Circle and Wave


In my GUT of Zen there are two phrases that suggest time and space:

You are the Universe unfolding

Mind evolving.

And there are two phrases that are outside of time and space:

No beginning and no end

No separation.

Time and space are deep and difficult. Don’t be seduced by clocks and rulers and your day-to-day experience into thinking you have any idea what they are about.  The 13th century Japanese Zen master Dogen famously spilled a lot of ink writing about time and change. Change is discussed in some of the earliest Buddhist writings we have. Scientists debate the nature of time and space to this day. In a recent review in the scientific journal Nature titled “Theoretical physics: the origins of space and time” (8/23/13) there is the lament that physics is incomplete without an explanation of time and space. There are seven competing theoretical models discussed, with titles like “quantum loop gravity” and “Holography”.

I will discuss time and space at another time and space.  For now I want to start to look at the identity of my two pairs of statements in my GUT of Zen, the identity of the pair of statements that are in time, the relative, and the pair of statements outside of time, the absolute. Embedded in this GUT is the tension between the relational, the relative, the contingent, the deep and abiding interconnectedness and interaction of what is in time and space, and the absolute, what is beyond our experience in time and space.

Readers who are Zen practitioners know I didn’t come up with this terminology on my own. There is an ancient Chinese Chan (Zen) poem that is foundational to our practice. In Mandarin it is called Cantongqi, in Japanese Sandokai. The poem was written by Shitou Xiqian (Japanese: Sekito Kisen) in 8th century China. The title is notoriously difficult to translate: “The identity of relative and absolute” is the version we use, and I like the kind of mathematical sound and unapologetic nature of  “identity.”

Books have been written and series of talks given about this poem. This is one of those places where both science and Zen converge in wonder and profundity. Sages, philosophers and scientists have grappled with this throughout the ages. The identity, or some say the harmony, of the world of the relative and the realm of the absolute is not very amenable to the intellect, to concepts and language, or even to mathematics, which gets lost in infinities. We can use some ideas at least as metaphors perhaps, even if just to get us started. For the time being lets start by looking at the universe embodied in the circle, and to do that, lets look at symmetry and the breaking of symmetry.

There are many types of symmetry.  You can see mirror symmetry in the fluke of a whale. The right half of the fluke is the mirror image of the left, and visa versa. That is why the water is so evenly dispersed in the photograph of the fluke at the start of the blog. Such symmetry is very functional for the diving whale.

Let’s look closer at just what symmetry is and how to spot it by looking at rotational symmetry as an example. If you close your eyes and I rotate an unmarked circle, you can’t tell I did anything when you open your eyes. This “can’t tell” test is a hallmark of symmetry.

The conservation laws of physics are also a symmetry. When we say the total energy of a closed system is constant, we are saying the total energy is the same before and after you do something to it. Look at the energy of the system before the experiment, and then close your eyes while somebody else does the experiment. When you look at a reading of the total energy when you open your eyes after the experiment is done, you can’t tell there was any change; it passes the “can’t tell” test. Actually we can’t measure total energy  directly (the slippery nature of energy is another discussion). We measure changes in energy. And that will be zero. If it’s not, you have some explaining to do. And that can be just where the real action is, say a new particle!

The symmetry of conservation laws has formal mathematical underpinnings that were defined a century ago by a woman who many physicists seem to adore, and for good reason. Emily Noether was by all accounts a brilliant mathematician, and a wonderful person who was helpful to all who came to her for mentoring and teaching. She couldn’t get a paying job  in a university  because she was a woman, and she had to flee Germany in 1933 because she was a Jew. Just thought I‘d introduce you.

The circle is infinitely, absolutely, symmetrical to rotation. There is a reason there is a  Japanese Zen calligraphy of circles called Enso: circles have no beginning and no end. But open up a circle and you break that rotational symmetry and get a different symmetry that is limited as to rotation, but can be repeated as infinite cycles. You get the wave. We can derive that mathematically, but let me just show you a picture:

Circle and Wave

We take a perfect circle and divide it in half. Now there is direction, duality, up and down, positive (+) and negative (–). Move the bottom half of the circle over so that the left tip of that lower half meets the right tip of the upper half of the circle and we get a wave!

In other blogs I will talk about quantum mechanics and field theory, Heisenberg’s uncertainty principle and particles and waves. I only want to point out here that mathematically we can sculpt waves. Just like additive sculpting (say in clay) or subtractive sculpting  (say in marble) we add and subtract waves to get new waves, even a sharp localized spike. That spike is the particle. The relative. And hidden in a spike there can be countless waves adding and subtracting, sculpting the spike mathematically. In the circle there is the wave, in the wave the particle, in the particle the wave, in the wave the circle.

The identity of the relative and absolute, symmetry and asymmetry, particle and wave. I love this stuff.


[The Tingari by Nanuma Napangardi, of Kintore, language Pintubi.]

I said in a Zen student talk I gave recently that as much as I love science, it always bothered me that there seemed to be the implication that all lives could be seen as in some way incomplete and misguided without the latest great scientific discovery that “changes everything.” It just didn’t seem fair! Even as a kid I thought, what if I die the day before the big announcement, whatever it was!

But the ancients knew about symmetry and symmetry breaking, circle and waves, the identity relative and absolute.

Just look at the Yin Yang symbol! Circle, wave, particle. Repeat.


They didn’t just know it in their GUTs. They experienced it in their beings, in their lives. That is what it is really all about.


Photos courtesy of Susan Levinson

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