I have a new interactive web page for exploring the 12 Tetrahelix that pass through a single Tetrahedron.

New interactive web page to show Octa- Cubocta- and Icosa helix within the Tetrahelix.

A new web page showing that the Tetrahelix vertex coordinates can be written as rational numbers. (From a comment by Gerald de Jong.)

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

r = (3 sqrt(3) / 10) EL

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

V0 = (r, 0, 0)

Then the next vertex of the tetrahelix (V1) will be at the coordinates

V1 = (r cos(theta), r sin(theta), h)

where theta is the angle around the z-axis and is given by

theta = arccos(-2/3) (approximately 131.8103149 degrees)

and where h is the distance in the z-axis direction and is given by

h = (1/sqrt(10)) EL

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

Vn = (r cos(n*theta), r sin(n*theta), n*h)

The coordinates for the vertices of a Clockwise tetrahelix Vn (n = 0, 1, 2, 3, ...) are given by

Vn = (r cos(n*theta), - r sin(n*theta), n*h)

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:

cos(n*theta) = cos(theta)*cos((n-1)*theta) - sin(theta)*sin((n-1)*theta)
sin(n*theta) = sin(theta)*cos((n-1)*theta) + cos(theta)*sin((n-1)*theta)

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

dist. = (sqrt(2)/10) EL

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 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.

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