6-Strut Tensegrity Sphere and its Transformations

Greg Frederick

The 6-strut tensegrity sphere is the spherical, or tensor equilibrium phase of the tensegrity tetrahedron and its dual, which is the same tensegrity tetrahedron but with a 90° rotation and with vertices oriented counter-clockwise to the other.

6-strut tensegrity sphere.
The 6-strut tensegrity sphere.

The dual of the regular tetrahedron is conventionally described as the same tetrahedron. Its dual is here described as the negative of the other.

Two tetrahedra, one rotated at 90° to the other, sharing a common center and sized so that the midpoints of their edges cross at the same point.
The positive tetrahedron (blue) and its dual, the negative tetrahedron (pink). Note that the vertices in one are the faces in the other, and vice versa.

Rectification of either (or the intersection of both) produces the regular octahedron.

The intersection of a positive and negative tetrahedron (left) results in the regular octahedron (right).
Rectification of either (or the intersection of both) a positive and negative tetrahedron (left) produces the octahedron. Note that the edges of the two tetrahedra cross at the vertices of the octahedron.

The 12 edges of the octahedron describe three intersecting squares. Connecting alternate vertices of these squares describes three intersecting lines aligned with the distribution of the struts in the 6-strut tensegrity sphere. For consistency, the three lines are here described as three 2-sided polygons, all sharing a common center and each intersecting the other at 90°.

Three intersecting squares (left) and three intersecting lines (right) sharing a common center and oriented at 90 degrees to each other and describing the regular octahedron.
The 12 edges of the octahedron describe three intersecting squares (left). Connecting alternate vertices describes three intersecting lines, or what we’re calling 2-sided polygons (right).

Each of the three intersecting lines (or 2-sided polygons) of the 6-strut tensegrity sphere consists of two struts and the triangular cross section of the valley formed by the tendons connecting each strut-pair to its dangler. The dimensions of this triangular cross-section are illustrated below.

  • n (number of sides of the polygon) = 2
  • δ (interior angle of the polygon) = 90°
  • S = strut length
  • d (vertical distance from polygon’s edge to strut) = S×tan(δ/2)/4
  • d’ (height of the triangle formed by the strut ends and the polygon’s vertex) = d×cos(δ) = 0
  • h (height of the triangle formed by the strut ends and the midpoint of their dangler strut) = (d+d’) = (d+0) = d.
  • gap (distance between strut ends) = S×sin²(δ/2)/2
  • dip (distance from strut end to midpoint of dangler strut) = S×sin(δ/2)/2
One of the three 2-sided polygons (and its strut pair) that together form the 6-strut tensegrity sphere, showing the cross section of the tension valley formed by the four tendons of each strut pair and their dangler.
Dimensions of the cross section of the tension valley formed by the four tendons of a strut pair and their dangler in the 6-strut tensegrity sphere.

In its spherical, or equilibrium phase, the dimensions of the cross section of the tension valley created by the tendons reach their maximum. In the polyhedral phases described below, δ approaches to 0°, and both the gap and dip go to zero (0).

The six unit-length struts of the 6-strut tensegrity sphere in relation to the unit-length edges of the three intersecting 2-sided polygons (left) that define its spherical or equilibrium phase (right).
In its spherical, or equilibrium phase, the struts are at their maximum distance from the unit edges of the three intersecting 2-sided polygons (left) of the 6-strut tensegrity sphere (right). For clarity, the 2-sided polygons are shown as transparent rectangles.

Transformation of the 6-Strut Tensegrity Sphere into Positive (or Clockwise) and Negative (or Counter-Clockwise) Tetrahedron.

From the spherical phase, the 6-strut tensegrity sphere is reduced to the tensegrity tetrahedron by sliding the strut ends along the tendon toward opposing ends of their dangler, as illustrated below.

Moving the struts in one direction results in the positive tensegrity tetrahedron…

Detail of a tensegrity sphere showing the struts moving clockwise along the tendons toward opposing ends end of their dangler.
In the transformation from 6-strut tensegrity sphere to tensegrity tetrahedron, the strut ends move along the tendon to opposing ends of their dangler.

…and moving the struts in the opposite direction results in the negative tensegrity tetrahedron.

Detail of a tensegrity sphere showing the struts moving counter-clockwise along the tendons toward opposing ends end of their dangler.
In the transformation to the negative tensegrity tetrahedron, the strut ends swap positions at opposing ends of their dangler

The surface of the 6-strut tensegrity sphere describes eight triangular tendon-loops. In the transition to the tetrahedron, four transform into the tetrahedron’s four faces, and the other four transform into the tetrahedrons four vertices. Their roles are swapped in the transformation to the negative tetrahedron.

A positive tensegrity tetrahedron, with three struts coming together at each vertex in a clockwise orientation (right), and a negative tensegrity tetrahedron, with three struts coming together at each vertex in a counter-clockwise orientation.
The oscillation of the 6-strut tensegrity sphere produces both the clockwise (or positive) tensegrity tetrahedron (right), and the counter-clockwise (or negative) tetrahedron (left).

The oscillation from the tensor equilibrium (spherical) phase, to the extremes of the two tetrahedron phases, may be stopped at any point and the model will hold its shape. As long as the elasticity and tautness of the tendons is maintained, the models do not show a preference for one state over another.

6-strut tensegrity oscillating between the positive and negative tetrahedron.
6-strut tensegrity sphere oscillates smoothly from its spherical phase, between the extreme of its polyhedron phases (the positive and negative tensegrity tetrahedron) all the time holding its shape and showing no preference for one phase over another. .

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