Highly Symmetric Tensegrity Structuresby Bob Connelly and Bob Terrell 

1. What is
a tensegrity?
It is a structure that joins nodes (points) with
inextendibile
cables and incompressible struts. The cable can be made from string,
wire,
or rope, and struts can be made from tubes, dowel rods, or just sticks.
See wikipedia
for a reasonable definition. 2. What is the
mathematical definition
of a tensegrity?
It is a finite configuration of points, the nodes, in space or
the plane where some pairs of the nodes are designated cables,
constrained not to get further apart, and some pairs are designated
struts, constrained
not to get closer together. Note with this definition cables and
struts are allowed to intersect and cross.
3. Who first created a
tensegrity?
If you don't count spider webs and constructions in nature, an
artist Kenneth Snelson is usually credited with the first
creations. He has an
interesting website here. 4. Who thought of the
name tensegrity? Buckminster Fuller. It combines the notion of "tension"
with "integrity",
creating "tensional integrity. You can see Bucky's musings here.
5. Where can I read more about this catalog and
tensegrities?
6. What are some links to other sites that show
tensegrities?
7. Can I build these tensegrities?
Why not? Once you have chosen a particular example, you
can read
off the lengths of the cable relative to the strut lengths in the lower
right
window. The picture should help you thread the cables and struts.
There
is only one way to make the structure with those lengths if everything
is
hooked up right. There are various materials that you can use.
Dowl rods
with nails in them are popular as in Pugh's
book or George Hart's page,
or Hugh Kenner, Geodesic Math and How to Use It, Univ. Cal Pr., 1976,
or RenĂ© Motro's book "Tensegrity:
Structural Systems for the Future". Yes. It is here . If you want to modify the program and recompile, then you will also need to get the render and jama packages listed in the references, or at least grab the render.jar and Jama.jar archives. There is also a modified version of the applet which is used to make the rotatable icons. For this you will need the source code . For each rotatable icon you want to make, run the main applet until you find an example you like, then read the required parameters from the applet and its source code, copying the example from our Home page. 9. What is the "tracecode" button? It has two uses: a) If you want the coordinates of all the vertices of the tensegrity currently displayed, those show up as the centers of the "sphere" commands. You might need those if you are planning to build it three stories high, for example. b) If you want to make a really sharp raytraced picture of the displayed tensegrity, copy and paste the displayed code into a file. (This works best on some platforms if you click at the bottom and drag upward.) Edit the file if you want to change the colors and textures, and run povray on it. The nonrotatable picture on the Home page was made in this manner. 10. Will the structures
shown in this catalog hold together?
Yes. They are all rigid, but not all choices give
interesting tensegrities.
Some are very degenerate.
11. How are the structures
chosen?
They must satisfy the following list of criteria: They
must have
the symmetries of one of the
six groups A_{4}, S_{4},
A_{5}, A_{4} x Z_{2}, S_{4} x Z_{2},
A_{5} x Z_{2}; There are two classes (each class
is designated
by its own color, usually red and blue) of cables such that there is a
symmetry
of the structure that takes takes any cable of one class to any other
cable
of the same class; For any two struts (colored brown or green),
there is
a symmetry of the structure that takes any strut onto any other;
The cables
are connected as a set; The whole tensegrity structure must be super stable. 12. What does it mean that a tensegrity is super
stable? This is a very strong condition on the geometry of the
structure.
There is a particular calculation that can be made in terms of
the stress
that exists in the structure. Certain numbers, called eigen
values, calculated.
If they are positive with altogether only 4 of them being 0, the
structure
will be super stable. A consequence of this is that the structure
is globally
rigid in all dimensions. This means that if there is any
other placement
of the nodes satisfying the cable and strut conditions, then this
placement
will be congruent to the original. It will be the same
configuration, possibly
rotated or reflected and then rotated. Another consequence is
that if you
build this structure it will stay stable and rigid, even when the
stress
is increased relative to the stiffness. This is very handy for
these tensegrities,
since the stiffness of the cables especially tends to be very soft.
13. If I have a tensegrity where all the nodes are
the same,
and it holds together, will it be in this catalog? Not necessarily. It could be rigid, but not super
stable. Or it
could have a different symmetry than one of the one of the six groups
in
this catalog. For example, the symmetric tensegrids described here have a
symmetry
group isomorphic a dihedral group.
14. What is a group? It is a set where any two elements g_{1} and g_{2}
of the set can be multiplied together to get another element g_{2}g_{1}of
the set. The multiplication operation (or group operation) must
satisfy the rules of associativity, inverses, and identity. See wikipedia
for
more details. Presented this way, this is a definition of an
abstract group. There are tons and tons of many different kinds
of examples of groups. Here
we concentrate on, permutation groups, symmetry groups, and groups of
matrices.
(You can look at a book such as "Introduction to Modern Algebra" by
Birkhoff
and Maclane for a nice introduction.) This approach greatly
simplifies the
cataloging of the tensegrities we consider. This is a set of functions, called permutations, that
permute the elements of some set. To be a group, any composition
of these permutations in the group is another permutation in the group.
We use the disjoint cycle
notation for permutation groups considered here. Our groups
permute the
symbols 1, 2, 3, ..., n. The symbol (a, b, c, ..., d) means a
> b >c
> ... > d > a. In other words these symbols are
permuted cyclicly.
The product of these cycles means the composition of the
corresponding
cycles. It is easy to see that any permutation can be written as
the product
of such disjoint cycles. For example, (12)(345) means 1 and 2 are
switched,
and 3 goes to 4, which goes to 5, which goes to three.
These are the alternating groups on 4 symbols and 5 symbols, respectively. These are all the even permutations of 4 symbols and 5 symbols, respectively. A permutation is even if when it is written in disjoint cycle notation, there are an even number of cycles of even length. For example, (12) is not even, but (12)(34) and (123) are even permutations. The group A_{4} has 12 elements, and the group A_{5} has 60 elements. 17. What is the group S_{4}? This is the group of all permutations of 4 symbols and has 24 elements. 18. What are the groups A_{4} x Z_{2}, S_{4} x Z_{2}, A_{5} x Z_{2}? The G x Z_{2} notation denotes the cross product which means that you take two copies of the group G, where one copy has a  sign attached. The group operation on the  acts like multiplication by 1. For example, (12) times (34) in S_{4} x Z_{2} is (12)(34). The G x Z_{2} groups have twice as many groups elements as G. So A_{4} x Z_{2} has 24 elements, S_{4} x Z_{2} has 48 elements, and A_{5} x Z_{2} has 120 elements. 19. Why are these groups chosen? 20. What does isomorphic mean?
If G_{1} and G_{2} are two groups, an isomorphism
is a onetoone function f (when g is not h, f(g) is not
f(h)) defined
on all the elements of G_{1} with values in G_{2}
such
that for every g, h in G_{1}, f(g)f(h)=f(gh), and every
element
of G_{2} is the image of some element in G_{1}.
This
means that from the point of view of formal group operations, G_{1}
and G_{2} are the same. If two groups are isomorphic they
must
have the same number of elements, but, for example, A_{4}
x Z_{2}
and S_{4} both have 24 elements, yet they are not isomorphic.
21. What is a symmetry group?
It is a collection of rigid distance preserving functions of
all of space, where each function transforms some set to itself.
The group operation is the composition of these functions, and so
to be a group, the composition of any two such functions in the group
is also in the group. For example the rotations of regular
tetrahedron, cube or regular dodecahedron are symmetry groups. 22. Does the tensegrity always have a group of
symmetries isomorphic to the group in the group window?
Yes, but the tensegrity itself may have other larger groups of
symmetries.
23. What is the order of a group element?
If g is a group element, then g^{n} is g multiplied by
itself n times. The order of g is the smallest n = 1, 2, ... such
that g^{n} = e the identity element. For example, the
permutation (12)(34) has order
is 2, and the element (12345) in A_{4} x Z_{2}, has
order
is 10. It is particularly easy to figure the order of a
permutation when
it is written in disjoint cycle notation.
24. What does the group inverse mean?
One of the defining properties of a group is that if g is a
group element, then there is a group element g^{1}, where gg^{1}
= g^{1}g = e, the identity element. g^{1} is
the inverse of g. For example, the inverse of (123) is (321).
If g is of order 2, then
g = g^{1}. 25. When I choose a group elements in the cable and
strut windows,
what does that mean?
When the situation is not degenerate, there is a onetoone
correspondence between the nodes of the tensegrity and the group
elements. Any node can
be identified with the identity element of the group, and the two group
elements
in the cable window and the group element in the strut window are
identified
with cables and struts that are connected to the identity node.
Note that
if g is a group element corresponding to a strut or cable, then the
inverse
element g^{1} also corresponds to the same cable or strut.
26. I want the struts not to touch each other.
How can I find
examples where that happens? If the order of a strut is greater than 2, then there are two
struts connected to each node. So you at least have to choose a
group element in the strut window that is of order 2. But it
still may happen that the
struts will cross somewhere in their middle. This tends to happen
for the
groups with a Z_{2} factor. (They have mirror reflection
symmetries,
and the struts will intersect their mirror image, if the mirror
intersects
the struts obliquely.) But in any case, you can look at the
picture and
see if they intersect. If you try a different stress ratio in the
last window,
that might pull some of the struts apart. In some cases, you might need
to
use the "thinner" button and "zoom in" to decide whether there is
really
an intersection or not.
27. What do those funny lines in the window on the
right in
the program mean?
Each curved line, or lines, of a given color corresponds to a
particular representation of the group chosen. When you click in
the right hand window,
a vertical line appears through the point you clicked on. The
horizontal
axis corresponds to the ratio between the stresses on the two cables
you
have chosen in the cable window. The vertical axis corresponds to
the stress
on the strut. When the strut stress is below the top horizontal
line, it
is negative. When the strut stress crosses one of the curved
lines, the
tensegrity becomes unstable, but right at that the point on the first
curved
line, it corresponds to an equilibrium that it needs for its stability,
but
the representation that corresponds to that line. We call that
first representation
the winner. 

FAQ

