We want to prove that the cylinder axis of the Tetrahelix passes through the Tetrahedron’s triangle face at triangle face coordinate (7, 3).
The radius of the Tetrahelix cylinder is given by
where EL is the edge length of the Tetrahedra making up the Tetrahelix.
The Tetrahelix can be positioned so that its (x, y, z) coordinates are given by the equation
where (approximately 131.8103149 degrees) and
Using the equations
it is easy to calculate values for and .
n 


0 
1 
0 
1 


2 


3 


The nth Terahelix vertex will then have the coordinate
For the “first” 4 vertices, with EL=1, we get
What (7, 3) means is that we travel 7 (out of 10) units along one edge and then 3 units (out of 10) parallel to another edge of the triangular face.
Define V_{a} to be the vector from vertex V_{0} toward vertex V_{1} but which is only 7 (out of 10) units in length. Then define V_{b} to be the vector from vertex V_{1} toward vertex V_{2} and which is 3 (out of 10) units in length.
Then the vector to (one of) the face point (7, 3) is given by V0 + Va + Vb. If this vector has no x or y components then it must lay on the zaxis, which is the symmetry axis of the Tetrahelix.
We calculate V_{a} to be
And we calculate V_{b} to be
Then the vector to the point face coordinate point (7, 3) is given by
This lays on the zaxis. So the symmetry axis of the Tetrahelix does pass through the triangular face coordinate (7,3).
There are 3 such face points depending on which of the 3 vertices of the triangular face is used to begin measuring the 7 (and then 3) units of length.
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