The tetrahelix of Fuller's Synergetics consists of face bond regular tetrahedra. The mathematics for this spiraling structure is quit interesting. Dispite the tetrahelix being composed of regular tetrahedra (the "simplest" polyhedron), I have not been able to find a simple way to calculate the information for the tetrahelix.
From the Zheng paper (see References below): "The tetrahedral helix is also called the 'Bernal spiral' in association with discussions of liquid structures in the physics literature."
The vertices of the regular tetrahedra of the tetrahelix all lay on the surface of a cylinder. Let us visualize this cylinder to be along the z-axis.
The radius of the cylinder will be
where EL is the edge length of the tetrahedra used to build the tetrahelix.
Let us put a vertex (call it V0) of one of the tetrahedra on the x-axis. That is
Then the next vertex of the tetrahelix (V1) will be at the coordinates
where theta is the angle around the z-axis and is given by
and where h is the distance in the z-axis direction and is given by
In the above figures, the yellow band connects a vertex to the "next" vertex, while the distance h is the distance between the 2 blue bands around the cylinder.
In general, the coordinates for the vertices of a Counter Clockwise tetrahelix Vn (n = 0, 1, 2, 3, ...) are given by
The coordinates for the vertices of a Clockwise tetrahelix Vn (n = 0, 1, 2, 3, ...) are given by
Note that cos(theta) = -2/3 and that sin(theta) = sqrt(5)/3. You can calculate exact expressions for the vertex coordinates by using these relations together with the following trig identities:
One of the reasons that deriving the above information is difficult is that the axis of symmetry of the cylinder (the axis through the center of the cylinder) does not pass through the center of volume of the tetrahedra. The distance from the z-axis to the tetrahedron center of volume is given by the equation
Therefore, all the Tetrahelix cylinder axis of symmetry pass tangentially by a sphere of radius (sqrt(2)/10)EL centered at the Tetrahedron's center of volume.
Here is a sequence of images to help visualize the transformation.
First, consider the Tetrahelix to be made up of a number of 3-Tetrahedra units. By folding up each of these units, the Tetrahelix is reduced.
We can fold up a 3-Tetra unit as shown in the following figures.
Not that only the C-to-D edge needs to change its length. All other edges of the Tetrahedra remain the same.
At this point, the 3 Tetrahedra are very close to defining an Octahedron. They actually do form an Octahedron as the vertices "A" and "B" are brought closer togther.
When vertices "A" and "B" are brought together, a double Tetrahedron is formed. That is two Tetrahedra face bound togther. (This figure only shows one of the 2 Tetrahedra. The other is hidden behind this one.)
The transformation from the 3 Tetrahedra to the Octahedron Fuller calls the "Richter Transformation". See Color plate 6 in Fuller's book Synergetics. I am not aware that Fuller continued the transformation described here to transform a Tetrahelix.
However, if we define "intersect" to mean that the 2 Tetrahelix pass through a common Tetrahedron and thier symmetry axes pass by each other at 90 degrees, then this is true.
See all the detail on the next Tetrahelix web page.
Here is a list of references. However, I did not use any of these references for my calculations.
I have not see the following references. These references are given in the Zheng paper.
Usage Note: My work is copyrighted. You may use my work but you may not include my work, or parts of it, in any for-profit project without my consent.