Putting It Together: An 8th Principle for Creating Geometric Art

A few weeks ago I wrote a post outlining seven principles for designing geometric art, illustrating those principles by “reverse engineering” an interesting model to see how I could design new works based upon it.  (For those who are interested, I’ve continued to explore those variations, and will reveal the fruits of those investigations in an upcoming post soon!).

Since then I realized that I neglected to mention at least one more important principle, because it didn’t arise in that context:

Principle 8: Combine known forms and techniques in a new way

I recently designed a new piece that illustrates this perfectly.

The Starting Point

My starting point for this project was a piece of pop-up kirigami (cut paper) that I designed and created many years ago:

Kirigami "Fractal Cubes"

(Sorry about the picture quality – my wife is on vacation with the good camera so I had to use my iPhone!)

This piece was an exploration of fractal cube stacking, and has always been one of my favorites. (Hey newsletter folks — anything here look familiar?!) :-)  I had always noticed that the envelope of all those stacked cubes — especially if one were to trim away all the extra flat paper on the sides — looked a lot like one quarter of an octahedron, but had never thought through how I might capitalize on that. Now, hold that thought for a moment…

The New (Old) Idea

As I discussed in another post two months ago, I recently attended an origami convention, which exposed me anew to many basic origami forms that I had not worked with actively in many years, including the traditional origami bases. One of these is called the “Preliminary Fold” base (because it is the starting point for several other classic bases), and looks like this:

Preliminary Base (flat)

Preliminary Base (folded)

The New Combination

So, recently returned from the conference, I looked at my old kirigami model with new origami-sensitized eyes and realized that my old model (again, with all that extra flat material on the sides removed) was exactly 1/4 of a preliminary fold! Combining four of my model on the four quarters of a full preliminary fold would make the complete octahedron that had always been implied in the original model. Once I had the flash of insight about how to combine these two concepts, generating the pattern for the full model was fairly straightforward:

Fractal Cubes Octahedron - Flat

The Result

And so, I proudly present (at various stage of  cutting and folding, the result of applying this principle: my newly designed “fractal cubes octahedron”:

Fractal Cubes Octahedron - partially folded

In this view you can clearly see two copies of the original model, laying on their side.

Fractal Cubes Octahedron - mostly done!

Getting all the folds to fold the right way is a delicate process — almost there!

Fractal Cubes Octahedron - Complete (angle 1)

Fractal Cubes Octahedron - Complete (view 2)


There are some minor problems with this (dare I call it “preliminary”?!) version, which I hope to correct in a future version, but all in all it came out pretty nicely.  Two fairly simple ideas, combined in a new and interesting way, yielded a much richer and more interesting piece of art. Please let me know what you think, and especially if you’ve ever used this principle in your own work.


About Phil Webster

Phil is the creator of GeometricArts.com. You can reach him on the Contact page.

6 Responses to Putting It Together: An 8th Principle for Creating Geometric Art

  1. avatar
    Barak September 22, 2011 at 3:20 pm #

    I love the photo of the partially folded model! That really is escheresque. I daresay the final version looks overwrought next to it.

    I’m sure the photo of the final product don’t do it justice. I need a stereogram. And, alas, the the edges of the final model look a little bent. Is that distortion of your lens or is it real? Is there a way to reinforce the edges before beginning folding to prevent the bend?

    • avatar
      Phil Webster September 23, 2011 at 7:39 am #

      Thanks for your thoughtful comments! You’re absolutely right that 2D photos don’t capture it well. Definitely giving some thought to adding short videos in the near future to give a better sense of it.

      And yeah, unfortunately those bends are real. :-( Those outer edges (which were the inner diagonals in the flat diagram) were a little too thin and got bent along the way. In my next version I plan to address that by leaving a little more material there and approaching the folding sequence in a different way so that those areas get folded later and don’t get put under as much stress. Reinforcement is an interesting idea, too…

  2. avatar
    Barak September 22, 2011 at 3:24 pm #

    I should add that I find this particular exploration very exciting. I’m sure there is more to find here on an artistic level by doing less or more and recombining things. I like the contrast of the flat paper and the 3d fractal.

    • avatar
      Phil Webster September 23, 2011 at 7:39 am #

      Agreed – I think there’s a whole potential family of shapes here worth exploring.

  3. avatar
    ET December 7, 2011 at 12:10 am #

    This reminds me of our recent rendering: http://www.youtube.com/watch?v=wmEcZ3H6pGc

    • avatar
      Phil Webster December 7, 2011 at 12:41 pm #

      WOW! That is truly mesmerizing, thanks so much for sharing with us! – Phil