Joining two segments

My how time flies…

The join of two topological spaces A and B is basically the space of all line segments between every pair of points.  As a nice embedded, visceral example of this, a tetrahedron may be viewed as the join of two skew line segments.

Indeed this is the illustration on Wikipedia.

It also arises naturally when you think about grid diagrams of knots.  Remember this from way back?

Side Trefoil Tetra Torus

Thinking along those lines, you might consider discretizing it a bit…  Instead of taking the join of two entire segments, just take the join of n points along each segment.

Tetrahedron Join

Here I’ve used n=7.   And that gives us the complete bipartite graph K_{7,7} naturally embedded as a subset of the tetrahedron.  Moreover, any grid number 7 knot is a cycle in this particular embedding of the graph.

While it’s a rather simple object, I’ve nonetheless found it to be a rather pleasing one.  Lots of emergent rhythms.

Tetrahedron Join

Tetrahedron Join

Tetrahedron Join

Kinda gotta hold it and spin it around to really appreciate it, so print one out if you’d like.   I’ll talk about some of the others models there whenever I eventually get around to it…

Of course I would be amiss without plugging Segerman’s fantastic body of  W O R  K with 3D printing.  If you haven’t seen it before, you’re missing out.

 

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~ by Ken Baker on December 20, 2013.

8 Responses to “Joining two segments”

  1. I should look into printing these joined segments – discretely – in Three.js

    • I looked. See my previous comment.

      Now you can have a look

      http://jsfiddle.net/theo/4AAS6/

      • Hey alright! That lets anyone rotate it around to see the rhythms.

        Perhaps you could make sliders to vary the numbers of discrete points being joined. After that maybe also vary the lengths of the two segments and their distance.

        The model shown was actually the output of a parametric design using Grasshopper with Rhino.

        At one point in time I made a Mathematica manipulation for this thing too… But I like how you’ve got it in the browser.

  2. Your wish list is my command line:

    Segments Joined Discretely R2

    Instead of varying lengths of segments – which would require a redraw – I added Scale XYZ. In this instance I believe the two achieve the same effect.

    **

    Rhino with Grasshopper is very strong – and very expensive if you are not in academia.

    Mathematica: ditto – in spades. << is 'in spades' an imaginary number?

  3. Ha! I thought you had found super-cool rendering software that emulates photographs of physical models.

  4. […] them out into the audience.  It was all fun and games until someone lost an eye. (The one from that last post has pointy […]

  5. Very intriguing…. great post! :D

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