tensegrity
definition
From a basic physics
perspective what is tensegrity? Is it just a variation of torque; some sort of esoteric innerseesaw? Why is it not found in physics books? A fundamental idea of tensegrity is the compression:tension
ratio. In this writeup, the mathematical framework and idea of tensegrity will focus only on the range of optimal configurations a shape can take, from full compression(flattened) to full tension(expansion).
compressiontension
 The spokes in a spokedwheel provide outward tension; thus, a structure, which is
lighter and more flexible
than a solid wheel, but provides equivalent compressive strength can be achieved.  A vertical length of column can hold up a compressive load. Based on material properties, there is an inverse relationship between length and compression described by a
force density (f/l)
. For an unloaded column, there exists a maximal length beyond which tension cannot expand.  Geodesic domes can be formulated at planetary size, thousands(km) in diameter. This planetary sized
leverage
results from smaller changes in tension over the outer diameters leading to greater compressive strength centrally (ie the roof does not collapse).
applications
 Joints, such as ankle, pelvis, and spine, are prestressed with tensive loads leading to increase compressive strength and mobility
 A manta ray’s wing can be modeled with several lines of tensegrity, described as a learning surface providing nonlinear fluid control, capable of firmly flexing to quickly glide away, or rippling inplace to hover.
 shape of cells and DNA
models
 Given overhanging blocks, tension exists in the regions where blocks overlay. The blocks will not fall over if each subsequent block’s overhang is half the one below it; this allows the combined center of mass to be supported. This can be described using a harmonic series sum,
H(n)
.H(2)
, case of 2 bricks: \( X2 \cdot 2M = X1 \cdot M \to (x1=1/2) \)
H(4)
, span of total overhang greater than length of a block: x1=1/2, x2=1/4, x3=1/6, x4=1/8
 \( \left(\sum x1+..+x4 = \frac{25}{24} \right) > 1 \)
H(n)
, can it span any distance?: \( X_n\cdot(nM) = X_1 \cdot M \)
 \( \sum\limits_{k=1}^n \frac{1}{2k} \) ; \( \int \frac{1}{k} \, dk \)
 H(k)  ln(k)
logapprox undercounts the series_sum
\( \to Euler_constant( \gamma ) \)
template shape
 struts (compression)
 a centered cube with sides
2d
 len = 2l, in each plane of cube
 strut(0,d,l)(0,d,l) strut(0,d,l)(0,d,l)
permute rotational symmetry over cube's diagonal
: strut_coords \( (0,d,l) \rightarrow (d,l,0) \rightarrow (l,0,d) , .. ) \)
 a centered cube with sides
 tendons (tension)
 \(s \cong d/l \) where s is len of tendon connect two vertices
 for len
s
:
 struts (compression)

torque:

interactive equilibrium modeling:
rungakutta
 constrain force density from network geometry
 calc nodal residuals and gradient
 update force densities and network geometry
terminate
linear least square:
Fundamental Linear Algebra
 space
rowspace
C()colspace
C(A)nullspace
N(A) contains all solution to
 dimensionality
dim r
 rank == dim C(A) == dim of C()
dim m
 dim C(A) + dim = m
dim n
 dim + dim N(A) = n
nullspace
 kernel = dim N(A) + rank = n
 dim = m
 orthogonality
 \( N(A) \cdot C(A^T) = 0 \)
 \( N(A^T) \cdot C(A) = 0 \)
 transform
 basis: nlinearindpt vectors; minimal spannning set; spans n dimension
reducible
if A(mxn) some rows linearindptinvertible
(unique solution) if A(mxm) rows linearindptoverdetermined
m>n, all m rows lin indpt, (usual case, sys not solve exactly, LS)underdetermined
m<n, infinite solutions
a toy model
refs
prestressed ankle: http://www.jbiomech.com/article/S00219290%2809%29003558/abstract
http://www.magicalrobot.org/BeingHuman/2012/10/tensegritysnakerobot https://github.com/kcaluwae/tensegrityelsimulator/blob/master/examples/cartesian_coordinates_simulator/show_6bar_18springs_minimal.py
http://graphicallinearalgebra.net/2015/06/09/matricesdiagrammatically/ http://bl.ocks.org/mbostock/7782500
Interactive Equilibrium Modeling, Lauchauer
http://fab.cba.mit.edu/classes/S62.12/people/keeter.matt/solver/index.html#Colors antimony https://getpocket.com/a/read/933453554