“Jessen's icosahedron”的意思、由来-开放百科全书

Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron. It was introduced by Børge Jessen in 1967 and has several interesting geometric properties:

Although a shape resembling Jessen's icosahedron can be formed by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral-triangle faces by pairs of isosceles triangles, the resulting polyhedron does not have right-angled dihedrals. The vertices of Jessen's icosahedron are perturbed from these positions in order to give all the dihedrals right angles.

Tension integrity transformations

The Jessen's icosahedron is one of a family of chiral dipolygonids described by H.F. Verheyen^[2] capable of twisting, expansive-contractive transformations termed Jitterbug transformations by Buckminster Fuller.

In particular, the Jessen's icosahedron is the geometry of the tensegrity icosahedon^[3], a physical structure composed of six rigid compression rods (forming the concave edges) and 24 tension cables (forming the other edges). Realized in this form, the Jessen's icosahedron's property of continuous but not infinitesimal rigidity (its equilibrium shakiness) is dramatically manifest. Only the rods must be rigid, to resist compression and keep the cables taught and the whole structure drum-tight, despite the fact that the rods do not touch each other: they float in a net of tension cables.

The 6 rods float in parallel pairs, one pair lying in each of the three orthogonal planes. The pairs rest {{sqrt|3/2}} edge lengths apart from each other. The rod length must be twice this distance (about 2.45 edge lengths).{{Efn|The 3/2 power also occurs, famously, in Kepler's third law, as the ratio of the radius of a planetary orbit to its period. This is not a coincidence, but a consequence of the similar spherical geometry of these two equilibrium systems, in which the 3/2 power ratio describes the balance point of the compressive (radial) and tensive (circumferential) forces at which the stable system rests.}}

Provided the 6 compression rods are spring-like and can be compressed slightly (down to √2 edge lengths), the 6 concave pairs of isosceles triangles can be folded open (simultaneously) into square faces, transforming the Jessen's icosahedron into a cuboctahedron. Or they can be folded closed (which also compresses the 6 rods to length √2), transforming the Jessen's icosahedron into an octahedron. Released, the compressed rods spring back to their resting length, returning the Jessen's icosahedron to its resting shape.

These two ways of compressing the concave edges amount to twisting the whole structure away from its resting shape in one of two opposing directions. The vertices move slightly all together in either outward or inward spirals, and the long radius (center to vertex) of the entire structure grows from its resting length to a maximum (the cuboctahedral radius of 1 edge length) when fully open, or shrinks to a minimum (the octahedral radius of √2/2 edge lengths) when fully closed. The compression rods move slightly as well. When compressed, they move either apart (outward) or toward each other (inward), but always remain parallel and in their original plane. The simplest way to force the structure to expand or contract (which it always does radially and symmetrically) is to grab any pair of rods, and push them apart or pull them together. The other two pairs will move the same distance, in concert.

When the parallel rods are exactly one edge length apart, the vertices describe a regular icosahedron. This is not quite the resting shape; it is reached when the structure has been twisted slightly inward (the rods have been forced slightly together from their resting distance of about 1.225 edge lengths).

As the isosceles faces fold outward or inward, they distort (because their long edge shortens). The equilateral faces displace also, but do not change shape during the twisting; they merely rotate around their centers and travel outward or inward slightly.{{Efn|The triangular faces rotate up to 45 degrees left or right of their resting positions, so their vertices travel a spiral path outward or inward on the surface of a cylinder whose axis passes through the centers of opposite triangular faces.}}. Therefore the physical structure can even be skinned with rigid equilateral panels, and with foldable or stretchable isosceles panels, without impeding its ability to traverse the entire range of motion.{{Efn|The concavities which fold open or closed are irregular tetrahedra, with four unit-length edges (tension cables) and two opposing unequal-length edges whose lengths vary with the motion. At equilibrium, the shorter unequal edge (the "invisible edge") is 1.225 and the opposing unequal edge (the compression rod) is exactly twice that. The shorter edge varies between 0 and 1.41 (the regular icosahedron appearing when it is 1). The longer edge varies between 2.45 (at equilibrium) and 1.41 (at fully open or fully closed).}}

Finally, if the compression rods are not springs but nearly rigid, the structure still exhibits its equilibrium springiness, even though the isosceles faces are not able to fold fully open or fully closed. To whatever extent the rods can be compressed, even minutely, a very evident small range of inward and outward motion (springy radial expansion and contraction) remains.

A tensegrity icosahedron capable of the full range of motion can also be constructed using fixed-length rigid rods which cannot be compressed, provided the cables can be stretched instead. The NASA SUPERball tensegrity robot is an example of such a structure that is capable of even more complex motions, since its cables can be elongated individually (by unwinding them from spools).

Tension integrity transformations

See also

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