Origami Meditation (OM)
Paper-folding can be more than recreational fun. As a mathematician, I see the
creation of an origami model as similar to a geometric construction. Others might
view the practice as a meditative way to “clear” the mind. What will you see?
Origami (ori- folding, kami-paper): is art and science,
can be recreational and analytical, concrete (model) and abstract (crease pattern),
visual, fun, challenging at many levels AND….. It can be spiritual. The end product is
not necessarily that important, it is the process along the way upon which we will
reflect. Let’s be mindful throughout the construction. What will you see?
We will be constructing a traditional butterfly. Instructions may be found at
https://www.youtube.com/watch?v=lNJ_-CKht3A
1. Unity
In the beginning , there was the Square. The Japanese word for paper is pronounced
Kami. The Japanese word for spirit or divinity which pervades every kind of object
also is pronounced kami, but has a different written character. Eric Keeneway sums
this idea up nicely. “To a few paperfolders the oneness of the square of paper (which
has the capacity to become all creatures, interdependent because the square always
remains a square) symbolizes their belief in the harmony of the universe and the
presence of the Buddha-nature in all things.” Similarly, through the Tao, all things
are given life and form. The square is the uncarved block. Tao signifies the
primordial essence or fundamental nature of the universe. In the foundational text
of Taoism, the Tao Te Ching, Laozi explains that Tao is not a 'name' for a 'thing' but
the underlying natural order of the universe whose ultimate essence is difficult to
circumscribe. Tao is thus "eternally nameless” (Dao De Jing-32. Laozi) and to be
distinguished from the countless 'named' things which are considered to be its
manifestations. Zen asks the question “Who am I? The square of paper asks the
question, ”What am I?”
2. Duality
The Bisected Square – first fold. The first fold emphasizes the duality of nature:
mountain/valley, male/female, yin/yang , dark/light, plus/minus. Origami is nonadditive like painting, non-subtractive like sculpture. It is Zero-sum. Hindus believe
that as creation began, the divine, all-encompassing consciousness took the form of
the first and original vibration manifesting as sound "OM". One fold is like the simple
melody from which more complicated creations, like sonatas, begin.
3. Mindlessness
The 4 x 4 Grid . Origami can also be “mindless” a necessary escape from our own
constant thinking. (Zen?) The followers of Zen believe that enlightenment comes,
not from the study of the scriptures or from metaphysical speculation, but from a
sudden flash of intuition which occurs during disciplined meditation. Life The
World’s Great Religions. Meditation – 1) emptying or concentration of mind, 2)
pondering of something, 3) serious study of topic. Logic is important in proving a
mathematical theorem, but “intuition” is where the inspiration of a proof begins.
What is intuition? Is it the “clear mind” of non-thinking found in zen? Is it that spark
of creativity, when one discovers a key insight in how to think about something
differently?
How many squares do you see? 1 + 4 + 9 + 16 = 30.
4. Diversity
Triangles – (make sure to flip over) - another way to fold – matching adjacent edges
instead of opposite edges, new bisected angles, new shape, mountains and valleys
on the same side. This step in the construction is revolutionary. It is conceptually
block-busting. The routine of how we see the world around us is shaken, awakened.
This step reminds me of the moment in math history when the Pythagoreans
discovered that numbers were not ALL expressible as the ratio of two integers or
whole numbers. Indeed, they proved that there were numbers, such as √2, which
were “irrational,” outside of the then accepted realm of rational numbers. The same
thing happened about two thousand years later when imaginary numbers were
“created” to help solve a mathematical riddle.
5. Memory
Shallow Bowl (flip back over). An extra dimension is emerging from the folds of
memory. The paper remembers its past. We can strengthen that memory by further
folding. Don’t be afraid to self-examine. We can unfold (rewind time) and easily
come back to the present. However, some things are hard to change, the past
persists.
6. Transformation
There is some sort of magic of touching the paper at a specific point and having it
spring into a different form. I wish I knew the physics of the situation better.
Human nature seems to require us to change things. Math is the study of how things
(elements of a set) transform (function rule) and create patterns. We want to
recognize and analyze these patterns.
Maekawa’s Theorem states that the number of mountain folds in a flat-folded vertex
figure differs from the number of valley folds by exactly two folds.
Kawasaki's theorem is the statement that the crease pattern may be folded flat if
and only if the alternating sum and difference of the angles adds to zero:
Most origami transformations are precise, like the straight-edge and compass
constructions of Euclidean geometry. Many practitioners or origami set boundaries
on what is allowed and what is forbidden to do with the paper.
Four-petal Flower – The paper has lost its abstraction and given birth to the
concrete. Can try to get it back to two-dimension, but paper resists. At a crossroads.
Can go in many directions from here.
Hexagon
Boat – Trapezoid
Triangle
Wing Modeling
Symmetric Fold-over
Wing Separation
Second Symmetric Fold-over
Final Product
Mecho (female) and Ocho (male) Butterflies are the earliest examples of Japanese
paperfolding. They are associated with the ceremonial drinking of sake which is part
of the wedding ceremony. The butterflies are made from the paper covers on the
flasks holding the sake.
David Lister would say that the butterfly “captures the joy of spring, the miracle of
metamorphosis, the wonder of life, and the joy of freedom.”
Conclusion
Vanessa Gould”s 2009 award-winning documentary “Between the Folds.” begins
with these ideas. “We are born of folding,” as we age our skin folds, our DNA is
folded. You can find it here.
http://www.youtube.com/watch?v=bJRBiIeFe7Q
Also, Sarah Singer sent me this short presentation on origami and mathematics. It is
worth a look.
http://www.kcet.org/arts/artbound/counties/los-angeles/an-origami-momentmathematics-meets-paper-folding-in-los-angeles.html
Akira Yoshizawa father of modern origami. (1911 – 2005) made 50,000 models,
none sold.
50s < 20 folds, 60s 20 – 60 folds, 80’s 70 – 90 folds, now 300 folds. Baroque or
rococo state right now.
Some visual samples of origami.
David Lister (Origami and the Spiritual), “I look at the plain square in front of me.
And I find that it is no longer a mere piece of paper. I find that it is a magic casement
through which I can gaze at enchanted landscapes and pass to worlds of a higher
experience and spirituality. So, I recall the words of William Blake, (also in the UU
hymnal - 398)
“To see the world in a grain of sand, and heaven in a wild flower, hold infinity in the
palm of your hand and eternity in an hour.”