b,e) and a square antiprism (c,f). Complete angles at each vertex shown by circular arcs must be 3π/2 as a necessary condition of existence of such weavings. Note that a realizable development combinatorially equal to that of a cube typically does not fold into a cuboid as illustrated by the 10-faced polyhedron in part (c); ‘unintended’ new folds are drawn in thick lines.

This paper aims to extend investigations to a class of objects with less symmetry or no symmetry at all: to cuboids that are combinatorially (but not geometrically) equivalent to the cube. There is a double problem to be solved: in order that a cuboid can be completely covered by a rectangular twofold, two-way weaving, it must satisfy two conditions of fairly different kinds: (i) the sum of angles that meet at each polyhedral vertex (called also the complete angle at that vertex [9, section 1.5.2]) must have a special value, because otherwise the neighbourhood of vertices could not be covered without non-rectangular crossings and (ii) the proportion of some distances must be rational which makes it possible that some closed strands of equal width together cover the entire surface exactly twice. First of all, an exact condition is determined under which a complete (rectangular) double covering exists for cuboids; then a construction to derive all cuboids that can be woven is also proposed. It will also be shown that if condition (i) is satisfied by a cuboid, then there exist other weavable ones within its arbitrarily small neighbourhood.

In order to simplify the forthcoming discussion, let some basic concepts be defined and relationships proved.

Definition 1.1 —

If the complete angle v at a vertex is a multiple of π/2, then the vertex is said to be v-rectangular.

Definition 1.2 —

A polyhedron with all vertices v-rectangular is also said to be v-rectangular.

Because any strand in the weaving pattern discussed here can be unfolded into a plane as a long rectangular strip, many of the forthcoming arguments will also be based on a planar representation of cuboids. A connected planar set of face polygons of a cuboid C together with rules of gluing of their edges is called a net or a development of that polyhedron [10] and is denoted by D(C) henceforth. Note that a development can also be defined in a more abstract way [9] as a unique set of (disjoint) face polygons and gluing rules but can be represented by several realizations of nets; however, that distinction between net and development is not considered in this paper.

Proposition 1.3 —

A polyhedron C can be woven (i.e. covered by a complete twofold, two-way weaving) by orthogonal strands of unit width if and only if (i) there exists a rectangular coordinate system and a development D(C) of the polyhedron such that all vertex points in D(C) have integer coordinates and (ii) C is v-rectangular.

Proof. —

If there exists such a weaving on C, then all vertices coincide with intersections of perpendicular boundaries of parallel strands. Let a two-dimensional Cartesian coordinate system be fixed to one of the faces of C such that their axes are parallel to one of the two directions of strands and the unit is set to be equal to the strand width. Consider D(C) in this two-dimensional coordinate system. It is clear that the image of any strand in D(C) is straight, and they can only intersect at an angle of π/2; therefore, all corners of individual unit square tiles of strand crossings (including those at vertices of C) are of integer coordinates. The proof in reverse order proceeds as follows: let a development D(C) of a v-rectangular C with integer coordinates of edge endpoints be given. Assume that D(C) is a single planar polygon, i.e. a polygon with a single closed boundary that can even be overlapping; a development composed of disjoint polygons can always be glued together to form such a single polygonal development. In this case, the graph of edges cut in C (those forming the boundary of D(C)) is a tree, so it is always possible to get from one side of an edge e to another on the surface of C without crossing the cutting line. The relative rotation of two edges to be glued together in D(C) (corresponding to edge e in C) is the signed sum of angles visited on the surface of C while getting along the cut from one side of e to another. This angle is obviously a multiple of π/2 because of the v-rectangular vertices, and thus positions of two images of e in D(C) relative to an unoriented rectangular grid are congruent: any strand has its continuation on the adjacent face of C. ▪

We note here that definition 1.2 could also be given as the angular defect or (curvature) at any vertex is a multiple of π/2. For any n-vertex polyhedron homeomorphic to the sphere, the sum of complete angles is s_n=(n−2)2π. For eight-vertex convex polyhedra like a cuboid, s₈=24π/2, so they can only be v-rectangular if complete angles at all vertices are equal: v_i=3π/2, i=1,…,8. In order to produce weavable polyhedra, it looks convenient to design a development according to prescribed values v, and if one succeeds in associating all edge endpoints with integer Cartesian coordinates, the job seems to be done. A major problem is, however, that a folding of a development into a polyhedron (topologically equivalent to the sphere) is not always realizable as a convex polyhedron. Realizability of polyhedra is a topic extensively studied in many works [11–13], but those studies mainly deal with realization of projections rather than with developments. The problem of realization of a development is addressed in [9,14]. Even if it is known from [9] that any intrinsically convex polyhedron (i.e. which has only vertices with v<2π) has exactly one convex realization, unfortunately, that unique realization does not always reflect the designed topology of the edge network in a development. For example, think of the development of a straight (metric) square prism, then distort four side rectangles around the bottom square uniformly such that they form a series of parallelograms: the new development will not fold into a hexahedron but a square antiprism as shown in figure 1c,f.

Thus, to be sure about the topology of the edge network in a development, let us try to produce the desired polyhedron rather by a ‘three-dimensional reasoning’: we will depart from an appropriate tetrahedron and truncate it into a cuboid that is still weavable. The proposed truncation process is illustrated in figure 2. The particular difficulty of our realization problem is that, unlike almost all realization problems in literature, there is a condition (for the cuboid to be v-rectangular) related to the complete angles, which is not projective invariant. This fact makes doubtful the success of a projective description, but elementary geometric considerations in two-dimensions of a problem that is essentially three-dimensional may also get fairly complicated. In the following discussion, a key property of v-rectangular cuboids is proved in a three-dimensional approach using spherical geometry (§2), whereas the truncation process is described with an elementary two-dimensional reasoning afterwards in §3.

Figure 2.

Derivation of cuboids via two successive truncations: initial tetrahedron with two v-rectangular vertices S and T (a), polyhedron resulting from the first (b) and the second truncation (c). Small triangles denote complete angles 2π/3 at the corresponding vertices. Complete angles at the last two (N and N*) will be proved to be equal in §2; geometric details of truncations will be given in §3, supported by figure 6b.

2. Spherical octahedra with equiareal triangles

This section contains the proof for a statement that will be found crucial in the last truncation of the entire truncation process of producing weavable cuboids (shown in figure 2): if a cuboid has six v-rectangular vertices incident to two faces sharing a common edge such that each of them has a curvature of π/2 (or a complete angle of 3π/2; see six marked vertices of figure 2c), then the complete angle is also 3π/2 for both remaining vertices (that are necessarily connected by an edge of the polyhedron; see vertices N and N* of the same figure). This statement follows from another one formulated in spherical trigonometry that will be proved in the next paragraphs. Note that discussion of this §2 uses its own notation: matching symbols in other chapters may have different meaning.

Any cuboid uniquely defines its spherical image [9] which is constructed as follows: consider unit normal vectors of all faces defining six points on the unit sphere, then connect any pair of those points if the corresponding faces of the cuboid are adjacent. The resultant object is a spherical graph defining an octahedron. Because the plane containing an octahedral arc is perpendicular to the corresponding edge of the polyhedron, it is easy to see that angles α,β,γ in a spherical triangle correspond to angles π−α,π−β,π−γ around a polyhedral vertex, hence the spherical excess ε of a spherical triangle corresponds exactly to the curvature of the polyhedral vertex in case (see figure 3a for illustration). However, spherical excess is also the measure of triangular area on the unit sphere. In this view, one can formulate.

Figure 3.

Spherical image of a cuboid. (a) Three faces sharing a vertex with a sphere tangent to them: dashed lines are perpendicular to edges, so a complete angle 3π/2 at the vertex is equivalent to the sum 3π−3π/2 of angles of the spherical triangle A12. (b) A three-dimensional view of a spherical octahedron with unit radius and equal triangular areas of π/2; vertices A and B are not connected, vertices 1,2,3 and 4 follow each other anticlockwise viewed from point A. (c) Construction of Lexell circles (dark): locii of vertex 2 for which, e.g. area A_A12 is constant, are on a circle passing through antipodals of A and 1 (from the two intersections of circles, the one with position vector $-{\mathbf{n}}_1$ cannot be vertex 2, because this point causes both triangles A12 and 1B2 to degenerate into lunes, which are impossible to appear as spherical images of convex polyhedral vertices). (d) Schematic of finding new vertices with pairs of triangles with given areas on the sphere: A,1,B are fixed arbitrarily, vertices 2 and 3 are found successively in ‘positive’ direction, whereas 4 and 3 are obtained in the opposite sense. Because the formal cyclic order of vectors (AiB,i=1,2,3,4) is constant in formulae, negative sense is introduced by taking the negative of $T=\tan \pi /2$ (it is shown by ±T in the figure). The proof proceeds by showing that vertices 3_p and 3_n coincide.

Proposition 2.1 —

If six spherical triangles in a spherical octahedron on the unit sphere have an area of π/2 each and the remaining two triangles share an edge (arc), then both of them are also of area of π/2.

Proof. —

First, it should be noted that any such octahedron with six equiareal triangles can be determined by three vertices: let A and B be fixed somewhere on the sphere as two octahedral vertices not connected by any edge, and let also vertex 1 be fixed such that A1 and 1B are both octahedral edges. Finally, let three missing vertices be labelled by 2,3,4 successively such that they follow each other in a positive (anticlockwise) sense viewed from point A (figure 3b). If edge A1 and area A_A12 of a spherical triangle A12 is given, then vertex 2 of the triangle lies on a one-dimensional manifold. It can be shown that it is a circle passing through antipodal points of A and 1; such a circle is also known as a Lexell circle [15, p. 114]. Given areas A_A12 and A_B21 locate therefore vertex 2 at the intersection of two Lexell circles (figure 3c). Note that vertex 2 is uniquely determined as far as cyclic order of subscripting of areas is considered: A_IJK means the area of the triangle where I,J,K follow each other in positive sense; another (concave) triangle with the same vertices has therefore an area A_KJI=4π−A_IJK.

The proof proceeds by showing that the position of vertex 3_p (defined by vertices A,2,B where position of vertex 2 has previously been defined by vertices A,1,B) coincides with 3_n obtained through similar triples of vertices A,1,B and A,4,B in reverse order (figure 3d). Some important ideas of the proof are given below; a detailed proof is available (electronic supplementary material: detailed_proofs.pdf) in an electronic format.

The area A_IJK and cosines of bounding arcs i,j,k are related as follows [16, p. 103]:

$$\cos \frac{1}{2}{A}_{IJK}=\frac{{\cos }^2(i/2)+{\cos }^2(\hspace{0.17em}j/2)+{\cos }^2(k/2)-1}{2\cos (i/2)\cos (\hspace{0.17em}j/2)\cos (k/2)k}.$$

With $\cos (I,J)={\mathbf{n}}_I\cdot {\mathbf{n}}_J$ ($|{\mathbf{n}}_I|=|{\mathbf{n}}_J|=1$) for any arcs IJ, trigonometric identities yield

$$\tan \frac{1}{2}{A}_{IJK}=\frac{({\mathbf{n}}_I{\mathbf{n}}_J{\mathbf{n}}_K)}{1+{\mathbf{n}}_I\cdot {\mathbf{n}}_J+{\mathbf{n}}_J\cdot {\mathbf{n}}_K+{\mathbf{n}}_K\cdot {\mathbf{n}}_I},$$

also abbreviated as T_IJK. Further transformations with (I,J,K)=(A,1,2) lead to the form

$$(({\mathbf{n}}_A\times {\mathbf{n}}_1)-T_{A12}({\mathbf{n}}_A+{\mathbf{n}}_1))\cdot {\mathbf{n}}_2=T_{A12}({\mathbf{n}}_A+{\mathbf{n}}_1)\cdot {\mathbf{n}}_1,$$

which is the equation of a circle as the locii of vertex 2 induced by vertices A and 1 indeed. Applying similar arguments in triangle 1B2, intersection (different from $-{\mathbf{n}}_1$) of two Lexell circles can be calculated from a vector–vector function in a form ${\mathbf{n}}_2=\mathbf{f}({\mathbf{n}}_A,{\mathbf{n}}_1,{\mathbf{n}}_B,T_{A12},T_{1B2})=\mathbf{f}({\mathbf{n}}_1,T_{A12},T_{1B2})$, as ${\mathbf{n}}_A$, ${\mathbf{n}}_B$ are fixed; and it remains to show that

$${\mathbf{n}}_3=\mathbf{f}(\mathbf{f}({\mathbf{n}}_1,T_{A12},T_{1B2}),T_{A23},T_{2B3})=\mathbf{f}(\mathbf{f}({\mathbf{n}}_1,T_{A14},T_{1B4}),T_{A43},T_{4B3}).$$

This (quite general) relationship can be specialized by considering T_A12=T_1B2=T_A23=T_2B3=1 and T_A14=T_1B4=T_A43=T_4B3=−1 (note the ‘clockwise subscripting’ of the latter four terms). The solution itself is rather technical and ends up with comparing two polynomials in four scalar variables on up to the fifth, 12th, 13rd and 14th powers, respectively. The check for equivalence of polynomials has been done using symbolic mathematics software MAXIMA (Maxima – a computer algebra system. http://maxima.sourceforge.net.). ▪

Two comments on proposition 2.1: (i) the proof given here is far from being elegant but reflects the strong nonlinear behaviour of spherical areas; nonetheless, the existence of a much smarter proof is conjectured. (ii) Numeric experiments support the conjecture that similar relationship holds for four pairs (Aij and Bji) of equiareal triangles when areas in pairs are ε,π−ε,ε,π−ε, respectively. Moreover, there is also numeric evidence about that if there are 2k−1 pairs of equiareal triangles with areas pairwise ε,2π/k−ε,ε,2π/k−ε,…, then the last pair is also equiareal; yet these conjectures are not proved.

3. From tetrahedron to cuboid

This section reviews the entire truncation procedure from an appropriately chosen initial tetrahedron to the final polyhedron as already illustrated in figure 2. The procedure is described as a series of transformations made on a two-dimensional development of the initial tetrahedron. In the first approach, only generic configurations are addressed, which means that the tetrahedron chosen for further truncation is assumed to be bounded: proposition 3.1 provides a uniform construction of developments of any such tetrahedra; proposition 3.2 proves the existence and uniqueness (up to translation) of each truncation and it also shows that a truncation introduces no new vertices with irrational coordinates in the modified development. After some conclusions having been drawn, the analysis is extended to truncation processes in different cases of degeneracy (interpreted as unboundedness owing to three parallel edges or two parallel faces) of that initial tetrahedron.

(a). Bounded initial tetrahedra

Let the term ‘appropriately chosen initial tetrahedron’ be understood as a special tetrahedron H containing two vertices with complete angles of 3π/2 each (for brevity, ‘initial tetrahedron’ henceforth). Any two vertices of a tetrahedron are connected by an edge, so let our two vertices be the endpoints of an edge ST, whereas two other vertices are P and V . Let us develop H in a plane (D(H)) as shown in figure 4a (see the grey shape delimited by a thick border only). Complete angles v=3π/2 are ensured by $\mathrm{\angle }{T}^{\mathrm{\prime }}ST=\pi /2$ at S and T′P′∥TP at T. Without the loss of generality, a≥b is declared. First, we prove the following.

Figure 4.

Development D(H) of a tetrahedron H. (a) Net with solid border of H having two vertices (S,T) of v=3π/2 mapped to the corners of the inner square; V ^′′ and V ^′′′ mark unfolded and completely folded configurations of quadrangle V SPT; (b) edge vectors of the development of trihedral vertex P (r and u are directed to edge midpoints for convenience but any length are arbitrary except for $\mid \mathbf{\text{r}}\mid =\mid \mathbf{\text{u}}\mid$): r^T=(−a,b), s^T=(−2a+c,d), t^T=(−2a+d,−c) and u^T=(−a,−b).

Proposition 3.1 —

Any initial tetrahedron H can be developed such that two centrally symmetric instances of the net of D(H) corresponds to a rhombus with a concentric square gap, determined by four scalar parameters (i.e. two diagonals of the rhombus and two coordinates of one of the vertices of the square in a Cartesian system spanned by the diagonals of the rhombus).

Proof. —

Let points S and T be of complete angles of 3π/2 each. Once a line T′ST in D(H) has been fixed (with the origin positioned to the midpoint of TT′), triangle STP will locate point P of the same development to the right of line ST, making in this way the axis x of the coordinate system fixed towards P. Let P′ be the antipodal point of P in D(H) and therefore PT∥P′T′ is a sufficient condition for v_T=3π/2. Vertex V can then be located freely only on the axis y by the parameter b in order that H remains realizable in three dimensions (i.e. $\overline{PV}=\overline{P^{\mathrm{\prime }}V}$), and the pattern of the figure for the development is complete. ▪

It must be observed, however, that such developments based on a rhombus of axes 4a and 4b, containing a square gap (concentric with the rhombus) determined by parameters c,d are not always realizable. A necessary condition for realizability can be found as follows. Let us develop tetrahedron STPV again, now in a way that edge ST is not cut. Imagine that triangle SV T′ is rotated by +π/2 about S (shown in dotted lines, the new position of V is V ^′′): one condition for realizability is now that distance PV ^′′ must not be smaller than PV . It can easily be shown that coordinates of $\overrightarrow{V^{\mathrm{\prime }\mathrm{\prime }}P}$ are (2a+2b−c−d,c−d), so the condition above is written as

$${(2a+2b-c-d)}^2+{(c-d)}^2>{(2a)}^2+{(2b)}^2,$$

that is

$$b>\frac{2a(c+d)-c^2-d^2}{2(2a-c-d)}$$ 3.1

noting that the denominator equals ${\mathbf{s}}^{\mathrm{T}}\mathbf{t}/a$ and is strictly positive. The other condition concerns the configuration when V ^′′ is completely folded into V ^′′′ (V ^′′′ also corresponds to the mirror image of V to line SS′): here, the distance PV ^′′′ must be smaller than PV . It can be shown that this condition is always satisfied as far as parameters b,c both remain positive.

(b). Truncating a trihedron

Here, we will present a method for cutting off the neighbourhood of a trivalent vertex (‘trihedron’) such that v=2π/3 for at least two out of the three new vertices. A point R is chosen on one of the three edges beforehand to be incident to the cutting plane; the truncation is done such that another two new vertices will be v-rectangular. From now on, the term R-truncation refers to this special kind of cutting. This is the key procedure in deriving cuboids from tetrahedra: two such cutting planes that intersect inside the tetrahedron produce a cuboid. Two different truncation modes can be distinguished, depending on whether the neighbourhood of a vertex can be cut by a plane through an internal point R of an edge adjacent to the vertex (termed internal truncation from now on) or it can be done by a plane through a point R on the prolonged edge beyond the vertex (external truncation). Out of the two cases, only the internal R-truncation is discussed in details, because a generic (bounded) initial tetrahedron shown in figure 4 can always be transformed into a cuboid by two successive internal R-truncations (of vertex P and V , see also comments of §4).

Two key features of R-truncation will be shown here: the first one is some kind of uniqueness, namely if a trihedron with a complete angle smaller than π is R-truncated by a plane through a point R on a given trihedral edge, then the plane of truncation is unique. Note that (i) if R is moved along the edge, the normal direction of the plane of truncation is preserved, (ii) obviously, that edge containing R can be chosen in three different ways out of the total of three. The second feature concerns the rational character of R-truncation: if the development of a trihedral vertex can be characterized by rational coordinates (including all slopes of edges and a given point R on one of them which will be incident to the plane of truncation), then all vertices obtained by the truncation have also rational coordinates. A proof will therefore be given for the following.

Proposition 3.2 —

Internal R-truncation of a trihedron C for which D(C) can be given with edge direction vectors of rational coordinates (and the truncating plane is set through an edge point R also with rational coordinates) results in (at least) two new v-rectangular vertices of rational coordinates as well (an R-truncation through R is unique).

Proof. —

Consider a development of a trihedron, similar to that of figure 4b, as shown in figure 5a. Let a point R be fixed on the edge where the trihedron (shown in three dimensions in figure 5b) is cut before unfolding: it makes both points R and U fixed in the development. We look for the two-dimensional trace of truncation (‘truncation polygon’; figure 5c) RS_αQT_βU with right angles at S_α and T_β, as well as with $\mid R{S}_{\alpha }\mid =\mid S_{\alpha }Q\mid$, $\mid Q{T}_{\beta }\mid =\mid T_{\beta }U\mid$. It is to be shown that vectors $\overrightarrow{P{S}_{\alpha }}=\alpha \mathbf{s},\overrightarrow{P{T}_{\beta }}=\beta \mathbf{t}$ are unique and values α,β are rational. The complete proof (electronic supplementary material: detailed_proofs.pdf) is based on elementary vector algebra, so only main ideas are presented here again. From the equality of two expressions of vector $\mathbf{q}$,

$$\alpha \mathbf{s}+\mathbf{k}\times (\alpha \mathbf{s}-\mathbf{r})=\beta \mathbf{t}-\mathbf{k}\times (\beta \mathbf{t}-\mathbf{u}),$$

where $\mathbf{k}$ is the unit vector perpendicular to the plane of the development, α can be eliminated by means of a cross product to express β and vice versa, giving final expressions

$$\begin{array}{rl}\alpha \mathbf{s}& =\frac{\mathbf{t}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{t}\cdot \mathbf{s}}\mathbf{s}+\mathbf{k}\times (\frac{\mathbf{r}+\mathbf{u}}{2}-\frac{\mathbf{s}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{s}\cdot \mathbf{t}}\mathbf{t}),\end{array}$$ 3.2

$$\begin{array}{rl}\beta \mathbf{t}& =\frac{\mathbf{s}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{s}\cdot \mathbf{t}}\mathbf{t}-\mathbf{k}\times (\frac{\mathbf{r}+\mathbf{u}}{2}-\frac{\mathbf{t}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{t}\cdot \mathbf{s}}\mathbf{s})\end{array}$$ 3.3

$$\begin{array}{rl}\text{and}\mathbf{q}& =\frac{\mathbf{t}\cdot (\mathbf{r}+\mathbf{u})}{2t\cdot \mathbf{s}}\mathbf{s}+\frac{\mathbf{s}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{s}\cdot \mathbf{t}}\mathbf{t}-\frac{\mathbf{r}+\mathbf{u}}{2}+\mathbf{k}\times (\frac{\mathbf{t}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{t}\cdot \mathbf{s}}\mathbf{s}-\frac{\mathbf{s}\cdot (\mathbf{r}+\mathbf{u})}{2\mathbf{s}\cdot \mathbf{t}}\mathbf{t}+\frac{\mathbf{u}-\mathbf{r}}{2}).\end{array}$$ 3.4

The structure of equations (3.2), (3.3) and (3.4) shows both the uniqueness and rational coordinates of position vectors of three new vertices in the development. ▪

Figure 5.

Internal R-truncation of a trivalent vertex P: it is aimed at finding the position of vertex Q (vector q), as well as scalar factors α,β such that vectors αs,βt point to vertices S_α,T_β in the development of the truncated trihedron. (a) Trihedron (trivalent vertex) to be truncated by a plane through a fixed point R≡U; (b) three-dimensional sketch of the R-truncated vertex (the tetrahedron PRS_αT_β is removed), two small triangles at S_α and T_β denote complete angles of v=3π/2; (c) development of the R-truncated trihedron ($R{S}_{\alpha }=S_{\alpha }Q,Q{T}_{\beta }=T_{\beta }R,\mid \mathbf{\text{r}}\mid =\mid \mathbf{\text{u}}\mid$), vectors s and t are edge vectors of arbitrary length.

These expressions can be specialized to the geometry of figure 4b where the R-truncation of the neighbourhood of P is started from a point R half-way between P and V for reasons to be explained later (see the comment after definition 3.3). Note also that $\mathbf{s}\cdot \mathbf{t}=2a(2a-c-d)$:

$$\begin{array}{rl}\alpha \mathbf{s}& =\frac{1}{2}\frac{2a+c-d}{2a-c-d}\left[\begin{array}{c}-2a+c\\ d\end{array}\right],\beta \mathbf{t}=\frac{1}{2}\frac{2a-c+d}{2a-c-d}\left[\begin{array}{c}-2a+d\\ -c\end{array}\right],\\ \overrightarrow{O{S}_{\alpha }}& =\left[\begin{array}{c}2a\\ 0\end{array}\right]+\alpha \mathbf{s}=\left[\begin{array}{c}a\\ 0\end{array}\right]+\frac{1/2}{2a-c-d}\left[\begin{array}{c}-c(2a-c+d)\\ d(2a+c-d)\end{array}\right],\end{array}$$ 3.5

$$\begin{array}{rl}\overrightarrow{O{T}_{\beta }}& =\left[\begin{array}{c}2a\\ 0\end{array}\right]+\beta \mathbf{t}=\left[\begin{array}{c}a\\ 0\end{array}\right]+\frac{1/2}{2a-c-d}\left[\begin{array}{c}-d(2a+c-d)\\ -c(2a-c+d)\end{array}\right]\end{array}$$ 3.6

$$\begin{array}{rl}\text{and}\overrightarrow{OQ}& =\left[\begin{array}{c}2a\\ 0\end{array}\right]+\mathbf{q}=\left[\begin{array}{c}a\\ 0\end{array}\right]+\left[\begin{array}{c}b\\ 0\end{array}\right]+\frac{1/2}{2a-c-d}\left[\begin{array}{c}-4cd-(2a-c-d)(c+d)\\ (2a-c-d)(c-d)\end{array}\right].\end{array}$$ 3.7

Now, let us apply the truncation procedure to the neighbourhood of vertex V . For this purpose, let us simply rotate the rhombus of figure 4a by π/2 clockwise, apply the transformation $a\leftrightarrow b$ and then cross-multiply the resultant vectors by $\mathbf{k}$ from the left to return to the original coordinate system. Using superscript ‘*’ for distinction, new coordinates are obtained as follows:

$${\alpha }^{\ast }{\mathbf{s}}^{\ast }=\frac{1}{2}\frac{2b+c-d}{2b-c-d}\left[\begin{array}{c}-d\\ -2b+c\end{array}\right]\mathrm{and}{\beta }^{\ast }{\mathbf{t}}^{\ast }=\frac{1}{2}\frac{2b-c+d}{2b-c-d}\left[\begin{array}{c}c\\ -2b+d\end{array}\right],$$ 3.8

etc.

Definition 3.3 —

A polyhedron resulting from two successive R-truncations of H (such that it has six vertices with v=3π/2 on two adjacent quadrangular faces) is called a 2R-truncated tetrahedron (H_2R).

The network of a 2R-truncated tetrahedron according to the above vectors is drawn in figure 6a: combinatorially, it is not equivalent to the cube, because it has only seven vertices and 11 edges. Recall that the construction method makes sure that v_i=3π/2 (i=1…6) for all six vertices except for R; however, it implies that v_R=π, and this polyhedron may therefore be weavable. Let now the two truncation polygons be rescaled by positive factors φ and φ* (φφ*>1) as shown in figure 6b. Within a certain range of scaling, the new truncation scheme will lead to a hexahedral development with two vertices (N,N*) instead of R. Although v_N+v_N*=3π is seen immediately for the new vertices, proposition 2.1 provides v_N=3π/2, v_N*=3π/2 separately. Keeping in mind that a point of intersection of two straight line segments of all endpoints of rational coordinates must also have rational coordinates (points N and N* of figure 6b), one can state the following.

Figure 6.

Developments D(H_2R) (grey) of 2R-truncated tetrahedra H_2R. Development of a six-faced polyhedron with seven vertices (a); development of a cuboid (b). Complete angles are all divisible by π/2 in case (a) but it is only justified by proposition 2.1 for complete angles at N and N* each as shown in (b).

Theorem 3.4 —

For any bounded tetrahedron realizable in three dimensions which (i) has two vertices of complete angle of 3π/2, and (ii) all vertices in its plane development have rational coordinates in at least one Cartesian coordinate system, there exist infinitely many sets of two R-truncations of the tetrahedron producing a cuboid that can be completely covered by a twofold, two-way weaving of strands of unit width.

Proof. —

By proposition 3.1, it is seen that any tetrahedron that meets conditions (i)–(ii) can systematically be developed into the plane (and such a development can be identified by four scalars a,b,c,d) according to the scheme of figure 4 in order to perform R-truncation procedures in two dimensions. Proposition 3.2 makes sure that if a double internal R-truncation results in a cuboid, then it is possible to set truncating planes (in infinitely many ways, governed by rational-valued scalars φ and φ*) such that two new vertices are also of rational coordinates in the development. From proposition 2.1, it follows that two new vertices are both v-rectangular; finally, proposition 1.3 implies that the development of the resultant cuboid (which can also be realized with all integer coordinates after an appropriate rational magnification by χ) is weavable. ▪

Because the set of rational numbers is dense in the set of real numbers, we can also formulate the following.

Corollary 3.5 —

For any v-rectangular cuboid C, there exist weavable ones C_w within an arbitrarily small neighbourhood of C (in the sense of either maximum or l₂ norm of difference of vertex positions in D(C) and D(C_w)).

Its importance lies in that a development of a v-rectangular cuboid generically cannot be fit (by rotation and uniform magnification) to integer coordinates in any Cartesian system, and random adjustment of vertices to integer (rational) coordinates will affect realizability. Nevertheless, corollary 3.5 proves the existence of such ‘close’ and realizable cuboids with arbitrarily small modifications of coordinates.

Now, a complete procedure of getting coordinates of vertices of D(H_2R) of all cuboids H_2R derived from non-degenerate tetrahedra H may look as follows:

• Step 1: choose a set a,b,c,d that corresponds to a realizable initial tetrahedron.

• Step 2: choose positive scalars φ and φ* (φφ*>1) still under the condition of realizability and do both internal R-truncations (coordinates are obtained from equations (3.5)–(3.8)).

• Step 3: calculate the coordinates of N according to figure 6b as the intersection of lines L*T_β* and LS_α (similar procedure applies for N*).

(c). Unbounded initial tetrahedra

Here, we will complete the proposed method with respect to degenerate (unbounded) cases: first, a simple degeneracy of one vertex (at infinity) of the initial tetrahedron (such tetrahedron marked by H* is called ‘simply unbounded’) is introduced, then a double degeneracy (two vertices at infinity, H^**, ‘doubly unbounded’ tetrahedron) is analysed.

Let first the simply unbounded case be dealt with. Consider now figure 7a as a network of an infinite triangular prism open at one end (obtained from a bounded tetrahedron in a limit $a\to \mathrm{\infty }$; mind the condition 2b>c+d of realizability from equation (3.1)). Even in such a special case, R-truncation of the neighbourhood of vertex V is done using expressions under (3.8) without any change, since both α* and β* are independent of a, so the limit $a\to \mathrm{\infty }$ does not influence the ratio α*/β* either (that ratio is sufficient instead of individual values, because a uniform scaling does not modify the direction of the new segment S_α*T_β* but rather results in a parallel translation of the truncating plane). The truncation itself can be interpreted as a degenerate internal one through a point P at infinity (figure 7b). Vectors $\overrightarrow{V{T}^{\mathrm{\prime }}}$ and $\overrightarrow{VS}$ are still given as ${\mathbf{s}}^{\ast \mathrm{T}}=(-d,-2b+c)$, ${\mathbf{t}}^{\ast \mathrm{T}}=(c,-2b+d)$, respectively. After re-evaluation of expressions under (3.8), it is obtained that

$$\frac{{\alpha }^{\ast }}{{\beta }^{\ast }}=\frac{2b+c-d}{2b-c+d},$$

always strictly positive with the above condition of realizability. The first R-truncation turned a simply unbounded tetrahedral prism into a simply unbounded four-sided prism associated with a spherical image of four spherical triangles sharing a vertex, and all other vertices being incident to one great circle. Because of the uniqueness of the procedure shown in §2 for generating new spherical octahedral vertices with pairs of equiareal triangles, the second R-truncation (i.e. of the neighbourhood of vertex P at infinity) can only result in a four-sided face as the mirror image of S_α*T′ST_β* with respect to a vertical plane in the development, as that solution is trivial by symmetry.

Figure 7.

Successive R-truncations of degenerate initial tetrahedra. (a) Simply unbounded H*: D(H*) provides v-rectangular vertices V,S,T by construction; (b) degenerate internal R-truncation by a plane through a distant point P with s^*T= (−d,−2b+c) (vector V T′) and t^*T=(c,−2b+d) (vector V S); (c) doubly unbounded tetrahedron H^** with $a,b\to \mathrm{\infty }$ resulting in a cuboid with two parallel faces: for reasons of symmetry, all eight vertices must be congruent.

Finally, figure 7c shows the neighbourhood of v-rectangular vertices S,T in a network D(H^**) in a doubly unbounded case when both a and b tend to infinity (a pair of opposite faces tends to have parallel planes in the final hexahedron, giving rise to a spherical image of two adjacent triangles). One can see again by symmetry that a net obtained by reflecting figure 7c about a right-hand side vertical and a top side horizontal axis (and then by cutting off infinite areas on the right-/left-hand side) is a solution (with symmetry C_2v), and no other solution is possible again.

4. Discussion

Some comments on the present construction of weavable cuboids are as follows

• — all angles of the final hexahedron are determined solely by the four parameters a,b,c,d of figure 4, because angles resulting from R-truncations are always unambiguous (see equations (3.5), (3.6) and (3.7), as well as comments given in figure 7); scalars φ,φ* do not influence angles.

• — Any final cuboid with no parallel faces can be generically realized from three different sets of a,b,c,d (i.e. different initial tetrahedra H). In the order of increasing degeneracy, some more exact statements are supported by the method of construction.

  • I If a cuboid has no parallel edges, then it has three pairs of opposite faces to intersect in a line corresponding to the edge PV of H in figure 4, giving rise to three different initial tetrahedra to be truncated into cuboids.
  • II If the cuboid has parallel edges (but no parallel faces), it can either be obtained with a special choice of c=d (and all parallel edges will be obtained from R-truncations) but only from a single initial H; or from two initial objects H* (i.e. an infinite triangular prism bounded only on one end, which is further truncated by a plane parallel to the axis of the prism; generically, it can be done starting from two different pairs of opposite planes of the resultant four-sided prism, see figure 8 for illustration).
  • III If a cuboid has parallel faces, it can be easily seen that it cannot be derived from a non-degenerate H: the proof is based on figure 4, where lines SP and TP can never be parallel. Because of eight congruent vertices and the C_2v-symmetry, they can be obtained in two different ways: either from two infinite triangular prisms H* bounded only on one end and having a mirror-symmetric pair of vertices or from a configuration H^** with a centrally symmetric pair of vertices.

• — all vertices of weavable cuboids in the present construction can be associated with integer coordinates (in the two-dimensional development) in a predefined basis xy. It is obvious that any inflation of that rectangular tessellation according to quadrangulation numbers (after the concept of triangulation numbers introduced in [17]) results in new weavings with longer strands (larger polyhedron). Nevertheless, it is also possible that vectors $\mathbf{r},\mathbf{u},\mathbf{s}$ and $\mathbf{t}$ written in another basis with integer coordinates produce another weaving (tessellation) with even shorter strands (i.e. they are themselves obtained by similar inflation of another configuration), see figure 8 also in this respect.

• — the proposed double truncation procedure allows the resultant cuboids to belong to the following symmetry groups: generically, C₁ (no symmetry) and specially C_s or C_2v (with four parallel edges connecting mirror-symmetric deltoidal faces) when derived from an initial bounded H; generically C_s and specially C_2v (either with four parallel edges connecting mirror-symmetric deltoidal faces or with two parallel rectangular faces of pairwise parallel edges) or D_2d (cuboid with two parallel, isometric and rectangular opposite faces) when derived from a singly degenerate H*; generically, C_2v (with two parallel rectangular faces of pairwise parallel edges) and specially D_2h (rectangular block), D_4h (square prism), O_h (cube) when derived from a doubly degenerate H^**.

Figure 8.

Example of a cuboid weavable by strands of unit width with initial parameters a=160, b=80, c=d=32, φ=φ*=1.05, χ=1: development (a) and three-dimensional realization (b). The cuboid has C_s symmetry with four parallel edges because of c=d, hence the configuration could also be derived from a simply unbounded polyhedron (triangular prism that is infinite along parallel edges and their planes contain either edges SS_α,S_αN,NT_β* or S_αN,NT_β*,T_β*S). Strands running along coordinate directions in the development do not give the smallest length here: in another rectangular basis with unit vectors $(\sqrt{2},\sqrt{2})$ and $(\sqrt{2},-\sqrt{2})$, all vertices also have integer coordinates.

Finally, we can conclude that the above construction (accounting for both metric and projective conditions) is suitable for proving the existence of weavable cuboids obtained from any initial (realizable, bounded or unbounded) tetrahedra, and also for finding all possible cuboid geometries that admit such a weaving (but not for finding all possible weaving patterns on a weavable cuboid). It is not applicable, however, to give a complete classification of those infinitely many solutions or to get ‘optimal’ weavings in the sense of shortest strand length (the smallest cuboid in a Cartesian system with fixed unit) either because (i) the size of the object depends on common divisors of a set of complicated expressions and (ii) even for a given set of parameters a,b,c,d,φ,φ*, a change of basis may result in a smaller polyhedron.

Data accessibility

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Competing interests

The author has no competing interests.

Funding

This paper was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the OTKA under grant no. K 100894.