These
tensegrity
models have been made by connecting together edge units to
form twisted polyhedra. The edge units have a notch
at each end, and a loop of cord running along the
length. They connect by slipping the cord of one edge into the
notch of another. Each unit connects with four others: one at each
notch and two along the cord.
The basic idea is explained nicely by George Hart in his page on
Soda Straw Tensegrity Structures.
I have built my units from two barbecue sticks bound by wire near each end,
which gives the notches. The cord is either a loop of string, or a length of
elastic knotted at each end and sliped through the notches.
The process of twisting transforms an equal edge polyhdron into an
equal edge dual form. In the case of the regular and quasi-regular
polyhedra that dual has the same shape as the "normal" dual (by
polar reciprocation). Not all equal edge polyhedra have an
equal edge twisted dual.
The regular and quasi-regular polyhedra are also linked with their twisting
mid-point by a jitterbug-like transformation, as described in
Polyhedral Twisters.
Twisted polyhedra are also popular as a base for wooden puzzles. A lot of
these puzzles can be seen at
Puzzle World.
A twisted octahedron.
The stuts come in four sets of three parallel struts which
run along the middle of equilateral triangle channels. These
correspond to the positions of a particular
packing of rods with a triangular cross section.
This helps make the twisted octahedron a popular base for
wooden puzzles
The mid-point of twisting between an octahedron and cube.
The cords form a cuboctahedron, having the six faces of the
cube and 8 faces of the octahedron. The struts form four
triangles inscribed in the equatorial hexagons of the
cuboctahedron.
The mid-point of twisting between a cuboctahedron and rhombic dodecahedron.
The cords form a shape close to a small rhombicuboctahedron. It has the
14 faces of the cuboctahedron (6 squares and 8 triangles) and the 12
rhombi of the rhombic dodecahedron. The struts are arranged as 6 squares,
arranged in parallel pairs offset by an 1/8 of a turn.
A twisted icosahedron. Like the twisted octahedron it has sets of
parellel struts - in this case 5 sets of 6 struts. The triangular
channels these run through are the shape of a golden gnomon, the
triangle made from joining three consequetive points on a pentagon.
This makes it another popular choice for wooden puzzles, as
is described in this discussion of the
Jupiter
The mid-point of twisting between an icosahedron and dodecahedron.
The cords form an icosidodecahedron, having the 12 faces of the
dodecahedron and the 20 faces of the icosahedron. The struts form six
pentagons inscribed in the equatorial decagons of the icosidodecahedron.
A twisted small stellated dodecahedron (viewed close to a three-fold axis).
One of the features of the twisted polyhedra is that each strut is
paired with a parallel line on the other side of the centre through
which lots of struts (possibly extended) pass. This feature is quite
striking in this view. Three parallel struts are seen end-on, each paired
with a line of crossing. Of the other struts, 24 pass through just
one line, and three pass through two. Here is another example showing a
crossing line in a twisted cube
The property is taken good advantage of in this
30 Notched Sticks
puzzle. The configuration of the pale sticks is a mirror image of
that of the hexagonal sticks, with the two being related by a central
inversion.
A twisted stella octangula (viewed along a four-fold axis).
The usual connection method was unstable in this polyhedron so
the 8-way vertices have been connected slightly differently. If
connected by the usual way the stella octangula would be the
twisted dual of the truncated cube.
A twisted stella octangula (viewed along a three-fold axis).
The usual connection method was unstable in this polyhedron so
the 8-way vertices have been connected slightly differently. If
connected by the usual way the stella octangula would be the
twisted dual of the truncated cube.
