The Fitful Flog

December 3, 2006

Stellated Curved Tetrahedron

Stellated Curved Tetrahedron on Flickr

Update: Better crease pattern and some nice hints can be found here.

Say what? Well, we must call it something and that name may be unlovely, but it is not unapt.

It’s in two pieces, the top wrapping over the bottom. Or the other way around, doesn’t matter much.

If you’ve spent anytime at all with circles, you’ve noticed that it’s very easy to fold a hex grid. Circles were built for this. But where, you ask, do the curves come from? Easy — the edge. Circle’s only got the one. Look at the CP and note where the curves begin and end. Now, fold the edge over, so it connects the two points. See? Just bend the paper a little and start pinching in the curve. As you practice this, you’ll see how much more accurate your lines will become.

The Crease Pattern

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27 Responses to “Stellated Curved Tetrahedron”

  1. 1
    christine Says:

    I owe you thanks for making me play with circles again. I love where you’ve been going with this. I can’t say I have made your rose yet, or this new creation (which makes me think of a trapped cube modular I made once), but I look forward to it.

    I started the rose, but used a soft paper which isn’t very effective. I will retrace later.

  2. 2
    Lorenzo Marchi Says:

    I finally fold it!! so difficult to lock for me but funny to make! thank you for sharing, as always…

  3. 3
    Tom Hull Says:

    You may catch some flak from your madcap use of the word “stellation.” The technical definition (which you can read here) is to take your polyhedron, extend each face to an infinite plane until they intersect, and see what you get. But most people, especially in the origami kingdon, use it to mean, “put pyramids on your faces,” which is a process that goes by the name “augmentation.” Still, “stellation” sounds a lot sexier than “augmentation,” so I know I’m talking into a stiff wind here.

    And your object doesn’t really look like an augmentation either. The mathematician in me looks at it and thinks, “Hmmm … dimpled sphere? Sphere with four 3D, double sinks? Ooo! How about “Double Non-flat Sunken Sphere with Tetrahedral Symmetry”?) Or you could go the David Mitchell route and give it a glamorous monicker like “Electra” or “Qntaira” or “Nosebleed.”

  4. 4
    oschene Says:

    Yeah, the nomenclatural aspects are a sticky wicket. I went with stellated because the flat surface tetrahedral thing does appear to a stellation of something else. And the curved faces don’t seem to describe one sphere, but an intersection of four, in a Venn zen sort of way.

    Nosebleed does have a nice ring to it.

  5. 5
    Boing Boing Says:

    Curved tetrahedron origami

    Spurred by my earlier post about Arthur Silverman’s tetrahedron-based sculptures, BB reader Philip Chapman-Bell points us to his beautiful stellated curved tetrahedron origami. He’s posted a crease pattern so you can fold your own! Link Previously o…

  6. 6
    Eric Gjerde Says:

    Philip, keep up the good work! you keep showing up on Boingboing- funny that your blog is in my morning reading material and yet you show up in the afternoon as well… :)

  7. 7
    Neatorama » Blog Archive » Stellated Curved Tetrahedron: Beautiful Math Origami. Says:

    […] Link – via Boing Boing   […]

  8. 8
    interesting things - kuratkull.com » Stellated Curved Tetrahedron: Beautiful Math Origami. Says:

    […] Link – via Boing Boing […]

  9. 9
    Pouet Says:

    Hi

    Maybe you can explain where to cut, and fold…
    I’m not sure but I think you have to make small cuts around the “pyramids” or it’s not going to fold well.

  10. 10
    oschene Says:

    No cuts — there are two halves, each a circle.

    I’ll see if I can make a more explicit CP with conventional dash-dot lines. This may take some time.

  11. 11
    Nota Mental » Blog Archive » Origamis extraños Says:

    […] Hay alguien como el/los autor/es de la página Fitful Flog que lo llevan al extremo. Unos ejemplos: el modelo de este en concreto lo encontrarás en esta entrada […]

  12. 12
    Nat Edgar Says:

    IF the surface has four circular dimples, and IF each dimple touches all of the others along a line from the edge to the centre, and IF you omit the pyramids, then the surface would be a nice model of Steiner’s Roman Surface, a surface wll known in topology, which has only one side (see Wikipedia).

  13. 13
    oschene Says:

    Wow — that might take some work.

    I’d have to do it in the morning — making an impossible surface will require a lot of coffee. Oh, wait — maybe you mean like this:

    Tetrahedral Bud

    It’s considerably easier to make.

  14. 14
    Nikhil Datta Says:

    Maybe a video on YouTube of you actually doing it ?

  15. 15
    Technology Insight » Blog Archive » Curved Tetrahedron Origami Says:

    […] If you’ve spent anytime at all with circles, you’ve noticed that it’s very easy to fold a hex grid. Circles were built for this. But where, you ask, do the curves come from? Easy — the edge. Circle’s only got the one. Look at the CP and note where the curves begin and end. Now, fold the edge over, so it connects the two points. See? Just bend the paper a little and start pinching in the curve. As you practice this, you’ll see how much more accurate your lines will become. [via] – Original link […]

  16. 16
    Taggy Says:

    Wow… I’d love to know *how on earth* you make this! Any pointers?

  17. 17
    Nat Edgar Says:

    Steiner’s Roman Surface
    Hi Oschene,
    I think you are almost there !
    Take a look at the Wikipedia reference.
    Nat Edgar

  18. 18
    thomas Says:

    Wonderful, maybe 2 or 3 more pictures would help me see how you start after the creasing is done! :)

  19. 19
    gennessee Says:

    This is lovely. I amazed myself by folding my way through most of it, but I’m stuck at how to attach the halves together. I think I may need to make additional folds to the surfaces that wind up inside the model, but I’m not sure where.

  20. 20
    Stephen Ross Says:

    Hey guys, I’m new to origami. In fact, this is my first attempt. How did I do?

  21. 21
    Stephen Ross Says:

    oops, forgot to mention there are pics on my Xanga Photoblog:

    http://photo.xanga.com/StevePoOo/albums/a3d0f504253834

  22. 22
    CDS Says:

    It looks like I’m the only one who can’t figure this thing out. Anyone want to post step-by-step pics to throw me a bone?

  23. 23
    Paul Hindess Says:

    Pointed in the direction of this by a kind (or perhaps malicious) brother. Having read nothing else here yet, I’m glad I decided to pre-fold all the lines and score all the curves carefully. The comments from others were helpful. I, too, came to a temporary impasse where I was saying to myself “this isn’t quite going to work without a few minor cuts”, but somehow persistence overcame this – in ways I could not possibly describe (for the benefits of others) even if I wanted to.

    I also thought “how the heck do I get these two halves together” (quite some time after the thought, “wow, I’ve done it . . . but that’s only HALF the model!!” and proceded to attempt the bottom half). Getting the halves together was a matter of trying it tentatively to see how they were meant to fit and then reinforcing creases in the right direction before making a proper attempt. It seemed necessary to me to accept that some of the major folds were going to be temporarily compromised whilst trying to get one piece inside the other, but all these folds came good with patient effort.

    An amazing invention this model and no doubt the result of numerous prior experiments into curved surface origami (that I will have to check out). I’m convinced that some alternative templates could be produced that might result in a final model that looks more presentable from any angle. . .

    One thought that crossed my mind is it might be possible to make the same model with more than two (smaller) modules (perhaps 4 and perhaps all identical). The hope would be that these might lock together in such a way that module 1 goes partly inside module 2 which goes partly inside module 3 which goes partly inside module 4 which goes partly inside module 1. This might produce a model that naturally pulls itself together a little more (though I have to say I am surprised and impressed that even my modest attempt at this is staying together so well!!!

    Thank you for an enjoyable evening’s entertainment!

    (-:

  24. 24
    Paul Hindess Says:

    PS Is gluing a good idea on these models? Or is that considered cheating?

  25. 25
    thomas Says:

    any tips on the tetras?
    The lines in the “bowls” are hexagons, I don’t see hot they can make the tetra.

    Maybe someone can post a picture of the two halves separated. That would help.
    (in 12 Megapixel, at least)!

  26. 26
    Cissy Says:

    Thomas — the hexagons turn into tetrahedron by joining two of the sides together. That makes the unit start to bend to produce the curve. I don’t have a photo, and I just made the bottom the other night for the first time. Working on the top still.

    Can anyone give me a clue on how to keep the tetrahedron together?

  27. 27
    oschene Says:

    There will be paper left over at the edges and it will wrap into the indentations on the other piece. I find it easier to wrap the top over the bottom, but I think it will work either way.

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