Large-displacement symmetrical foldable cones

A foldable cone, or f-cone, is a discrete vertex folded from flat with curved faces, and is an exemplar of nonrigid origami. We determine the exact equilibrium shape of a well-folded symmetrical f-cone by geometrical considerations alone, by treating as cones intersecting along original fold lines expressing equal fold angles. When moderate displacements prevail, the shape is found alternatively from the spherical image of the vertex, from deploying Gauss's mapping. The analytical working is much less compared to direct linearization of the exact solution framework; expressions for shape parameters are availed in closed form, and some of their variation is accurately retained compared to the exact case.

Figure 1

Foldable cones, or f-cones, made from disks of steel shim (a), (c), (e) or paper card (b), (d) folded once, (a)–(d), $n=2$, or twice, (e), $n=4$. In (a) and (c), the planar folded disk is pushed through and upwards to form two conical portions on each side of the fold line; (c) is (a) in upside-down views. The fold line in (b) is fully creased, with the conical halves meeting tangentially on the line; (d) has the same shape as (c), in paper card. The twice-folded disk in (e.i) is pushed through and downwards to form the shape in (e.ii); both vertices are conically curved.

Figure 2

Beam analogy for combined folding and curving of an f-cone vertex: (a.i)–(a.iii); a pair of uniformly curved beams locked together in rotational end fixtures, equivalent to $n=2$; (a.iv) local geometry of a general beam segment where $\mu$, in particular, is the fixture semiangle. (b.i), (b.ii) Two geometrically compatible configurations when $\mu =\pi /2$ (fully creased) for $n=4$; (c) a beam reconnected to itself ($\mu =\pi /2$, arbitrarily) has nonuniform curvature, equivalent to a single radial fold line in an f-cone ($n=1$).

Figure 3

Conical geometry of a singly folded f-cone vertex ($n=2$). (a) Conical half, with its base resting horizontally and subtending $\pi +2\theta$ radians; $\mu$ is the semiangle of the fold line, and $\lambda$, the vertex semiangle. (b) Vertical ($\mathbf{i}$, $\mathbf{k}$) plane through cone indicating apex semiangle, $\alpha$, and other auxiliary parameters. (c) PQR plane isolated from (a); (d) conical base locus from (a) showing tangential unit vector, $\mathbf{t}$.

Figure 4

Spherical image construction for singly folded f-cone vertex. (a.i) Some unit vectors, 1–6, normally appended to the vertex surface, circulating anticlockwise; (a.ii) schematic orientation in plan view. (b.i) Planar projection of corresponding spherical image for shallow displacements: the north pole (black dot) is the vertex apex; each curved path is semicircular, and the circulation follows the unit vector sequence in (a). Three areas enclosed by the profile in (b.i): $A_1$, twice (white), and $A_2$ (gray); (b.ii) is half of the image again, $A_3$ is an auxiliary area.

Figure 5

Conical geometry of f-cone with $n$ radial fold lines ($n=4$, arbitrarily). (a) Single conical portion, (b) vertical slice plane, (c) base locus; and various auxiliary geometrical parameters.

Figure 6

Geometry of prism from Fig. 5. Note that ${\mathrm{S}}^{\prime }$ differs from S because the plane ${\mathrm{PRS}}^{\prime }$ is normal to QS.

Figure 7

(a) Vertical slice through conical portion, which subtends more than $\pi$ radians; $\beta$ and $\varphi$ differ compared to Fig. 5. Restoration of conical portion, (b.i), by rotating its axis back to vertical, (b.ii), for an upright f-cone; (c) as (b) for inverted f-cone, ${\varphi }^{\prime }$ is equal to $\pi -\varphi$.

Figure 8

(a) Fold lines of an f-cone forming the edges of a regular pyramid ($n=4$, arbitrarily). Vertical pyramidal planes, (b.i) and (b.ii), for relating the vertex geometry between $\varphi$, $\lambda$, and $n$.

Figure 9

Shallow spherical images for a general f-cone ($n=3$, arbitrarily) when, (a), upright, and (b), inverted. Overall images are (a.i) and (b.i), with local areal detail in (a.ii) and (b.ii) respectively.

Figure 10

Variation of f-cone vertex semiangle, $\lambda$, with fold angle, $\mu$, for specified $n$ (blue-to-red, $n=3...15$): (a) upright f-cones, (b) inverted. The solid black line, (a), is for $n=2$; dashed line is Eq. (6), linear solution. (There is no inverted solution for $n=2$.)

Figure 11

Rendered shapes compared to models from Fig. 1. (a) $n=2$ and $\mu =\pi /2$; (b) is much less folded; (c) and (d), the same upright and inverted f-cone.

Figure 12

Variation of final (subscript “f”) parameters with $n$ for specified fold angles ($\pi /12...\pi /2$) for upright f-cones. Dots are exact geometrical solutions, and curves are linearized solutions via Eqs. (20), (19), and (18), respectively.

Figure 13

As Fig. 12, for inverted f-cones.

Figure 14

Fabric f-cone artwork hanging in the Royal Victoria Hospital, Belfast (photographed by author circa July 2019). (a) Gently curved rods emanating from the apex, support a heavy fabric, hanging in conical sections; (b) base outline resembles the beam analogy from Fig. 2; (c) apex structural details; (d) axial view of pillowed interior.

Figure 15

Buckled f-cone made from thin copper beryllium sheet, whose unique ductility and hardness allow for many close fold lines ($n=24$). No longer uniform but nearly cylindrical about a given axis, (a). This axis can be cycled around the disk almost neutrally, (b.i)–(b.iii); compare the cylindrical orientation to a fixed marker line (black) on the underside of disk (shown inverted).