Tensegrity: Geometrix

Tensegrity geometry involves fairly complex mathematics and it is the work of Robert William Burkhardt,that I continually refer to. A detailed account of his mathematical research and design theory is presented in the publication: A Practical Guide to Tensegrity Design.

In this publication he starts with the most basic of Tensegrity forms the T-Prism also known as 3-Prism.
 
These are just some of the images; captured from his publication; that illustrate his theories.
The mathematics are well documented but are too complex to discuss in detail here. He also covers how to adapt the formulae depending on what parameters are sought or known.

A recommended read for anyone interested in Tensegrity structures.
I would like to explore an alternative geometrix theory for this basic form of Tensegrity. If we consider a number of factors relating to this form, we find that for all compressive struts being equal in length the top triangular plane is parallel to the bottom triangular plane. We also know that for any given situation the rotational symmetry; that is the rotation about the centre axis (z-axis in above); has limitations due to the length of the struts assuming a fixed point at the base.

Essentially if we take the static vertical arrangement as per the first illustration above and rotate the top triangular form we reach this limitation at approx 180 degrees or the perceived point of intersection of all 3 struts.
 
These illustrations are from my own work with the first one showing the final 3 Prism arrangement and the second showing the theoretical graphical geometry for determining the location and size of the vertical tension members.
What I have done is create 2 circles defining the perimeter transcribed by the rotational movement assuming a fixed point at the base. The top circle is located at the strut length fully vertical with the lower one at the elevation on the point of maximum rotation. The path of movement from the vertical position to the maximum rotational symmetry achievable is shown as the Radial Path.

The radial path is determined by plotting a series of points at intervals tracking the motion of the strut from the vertical position. This path is transcribed on the perimeter surface of the 2 projected circles. This path passes beyond the minimum distance between the rotated strut and the eventual base point that would be connected by a tension strut.
So using the measuring capabilities of Solidworks it is simply a case of selecting the Radial Path and the associated connecting point at the base to determine the minimum dimension and thus we derive the ideal tension member length.
My findings at this stage are theoretical and I still need to prove the results with further experimentation. In this case I compared my theory with the model created using a dataset conceived by Robert William Burkhardt and there was a variation of only 0.04mm.
This variation may be attributed to the spline created from the series of points defining the radial path. The interpolation of this spline is dependant on the number of points (6) selected and consequently proportionally accurate.