Small Tensegrities

Here I am looking at the smallest rigid sticks'n'string framework. It has to have 3 or more stick otherwise it would lie flat. The well known and documented simplest tensegrity is similar to the one in my hand on the right except that the 3 cornered stars of string are replaced by triangles.

I think my construction is more satisfying because each stick is only pulled by 4 strings instead of 6, and less string is required.

This means that each pair of strings meeting one end of a stick are coplanar with the stick.

Here are two. The one on the left is kind of unsatisfying because the lengths of string arent especially interesting, and the sticks are at odd angles to each other.

The one on the right is much better because the lengths of string are in a very specific ratio to the lengths of the sticks, and the sticks are at right angles to each other. ( and it has more symmetries ).

How to calculate the relative lengths in the right-angled simplest tensegrity ? Normally I wouldnt think this is interesting enough to take note of, but I am writing it down because it isnt immediately obvious, and I want somewhere to keep the calculations that wont make me clutter up my house with scribbly papers.

To start off with notice that the reason the construction is rigid follows from the way each stick is held by 4 strings, as in the picture below, and also notice that the pair of strings at the end of a stick have to be coplanar with the stick. The stick in the picture below may wobble when pushed but if the strings are tight enought and the nails well fixed, it wont.

Next imagine the construction viewed from above and imagine how the sticks can fit around an imaginary cube. X is the length of the string from bottom to top, Y is the other distance between sticks' endpoints, and Z is the length of the strings that join in a group 3 spoked star. The length of Z can be calculated from Y using triangle properties, and is Y/sqrt(3).

Viewed from the top, the construction looks like a 3 spoked windmill!

Suppose the length of the sticks is 1.

Viewed from the side, along the axis of one of the sticks, we can cut the stick length into g and 1-g, where the other stick crosses.

Given that the length of a stick is 1, and this is divided into g and g-1, the diagram on the right shows where various lengths appear. Note the vertical/horizontal distance 2g-1 between the sticks we see from the side and the one pointing at us.

Viewed from the side, the line a string follows from the bottom right hand corner to the center stick can be extended in a straight line by the other string tied to that end of the stick ( because they are coplanar, remember !), and because of triangular properties, if extended further, it will bisect the top-left diagonal, as in the picture on the right.

Therefore, in the picture on the right, C/D = A/B.
We can substitute to get the value of g.

A=2g-1 B=1-g C=g/2 D=g+((1-g)/2)

etc.. giving...

g = 1/sqrt(3). Pretty fab eh!

Now we know what g is we can PYTHAGORAS theorem for 3d to get:

X^2 = (2g-1)^2 + (1-g)^2 + (1-g)^2 to give X = sqrt(5-8/sqrt(3))

Y^2 = (1-g)^2 + (2g-1)^2 + g^2 to give Y = sqrt( 4 - 2sqrt(3) )

and Z = Y/sqrt(3).

Easy !!

The tensegrity in my hand is rigid, but the dangling one isnt. They both consist of 9 tendons and 3 sticks.

The only difference is the length of the strings. The one on the right has strings just long enough to cause the framework to collapse ( must explain dimensions and relevance to VE )