A mathematical investigation of the Tetrahelix and Boerdijk-Coxeter helix, which provides a new formulaic way of producing a continuum of untwisted tetrahelices.

All of the code on this site is released under the GNU General Public License, and I hope you will reuse it.

You can make this page render a wide variety of 3D objects. Pressing the "circle", "sine", "spiral", "cone" or "helix" buttons will render those objects based on an algorithm for following parametric curves.

To create a new parametric curve at present, you would have to hack the code of this page, which we encourage you to do.

By enterring in the text area a short snippet of JavaScript defining an anonymous function which takes an integer and returns either -1, 0, 1, or 2, you can produce an unlimited number of shapes. Just enter such javascript and hig the "GO!" button. -1 means "stop." 0, 1 and 2 desrcibe the face to which to add the new tetrahedron.

Perhaps the simplest recipe is a purely periodic generator. An example is:

```
(i) => { return i<20 ? [ 2,1,0,1,2][i%5]: -1; }
```

Which simply cycles through an array of 5 values. You can change the numbers (always keeping them 0,1, or 2---go ahead ant try it.

This is a flattened view of a tetrahedron in the 3D generator, as viewed from the outside.- The base (face 3) of each tetrahedron is placed against a face of the previous tetrahedron, as determined by a generator.
- Edge 1 of the base of the new tetrahedron is matched against an edge 1 of the previous. Depending on which face it’s placed against, this may be an edge along the base (faces 1 and 3), or one adjacent to the apex (faces 0 and 2).
- Edge 0 of each tetrahedron is matched up with edge 2 of the tetrahedron it is placed against.
- For a chain of direction 1, no matter what the edge lengths are, the entire chain shares a single edge 1 along the base. This causes tight turns in the resulting constructions.

THIS IS A CLOSE APPROXIMATION TO A TORUS:

THIS IS A VERY NICE APPROXIMATION TO A HELIX WITH A TIGHT CAVITY

THIS IS LIKE K = 9 BUT SIGNIFICANTLY MORE OPEN

FROM K = 9 to K = 18 you produce helixes of various torsion more or less smoothly.

THIS PRODUCES A NICE BROADLY OPEN HELIX

THIS ALSO PRODUCES A NICE BROADLY OPEN HELIX, EVENLARGER

“Nearly Flat Sawtooth”

Alternate a/b = true; 2b = .95

These numbers (put into the same was as above for 2b, produce different polygonal toriods.)Triangle: 0.29

Square: 0.648

Pentagon: 0.835

Hexagon: 0.95

Setting 2b to 1.414 produces a straight "beam" different than the Equitetrabeam.

In fact, this method is a basic mechanism for producing toroidal forms are any lengh, which is very useful.

Modify edge lengths thusly produces a square "box beam":

This setting produces the perfectly triangular "Equitetrabeam"0b = 2a = 1.1547 (2/sqrt(3)

This setting produces a funny "sqare-hole helix" whose central cavity is square: 2a = 0b = 1.1547, alternate ab = true

This produces a similar "octagonal helix": 1a = 1b = 1.1547, alternate ab = true