Just Twist, About Minimal Origami Models Based on Polyhedra Structure
Year: 2011 Authors: Krystyna Burczyk; Wojciech Burczyk
Core claim
Unfolded paper already provides flaps, so minimal twirl origami models can be constructed without any creases.
Topics
minimal origami, modular origami, twirl models, polyhedra structure
Domains
geometry, polyhedral combinatorics, parameterized model families, origami, paper art, mathematical art, design of forms
Methods
twist-based assembly, paper tension technique, shape parameter variation, module-free construction
Media
paper sheets, paper strips, folded modules
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Bridges 2011: Mathematics, Music, Art, Architecture, Culture
Just Twist, About Minimal Origami Models Based on Polyhedra Structure
Krystyna Burczyk, Wojciech Burczyk ul. Konwaliowa 22 32-080 Zabierzów, Poland E-mail: burczyk@mail.zetosa.com.pl
Abstract
Origami is usually recognized as an art activity that requires a lot of folding to convert a plain sheet of paper into a piece of art. As mathematicians we asked a question what is the lower number of crease lines that create a nontrivial origami model. And we have found that the answer is: zero lines.
The paper presents a method that leads to astonishing origami models without any single crease line.
Background
Origami, the art of folding paper into a piece of art, meets mathematics in many places. An origami purist says that only a model starting from a square piece of paper and made without cutting and glue is a true origami model, however such approach is not justified by tradition. Modular origami is based on two-step approach. First a sheet of paper is folded into a module according to purist rules. In the second step many modules are assembled without glue into a final model, usually based on a polyhedra structure. A single module may correspond to a vertex of a polyhedron (vertex module), an edge of a polyhedron (edge module) or a face of a polyhedron (face module). Most modular models can be described by a set of parameters, see [7] that generate broad families of similar, but sometimes surprisingly different origami models.
Below, we will discuss effects of two parameters: shape of a paper sheet and location of flaps on a final model in case of a minimal edge module.
Just Twist
There are many different techniques to join modules without glue into a stable origami model, usually based on the friction of paper. Herman van Goubergen introduced a paper tension technique based on flaps of a module twisted into conical spirals [8]. We have developed his idea for the last 10 years [1-7] and called the resulting models twirls. After designing many complex modules and models, we asked a question: how many creases do we really need to make a twirl model. We were surprised by the answer: none. And we were surprised by the huge variety of such models as well as the appealing visual effect (see cover of [9]).
We discovered that we do not need any crease to create a flap to be twisted as unfolded piece of paper has already flaps. Moreover, unfolded paper gives us more freedom to position flaps as there are no lines that bound flap area.
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Burczyk and Burczyk
Results
Figure 1: Just squares.
Figure 2: Squares.
Figure 3: Rectangles.
Figure 4: Geodesics.
Figure 5: Parabolas.
Figure 6: Polynomial shapes.
Just Twist, About Minimal Origami Models Based on Polyhedra Structure
Figure 8: Polynomial shapes.
Figure 7: Cube.
Figure 9: Rectangles and squares.
Figure 10: Twelve strips.
Burczyk and Burczyk
Figure 11: Loops.
Figure 12: Triangles.
References
[1] K. Burczyk, Kręciolki. Twirls, Zabierzów 2003. [2] K. Burczyk, Kręciolki kręcone inaczej. Twirls Differently Twisted, Zabierzów, 2003. [3] K. Burczyk, Kręciolkowe kusudamy 1. Twirl Kusudamas 1, Zabierzów, 2008 [4] K. Burczyk, W. Burczyk, Kręciolkowe kusudamy 2. Twirl Kusudamas 2, Zabierzów, 2009 [5] K. Burczyk, W. Burczyk, Kręciolkowe kusudamy 3. Twirl Kusudamas 3, Zabierzów, 2009 [6] K. Burczyk, W. Burczyk, Kręciolkowe kusudamy 4. Twirl Kusudamas 4, to be printed [7] K. Burczyk, W. Burczyk, A Systematic Approach to Twirls Design, in P. Wang-Iverson, R. J. Lang (ed.) Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education, A. K. Peters 2011 [8] Herman van Goubergen, Curler unit, British Origami 205 (2000), pp. 17-18. [9] G. W. Hart, R. Sarhangi (ed.) Bridges Pecs. Mathematics, Music, Art, Architecture, Culture. Proceedings 2010. Tesselation Publishing, 2010.