Generate an appropriately facetted helix or spiral around a triangle path and discover that it almost generates a tetrahedron! A "stick tetrahedron model" actually forms. It seems to conform to preliminary requirements of stitching geometric faces together (according to popular geometric software of this day).

The helix diameter in this example is about 24 times larger than the triangular path diameter. (As modeled inside the comprehensive software program used, which is formZ . It was rendered in bonzai3d which is abbreviated for speed and focus rather than to include the highly detailed tools of formZ.)

Yet this primal terahelix does fall short in several respects.  It is not clear why the helix tool used in formZ produced facetted segments which are not equal in length. (I have asked formZ tech support for an oppinion on this. I will report their explanation here, if one is forthcoming. Yet obviously, one cannot expect much support regarding a theoretical question. Also results are somewhat variable depending upon experimental setup. Increasing the helix radius in ratio to the triangle radius, does seem to improve the "precision" of the resulting "stick tetrahedron", which is actually a helix, as noted in screen-shots below.)

The image above shows some settings and data concerning the model generated in formZ.  The red object (above) is the "Wire Helix Along Path". The small black object (barely seen in the middle of the image) is the flat triangle. I understand that the nomenclature of formZ is scholarly. My own wordings span various fields of endeavor, and less formerly so.

Is there a significant mathematical relationship which "snaps" conventional helix formulae into a stick-like tetrahedron, at very specific settings? I also wonder if there might be an adjustment to the math formulae for helices, wherein a "more perfect" tetrahedron will generate. In my model, the helix segments actually varied slightly in segment length. Next image below was an earlier effort to convey what it means to form or wrap a facetted spiral around a triangular path. A smaller scale helix is used and the helix itself is shown with a variable number of turns and variable segmentation.

For years i was perplexed as to how to present the relevance of this geometric discovery of a particular relationship between the helix, the solid tetrahedron and a simple, flat triangle. It almost seemed some sort of greater discovery associated with it might transpire. Later on, making a living distracted me from my favorite studies which relate to geometric patterns and applications. The next image views the same helix object with variations of steps-per-cycle, (from differing view angles) and includes a solid tetrahedron for comparison, (in green). (The image includes variations of helical frequency and smoothness, wound around a triangle).

 

I additionally wonder if this model might relate to toroidal geometry. In that by adjusting the ratio of diameters, (path diameter to helix diameter), tours-related objects are generated. Morphing relationally, also can manifest self-interfering tori. I further visualize relationships of rings, tori and helices. This link will show an experimental concrete rendition.

This additional link discusses many of my structural uses of rings.

Hire Bo for geometric modeling.

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Enersearch was incorporated by Bo Atkinson in 1980 but never materialized financially.  A  synergy of concepts were developed and are reflected in the pages of this series.  The synergy continues as a single handed effort of Bo Atkinson, in Maine, USA.