Untwisting the Tetrahelix

trial optimal rho r len d one-hop two-hop pitch inradius minmax ratio (%)



Untwisting the Tetrahelix

This is an interactive 3D simulation. To change the view, place your mouse in the view area, and hold and "drag" the mouse to rotate the image. The mouse wheel or a drag on your trackpad should zoom you in or out.

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.


Tetrahelixes are cool. R.W. Gray has a great site about them. Further motivation is described in an academic paper we are preparing that will be linked in draft form here.

The math on this site creates three novel things:

  1. The ability to untwist adn retwist the Boerdijk-Coxeter tetrahelix (BC helix) in an way that is smooth and optimal in terms the difference between the longest member and the shortest member.
  2. A new structure, the equitetrabeam, which is a fully untwisted BC helix.
  3. Both the mathematics and an interactive GUI for designing tetrahelices to match a desired pitch.
This is valuable because architects, engineers, and roboticists can now use the beautiful and regular structure of the BC helix. In fact, this application was driven by the Tetrobot project. The BC helix happens to have an irrational pitch, which means that an architect of game designer who wanted to use this beautiful structure in a periodic and symmetric way could not do so. However, using the software here they can use a slightly irregular but still in a sense optimal tetrahelix with a pitch of exactly 12 tetrahedral to a singel revolution, for example. Robotocists such as myself can dynmaicall make a tetrobot untwist, in order to lay flat on a plane, but still build the robot out of actuators of exactly the same size.

How to Use

The site uses the so-called THREE.js "orbit" control. Click on the scene and drag and you will find your self rotating, always looking at approximately the center point. Moving the middle mouse wheel or the Mac scrolling pattern will zoom you in or out.

Accessing the Math

This site is coded in the file index.md, which has the THREE.js rendering code and code to to use the fundamental math. The math from our paper, however, is in tetrahelix_math.js. You are welcome to use tetrahelix_math.js in for your own purposes, such as in a computer game or to design your own physical structures.

Running the Test

Although it is currently a work in progress, my goal is to make tetrahelix_math.js testable via Mocha. You will need to install npm, node, and mocha for this to work.

I am not a node expert, but basically I did:

npm install -g browserify
npm install -g mocha
In order to use the tetrahelix_math module in the browser. I build the browser ready file with:

browserify tm_shim.js  -o bundle.js    
In order to run the tests, in the main repo, type:


Thanks and Open-Source software used

This page is using a lot of open source software, in particular Three.js. However, special thanks to Ricky Reusser for a simple little Newton-Raphson solver.

trial optimal rho r len d one-hop two-hop pitch inradius minmax ratio (%)

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Creative Commons License
This file and all PubInv materials by Robert L. Read by default is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.