By on August 4th, 2013


0352 Part of Fire Series by Erik Demaine

Three views of Fire Series [0352], computational origami by Erik Demaine and Martin Demaine            via

Since the word computation harks back to the 1400s you might reason that the modern efflorescence of “computational” arts and sciences was based in mathematical calculations.  Half right.  Other half is the extra-human capacity of computers which now take those calculations to an nth degree.

Mathematically possible but previously unattainable numbers are now being plumbed. Computational has come to mean math-based at a new and as yet unfamiliar scope: titanic. Computational origami, computational botany, computational medicine.  Name your subject and slap Computational on the front.  Now you’re an explorer who asks where you can travel in your field powered by rocket math.  Boundaries in many fields will change.  Terra incognita: reshaped.

What’s the nth degree?  For our purposes it’s a number that requires a computer’s power to investigate. Say you want to add 1 + 1 + 1 + 1 + 1 + 1 + 1…  If you hypothesize doing this six trillion times you don’t need a computer to tell you the answer is six trillion.

How about the old tale of the rice grains and the chessboard?  The king is owed a lot of gold by an enemy but he offers to let the debtor off the hook thus: he drops a grain of rice on the 1st square of a chessboard, then two on the 2nd square.  All the debtor must do is to continue the pattern until the last square is filled.

Will the debtor be fool enough to agree?  Here is an illustration showing the one grain as it doubles around the board.  The grains will ultimately total to 18,446,744,073,709,551,615.  Eighteen quintillion, four hundred forty six quadrillion, seven hundred forty four trillion, seventy three billion, seven hundred nine million, five hundred fifty thousand, six hundred fifteen — reached by an exponential growth that starts at 2 grains and piles up to 263.



Grains and Chessboard Problem        via Wikipedia       [More about the red line at the bottom of this page.]

Think of having to pay for the oxcarts alone.  Yet this is a number first recorded by the Persian poet Ferdowsi before the year 1011.  Recorded.  Already humanity had developed sufficient mathematical tools to solve the problem.  (How the poet got it to scan inside his poem I don’t know.)

Consider this when you approach the new computational arts and sciences.  They’re truly pushing into territories where humans have never been able to nose around before.


Computational Origami is for modeling the ways in which various materials, including paper, can be folded.  [see more below]

Erik Demaine and his father Martin Demaine, both faculty at MIT, create Curved Crease sculptures which are shown by galleries and museums.

5a_medium 550 px

Pushing Curves to the Limit, #5 by Erik Demaine and Martin Demaine     via


Earthtone Series, [0262] by Erik Demaine and Martin Demaine       via


Simons Series [0238] by Erik Demaine and Martin Demaine     via



From Kentucky Series by Erik Demaine and Martin Demaine     via

Anni Albers, pioneering textile designer and wife of Bauhaus luminary Joseph Albers:

“Being creative is not so much the desire to do something as the listening to that which wants to be done: the dictation of the materials.”             Anni Albers quoted on improvisedlife

4b_medium 600 px

Pushing Curves to the Limit, #4,  by Erik Demaine and Martin Demaine          via

Each piece is a combination of between two and four units, where each unit is hand-folded from one circle of paper with a circular hole and hand-scored concentric circular creases. The pieces are combined by joining a few key points together; the rest of the shape is formed from the natural equilibrium of the paper.         Erik Demaine, Pushing Curves to the Limit

The dictation of the materials and the natural equilibrium of the paper.  Yes.  The origamist’s choice of paper influences the form and feel of every finished piece.  Does it slump gracefully? Hold a pose with military starch?  Is it translucent, thick and grainy, Kraft-paper brown, flecked with mylar sparkle?

Imagine how future computational origamists will use math and eye to incorporate patterned paper into an even higher order design.


Hexagon wallpaper design, Cole & Son     via


Michael Hansmeyer’s computational architecture inhabits a whole different realm.  Compare the wallpaper hexagrams just above with his Stacked Hexahedra below.

Computational Subdivided Platonic Solids - Michael Hansmeyer 05 b

Stacked Hexahedra by Michael Hansmeyer         via

This project begins with the most primitive forms, the platonic solids, and repeatedly employs a single operation – the division of a form’s faces into smaller faces – until forms of an astounding complexity are produced.       Michael Hansmeyer

Computational Subdivided Platonic Solids - Michael Hansmeyer 01 600

Dodecahedron by Michael Hansmeyer          via

For the first time, complexity is not an impediment to design and fabrication. Rather, it is an opportunity that is waiting to be explored. For years, it was information technology that constrained architects. Arguably, this relationship has reversed: it is now architects who are constraining the possibilities of information technology.     Michael Hansmeyer

Computational Subdivided Platonic Solids - Michael Hansmeyer 04  600 px

Tetrahedron and Hexahedron by Michael Hansmeyer       via

These last two by Hansmeyer are obviously from other series.

pavilion4  600 px

Pavilion 4 (Catmull-Clark subdivision), by Michael Hansmeyer        via

1-10 Prototype Gilded by Michael Hansmeyer

1:10 Prototype Gilded by Michael Hansmeyer            via

Hansmeyer’s work brings neo-Gothic and neo-Gaudi structures closer to our built reality.


look further

Red line on the chessboard diagram 

  • •  “Second half of the chessboard.”  In technology strategy, the second half of the chessboard is a phrase, coined by Ray Kurzweil, in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization’s overall business strategy.  
  • •  While the number of grains on the first half of the chessboard is large, the amount on the second half is vastly (232 > 4 billion times) larger.


  • •  Hakīm Abul-Qāsim Ferdowsī Tūsī wrote the Shahnameh, Book of Kings, an epic poem and masterpiece of Persian literature.

Relevant rice

  • •  MASS MoCA Installation Comprises more than 800 Million Grains of Rice, via
  • 875,000,000 grains… will be used to dramatize the enormity of the population of the Americas and the miniscule place of one person in it. (All the rice will be recycled, and will re-enter the food chain upon completion of the performance.) Far from a static display, this landscape will constantly change as performers from the UK artists collective pile, measure, sort, and re-pile the rice to represent different comparative statistics, population changes, and demographic anomalies.

Computational origami

  • •  (continued from description above)The principles of geometry were first applied to origami around the mid-twentieth century, when Japanese physicists and mathematicians began to formulate axioms… that explain how folding creates three dimensional objects from a flat material. Humaiki Huzita, an Italian-Japanese mathematician, developed a sequence of six increasingly complex origami axioms that describe, at the most basic level, how any two points on a flat surface can be connected in a single line fold, and at the most complex level, the ways that four points on a flat surface can be related.Computational origami has been used to create complex paper objects, such as insects, that were once thought to be beyond the medium’s capacities. In addition… how to most effectively fold a roadmap, an airbag, and computer processors. The latter purpose was one of the original driving forces behind the development of computational origami: researchers believe that by folding processors most efficiently, they can fit the maximum amount of information into the smallest possible area.    Margaret Rouse, via WhatIs.
  • •  Robert J. Lang Origami  Generous and broad information on computational origami.
  • •  Robert J. Lang Lectures and Workshops

Erik Demaine

Anni Albers

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7 years ago

I can’t help but see Nodewerk here. Amazing to see such designs in 3D. How big are they?

Terry Hale
Terry Hale
7 years ago
Reply to  tricia

I was thinking Nodewerk, too, but it would take a lot of CPU power to create some of these. Ice cream, Chinese fans and frosting were others that surfaced. 😀 The *-hedra are stunning. I want to explore every surface to see if I can get the whole complexities to fit inside my mind, building it a layer at a time, not unlike a 3D printer (which is another thing I thought of), until it is firmly fixed and re-callable at any time . A sadly impossible task, but the appreciation is sure fun, and the pretty pictures are always… Read more »

Terry Hale
Terry Hale
7 years ago

Sloan, your dry yet savory and jaw-dropping, guffaw-inducing humor is such a treat. It tempers so well with all the information you present. And the presentation is done in such a good informative way! LOL! How’s that for redundant? I mean to that your teaching sessions (that’s what they are to me, anyway) are so personal, like you were sitting there across from me explaining things in a natural, conversational way that makes it easier to absorb. Your passion shows and makes it so interesting, even beyond the appeal of the content itself. The grains of rice offer is the… Read more »

Terry Hale
Terry Hale
7 years ago

Turns out my “couple of days ago” in my computer time was July 23rd. I really like Chrome’s history search feature! 🙂