Math Koans

Darwin worm stone

OK, not quite koans, but worth some time if you are so inclined:

Where were the 1 million digits of pi that we have calculated (or for that matter the unending string of numbers that are embedded in pi) before we invented numbers?

Same koan:

Where was the unending number we label “e” that is it’s own differential and integral before we invented calculus?

Similar koan:

For both pi and e, where are all those unending digits now?

Similar koan:

How can there be a square root of -1 when any two negative numbers multiplied is a positive number? We call this number “i” and it is essential in the mathematics of quantum physics that is at the heart of all scientific thinking.

I can go on and on, but maybe you get the point.



Entropy, Ego, What’s the Point?


Rather than launch into a technical description of entropy and the relationship of energy and entropy lets try this first.

More entropy means more disorganization and more ignorance. Low signal to noise. Less information. Like static preventing the faithful transmission of data. Think of loud static on a radio when you are trying to listen to music on your car radio.


If I tell you I mixed up the numbers one through ten and put them in a bag, then I picked out two, say a 3 and a 7, all you know about the next one I will pick is that it is not a 3 or 7. So they are mixed up, disorganized, and we have a bit of ignorance about some aspect of that system. Relatively high entropy. If I throw in some letters or blanks into the bag along with the numbers, i.e. static, you are even less able to predict the next thing to come out of the bag!

Now I tell you I ordered the numbers from ten down to one. There are no blanks or letters. I picked out a ten. Next picked will be… nine! Very good. You had little to no ignorance. But I had to put extra energy into ordering the numbers compared to throwing them in the bag. I had to have some way to assure they stayed in order as well. Low entropy, but it took more energy.

Meditation can be seen as aiming for high energy, low entropy. But I am not sure that’s quite true for zazen. You’d have to ask a Zen teacher. Certainly “mindfulness” is like that.

A circle is low entropy. You know everything about it and it took energy to create it (minimally mental energy, in addition perhaps energy to move the pencil or program and run the computer).

Symmetry is not ignorance. True, by definition symmetry is present when you can’t tell something has changed, like someone else spinning a circle while your eyes are closed, so that seems like ignorance. But to do that experiment, you need to know that the experiment was planned and then do it! That’s a lot of knowing, organization and energy!

Information is low entropy. It takes energy to put 0’s and 1’s in some order and that is one aspect of what information is. Ordered dualism.

Meaning is how we interpret and experience information. It is our perspective on it. It is contingent to the max. It is easily colored by our wishes and desires, by our egos.

I just read that the Nobel Prize winning physicist Steven Weinberg, who unified the electromagnetic and weak nuclear forces (along with others, of course; anyway major physics achievement) wrote: “The more the universe seems comprehensible, the more it also seems pointless.”

That seems very nihilistic and depressing. Perhaps that’s how he meant it. If so, somehow he had dealt with it because some four decades after writing that he is still writing books!

On the contrary, that seems very Zen to me. And liberating. It relieves us of arbitrary values and goals. The kind the ego sets up to measure ourselves by, so we can achieve them and reassure ourselves. Except when we don’t.

What ultimate, objective, cosmic, universal, non-dualistic “point” could there be? Any point we could articulate would be a human construct, limited and contingent, a dualistic notion of use in only a very small corner of time and space.

Matthieu Ricard writes in his book “Altruism” that the ego is the crystallization of our identity. He writes that we try to protect it. That’s pretty good, but I am not sure that it is quite right. There is no single anatomic brain space that houses the ego. I think the ego is the process by which we protect our identity. The identity is our sense of who we are based on our conditioning (biologic and psychological, contingent on where and when we are). It is how we organize our sense perceptions and react to them. It is our karma, if you will. It is how we try to make the world comprehensible, to find a point. The ego is the process of having and wanting there to be a point. A point is like a location, a beacon, a polar star that the ego can refer to on the horizon to measure itself and its position by so it can better protect us as we cruise through the world of time and space, the world of the six senses.

So as the universe becomes comprehensible, what we comprehend may not be to our ego’s liking. It may not put our bodies (brains included) at the top of the heap. It may remind us that our limited sensory experience is a pretty pale reflection of the vastness of the universe. Of course comprehensible in this context means the forces of nature. The things physics studies. That which can be measured. It does not mean the whole shebang.

To be clear: I am not suggesting a lack of values. I hope you value compassion. I hope you don’t value your suffering and especially not the suffering of others. I am only suggesting not being seduced into thinking that is the “point.”

Or is it? We can chose to embody compassion, we can aspire to the low entropy high energy state. Is that the “point” of our lives, our minds, the dream, the whole show? Some think so. I admit to liking that view. But maybe that’s the point! It is a goal to like, admirable to be sure, but do I like it because it makes me feel better about myself? Is that my ego protecting me?

No “point”? Perhaps that’s kind of like “ordinary mind is the way.” Or the miracle is chopping wood and carrying water. You don’t need a “point” writ large to the universe to eat when hungry, or to be compassionate. That is the functioning of the universe. What needs to be added? What would be the point?



Information Is The Dreams Stuff Is Made Of


Recently Stephen Hawking announced a new theory about what happens to information at the event horizon of a black hole.

Some scientists took him to task. They said in effect: isn’t it a bit of grandstanding to announce such a thing without showing your work?

I like that. Hold authority’s feet to the fire! That is the scientific way!

The question is: why do scientists care?

It turns out to be a question that is basic to the scientific view of how the universe is put together. Leonard Susskind wrote great book about it called “The Black Hole Wars.”

You see, information is conserved.

Like energy over all is conserved, is the same at the beginning of a process as at the end, though not the specific forms of energy (e.g. chemical energy becomes heat).

And most definitely not like entropy. Entropy is not inherently conserved!

Information that is conserved is not exactly the same as “meaning.” It is the possibility of different states. You know, like 0 or 1 in the binary code that the computer uses.

Or the letters of the alphabet. If you see:


in an e mail you think “laugh out loud.” Heck, you can program a robot to recognize it and make “ha, ha, ha” sounds.

Doest the robot know mirth? Joy? What it is to laugh?

Do you?

Is that information?

No, the idea that LOL means “laugh out loud” is meaning gleaned from information.It is not inherent in the information. We supply the meaning. Conscious, sentient beings do. If someone finds the letters LOL in a message many years from now, odds are it will not have meaning to them. Maybe not in very many years; I understand LOL is going out of fashion already. But it will have information.

LOL could have been randomly generated (a complex thing to do) or it could have been from a program that says: insert consonant-vowel-consonent.

Take a circle. Little information is needed to generate the circle:

  1. a definition (all points equidistant to one given central point)

2. and the variable (the distance).

The circle is symmetric. If you shut your eyes and I spin the circle around the central point, when you open your eyes the circle looks the same. No change. Symmetry.

But the universe has changed. The energy I used to move my muscles to move, say a cut out circle, thus spinning the circle, or by tapping circle moving instructions on a computer keyboard to spin a computer soft ware generated circle, comes from energy stored in my muscle cells.  The cells take glucose and break it down to CO2 and H2O molecules and use the energy released from the chemical bonds to create high energy bonds in ATP  (adenosine tri-phosphate) molecules, then the muscles use the ATP molecules for energy, breaking the ATP phosphate bonds (creating ADP, adenosine di-phostphate and then passing on the phosphate released from the ATP; don’t worry if this doesn’t mean much to you. The details aren’t critical) and thereby changing the energy state in the ATP/ADP/Actin/Myosin structure and thus changing the molecular structure of actin/myosin in muscle to create movement.

This chemical/mechanical process resulted in more molecules with less energy in their chemical bonds than the original glucose molecule, and released excess energy as heat. Also heat is generated by my fingers moving the circle or pressing the keys (friction and the energy of my fingers interacting with the molecules in the paper as I move the paper circle or computer keys as they crash into each other). The change in the molecules and the cells and the infrared photons (the heat released) pinging around create a less organized, higher entropy situation.

So the circle is unchanged, it is symmetric, it is in the same state after we spun it that it was  before we spun it, but the entropy of the universe has increased. We can re-create the glucose molecules, but it takes CO2, H2O, cellular organization and energy, for example in  the complex biological process of photosynthesis. But there will still be the same or more entropy each time we go about making any change.

So even a symmetric situation in the “real world” is not totally symmetric.  Even if we do the circle spinning as a thought experiment, where you don’t actually move the circle, as you did when reading this pretty much, takes energy! The energy of the chemical reactions and electrons moving about in your brain when you think generates heat and entropy.

Which leads us to thermodynamics and Maxwell’s demon.

But I digress; lets hold on doing more thermodynamics and Maxwell’s demon for this post. I will do more on that later.

For now, let’s get back to the idea that in the world of change and movement, the world of the senses (themselves of course information processors) information is conserved. Not meaning, just information.

Meaning is contingent. It is not conserved. It is relational, and generates entropy or uses energy to decrease entropy. Either way, energy and entropy are involved in meaning, playing off each other, perhaps. Energy is conserved. Information is conserved. Entropy is not. Meaning is not.

I find that very hard to get my head around. Why should that be? For that matter, why should it be that energy and information are conserved?

Perhaps it is because those conserved elements of reality were never created and can’t be destroyed, no beginning no end, so how can they fundamentally change?

Meaning is dualistic. It is not conserved. It is contingent on context.

Perhaps the universe at its core IS information. Some physicists think so. Every aspect of the universe that is, well, an aspect, is an aspect because it could have been otherwise (not necessarily just any old otherwise, perhaps a specific set of otherwise consistent with the laws of string theory, quantum field theory, whatever). Otherwise it isn’t an aspect.

0 and 1. Yin and Yang. Duality. That is what physics studies, after all. That is the core of our experienced universe of the senses.


Remember: information is not meaning, It is not essence, noumena. It is phenomena. It is occurrence.

Information is the dreams stuff is made of.

Meaning is determined by sentience.  Consciousness. Is there silicon sentience? If so, that robot will know mirth. And why not? Why should we be carbon chauvinists? Perhaps the very quantum fields can coalesce in many ways to find mirth.

When communications scientists developed the idea of information, it was to quantify the fidelity of communications. Does a phone message get through ungarbled? Not whether the speaker or her message was coherent. Do the 0’s and 1’s that make up your e mail message stay the same, or are some lost in  the “tubes” of the internet? It doesn’t matter if the email is LOL or a consonent-vowel-consonent randomly generated, whether or not it has linguistic or human intellectual or emotional meaning, if at each slice of time and space there was an either/or, a 1/0, there was information.

I will elaborate later. But I don’t know that I will get past the following no matter how hard I try:

Everything that happens according to scientists does not change the total information in the universe (though you can rob Peter to pay Paul energy/information, more here less there. Shuffle it around. That takes energy if we are talking about information). Information cannot be irretrievably lost.

This relates to symmetry (lack of change in some element of a system even if something somewhere else changes)

This relates to energy.

This relates to thermodynamics: what is likely to happen, and the role of entropy.

There is relationship between entropy and ignorance (another post; this is one of the technical definitions of entropy: how we can know the state of the components of a system) but as implied here, there is a connection between a type of contingent sense of meaning, meaning as we ascribe it to the stuff of daily life, meaning as motivation in our world of the senses, of our karmic experience, that is also part of entropy.

There is an Akashic record. Information in the universe is never totally lost. If you could lose information, modern physics collapses. Ask Dr. Susskind (or read his book!). This akashic record is not about some mystical new age vision of some grey haired old guy writing in a large parchment book with a quill pen somewhere or Santa Clause remembering if you were naughty or nice. In theory you could piece back all of the energy and information transitions and reclaim the original. Sure it may take time and energy without beginning and without end, so our technology may not be up to it.

Perhaps this “akashic record” is the manifest mind of the universe. It doesn’t have to track information back, put it back togeher. Perhaps it is the process, the functioning. It is not dualistic. It isn’t stuck in meaning in things like “LOL.” Maybe that is our dualistic perspective.

The process of oneness, of unfolding, of compassion, that is the flavor I suspect of this akashic record.

It is kind of fundamental and I find it kind of interesting!





Where the Rubber Meets the Road; Lessons From a Busy Month


I haven’t written on this website for about a month. I have been doing a lot of reading (non fiction mostly) and writing (trying fiction mostly) while keeping up my practices (medical/scientific and Zen). Very invigorating.

Three themes kept reappearing this month.

First, it is fun to have fun, and to share my enthusiasm, which I often have in abundance, but ego, praise and blame, and the need to “do” sneak in so easily. I set myself up for that!

Second, be careful about the stories you tell, they tend to come true in ways that may be unexpected or in ways that are not literal, but true nonetheless.

Third, when looking at how science describes the way reality functions, whether by studying biology and neuroscience, peeking into into the standard model of particle physics, quantum mechanics, string theory (metaphysics or physics? I am in the camp of those who think the former, but that is for another post), the cosmos as information or hologram, multiverses, multiple layered realities, computer metaphors, or whatever big picture cutting edge science and the various interpretations of science (metaphysics) can offer, it seems to come down to:

Is mind an epiphenomenon arising from evolved brain tissue, itself congealed energy, and that’s as far as it goes, or is Mind primary?

Does Mind arise from energy or is Mind the field in which energy and the organization of energy flows?

Does Mind need another field to maintain it, like a quantum field, or the vacuum with it’s teeming sea of virtual particles and energy without beginning or end, or is Mind a name for the ultimate field that, while still dualistic in a way, is an appropriate term to use because it reflects our experience, that is, is our mind, as we live it?

Is what we can measure and perceive primary or is consciousness primary?

Do we really describe Reality with the tools of the intellect, the mathematics we invent, the changes in energy we perceive with our senses, or do these tools of the mind just provide a great quantitative look at one layer that our monkey brains can handle, at the scale we evolved to live in, even if we push that out very far with very clever instruments and experiments, with the underlying energy and principles arising from Mind rather than the other way around?

Even that is of course a story, a concept, but I think when talking about science and practice, about how it is, that is where the rubber meets the road.

It isn’t whether I think I can prove Mind is primary. That’s exactly my point. It has been said that it is like a fish trying to prove water.

That’s why as busy as I get, and as interesting as I am to myself (I amuse myself greatly though it gets a bit much even for me sometimes), I keep up my practice.

I’m kind of curious.

Quantum Peak: Where Are You? Where Are you Going? Are you Sure?


Hakuin Zenji occupying the ground he sits on. Where is he?

Heisenberg’s Uncertainty Principle!

Most of us are uncertain about this or that. In quantum mechanics uncertainty isn’t a matter of confidence or knowledge, it is in the nature of the beast.

I am often amazed how this uncertainty principle is seen by scientists as such a strong principle that observations and outcomes must obey it. No questions asked, no reservations.

Here’s what it says:

There are measurements, things you can know about a particle, say a photon or electron. Some of these come in pairs such that both cannot be known to the same degree of certainty at the same time. Period. Our ability to measure the universe with our senses (and our devices which are extensions of our senses), what we can know by observation, is fundamentally limited.

Often it is said that this is due to the clumsiness and coarseness of our measuring devices. Send in a photon to “see” where the electron is by pinging it, and you now have an interaction that changes things. The size and energy of what you use to “touch” the world of particles is so large proportional to the particles, you can’t help but disturb it, to change it as you measure it.

Fair enough.

But it in fact goes more deep than that.

Lets look at momentum and position.

Momentum is how much oomph something has when it is moving, how much bang it would have if it hit something. If the object has mass, momentum is simply mass times velocity. The more massive the object and faster it is going in a specific direction (velocity is speed and direction, a very important point), the more momentum it has. Since photons have no mass, the momentum is a function of its energy, or wavelength, but that matters little to us here. The idea is the same, directed energy, how much oomph it has in a specific direction.

Lets look at an experiment, shining light at holes in the screen. The light is represented by the golden arrows going left to right.


If we shine a wide beam of light with many photons against a screen that has a hole in it, most of the light is spread out pretty evenly along the screen and will hit the screen pretty evenly all over (well, an area of the screen as large as the beam is). We don’t know where in that beam a given photon is exactly. It is in the room between the light source and the screen it was aimed at, but a given photon can be positioned anywhere in the beam of light (the straight arrows to the left of the screens in the illustration).

But assuming we know the wavelength of the light, and the direction the beam was pointed, we know the momentum of any photon in that beam with a great deal of accuracy. The beam was directed toward the screen, and so if undisturbed should be going pretty straight on (except for the stray cosmic ray or atom in the air hitting the beam, for example, pretty small effects here and they can be minimized), and at the speed of light in air, and so we pretty much know speed and direction pf the beam and so all the photons in it. So while the momentum of individual photons will vary a bit, it won’t be by much.

We can say then that before the light gets to the screen we have little (but some) information about position of the photons in the light, but a lot of information about momentum of the photons.

Next, some of the light goes through the hole on the screen at the left in our illustration. There is a phenomenon called diffraction. When the light leaves the hole, it bends out at the edges. The larger the hole, the less relative bending, the smaller the hole the more bending. Picture a broad water wave going through a small opening in a jetty. On the other side of the hole in the jetty the wave will expand. If it is a big hole, most of the water wave just goes right through undisturbed, only the part of the wave right at the edge of the opening in the jetty is going to spread out again after passing through. So big opening less relative rate of spreading.


A broad wave on the left goes through the holes and then spreads out. This is another way to see diffraction. In this case there are two holes and so the diffracting waves interfere. We will limit ourselves to one hole this post!


Our light now goes through the hole in the screen on the left below.


Only about two arrows from the light on the left enter the hole. We know where the hole is, so now we have a lot more information about position of photons just after they enter and and right after they exit the hole than we had before the light entered the hole. We know pretty well where that light (and any given individual photon in the light beam) in the hole is when it is in the hole or just after it exits the hole so we know with a high probability where a photon that is going through the hole or just exited will be found, much more so than before the light entered the hole.

But due to diffraction induced by the hole when the light exits the hole (to the right in the illustration) the beam spreads out. But at that point, at the exit of the hole, that tails of the arrows are close together, and the area the photons can likely be found is about the size of the hole, so we still have information about position that is much more precise than before. An important point is that it right after the hole at the base of the arrow that matters. It is the direction of the arrow, not what is happening at the tip that counts here. What we see though is that only the central arrows of light are still going in the same direction that they were before entering the hole, as they were not affected by the edges of the hole (really mostly the most central arrow) and are not diffracted. So while we know we will find a given photon in the area about the size of the hole, if it was at the edge of the beam its direction (hence momentum) will have changed considerably. So some photons have the same momentum, but many have changed. We are less certain about momentum because remember, momentum isn’t just speed (the speed of light didn’t change) but also direction (and that for many photons that has changed due to diffraction at the edge of the hole).

We went from knowing little about the position, and a lot about the momentum, to knowing a lot about the position and much less about the momentum of a given photon. The possibilities for position have decreased, the possibilities for the momentum have increased.


On the illustration above, we made the hole in the screen on the right smaller. Now only one arrow from the light coming in from the left  gets through. You guessed it, we then have more information about position on the other side of the hole. It is confined to a smaller area due to the smaller hole. But since the hole is smaller, on leaving the hole there is more diffraction,a large proportion of photons are diffracted (there is less “middle” of the beam for them to avoid being diffracted; by the way it is of course much more compacted than that; but it is a good enough model to have in our heads for us to see what is going on), and the arrows are more widely directed, pointing at more of an angle from the smaller hole than the larger hole (now rather than three almost undisturbed as in the screen at the left, only one goes through unscathed) as there is more hole edge effect (diffraction) for the size of the hole. That is how diffraction works, it increases the smaller the hole.

More diffraction, more range of momenta.

In fact a door in a room diffracts light coming through it and bends it, so light goes around corners just like sound goes around corners. In fact, YOU diffract! But the effect is so small we can not perceive it.

Now with the smaller hole we have even more information about position but less about the momentum. We know with greater certainty where a given photon is likely to be, but even less about what its momentum is. We still know something about the momentum, we are just less certain for a given photon use precisely what it is.


We see this in the graphs. The up axis of the graph (the thin axis arrow pointing up) is the spread of possible momentums, higher up is more momentum. The axis going left to right (the thin axis arrow pointing to the right) is the spread of possible positions. It is simply where the beam is, so where a photon may be found. So the larger our rectangle is up and down, the larger our spread of possible momenta (our uncertainty for a given photon is larger) and the wider the rectangle, the larger our spread of possible positions, (our uncertainty about position is larger).


Graphs of possible states of momentum (up and down), and position (left to right), for the light before it enters the hole (left graph), the large hole (center graph) and the small hole (right graph) .


In the graph on the left, we see a wide spread of potential positions, but a narrow band of momenta. This is the beam before it goes through the hole. So we end up with a narrow rectangle in blue; narrow up and down as momentum is pretty well known (reflecting little uncertainty about the momentum of any given photon in the beam) but very long left and right (reflecting great uncertainty as to just where a photon may be as the beam is wider than the hole before going through the hole).

In the middle graph, we see the situation as the light exits the larger hole. We know less about momentum, so the square is larger up and down, reflecting more uncertainty about momentum due to diffraction and the new direction the light can take. There are now more momenta a photon can have, more directions. New directions means new momenta. On the other hand, we know more about the possible position of the photons because where they are as they exit the hole is limited by the size of the hole, and this is a smaller hole, limiting where they are likely to be, so the rectangle is narrower left and right. We are less uncertain as to where the photon is; it just left the hole so that limits where we are likely to find it, outside of effects like quantum tunneling, a subject for later!


The graph on the right is what happens after light passes through the smaller hole. We are more certain about the possible positions of the photons as this is limited by the smaller size of the hole, so the rectangle is narrower left and right, but we are more uncertain about momentum (more diffraction changing the direction) so the rectangle is wider up and down.

The area of the square and how this area is distributed is the critical thing to look at. Areas in calculus are the “integral,” in this case “integrating” (summing up) our knowledge of possible values for momentum and position in our experimental set up, as it were. Making them squares of one density is too simple of course. The potential state of the photon may not be equally likely to be anywhere in the square. Some states are more likely than others. The likely position, for example, may be more concentrated in the center just opposite the hole. But I wanted to introduce a way to see very important and mathematically sophisticated quantum ideas. The area of the square is the “probability density” of where you will likely find the photon and what its momentum may be in this “space of states,” (that is official quantum jargon) that is, the space, or dimensions, of momentum and position in our experimental set up.

A quantum scientist can never speak about how it “REALLY is” just what is the range of possibilities given your experiment. This relates to integral calculus and Fourier transforms. It relates to the very heart of quantum mechanics. (Congratulations). Much of a course in quantum mechanics is solving such problems of the space of states in a given situation and the areas that reflect probabilities.

These quantum effects, this uncertainty of the “material” world, just like diffraction at a doorway, are real for you and me and cars and galaxies. We can’t see them, as they are very small at the scale of our sensory apparatus (eyes). We think we can look at the speedometer of our car and the direction we are driving and where we are on the road and know both momentum and position, but even there, as soon as we note all that, it has changed. But even if we have a set up that can look at all of this data simultaneously (a whole discussion right there) it would be changing not only because it takes time to observe and note all of these things, a computer can do that very quickly, but because there is no difference between us and the quantum world other than what our limitations as embodied beings relying on sense impressions at our scale imposes.

That is, you don’t know both your position and momentum with 100% accuracy. Just well enough to get through the door (well, and then some).

An interesting implication of this is quite consistent with the Buddhist teachings about change and impermanence. There is never no movement. Not at absolute zero, not ever. If there were no movement it would violate the uncertainty principle. We would know position exactly (wherever we froze the particle) and momentum exactly (no momentum if it isn’t moving!). Really, that’s what I meant at the beginning. This principle is so basic, so essential in the math as well as our observations, that scientists will not allow it to be breached. Like conservation of energy, it is foundational in science.

So what does it mean to me? Is it cool that some aspect of Buddhist philosophy has scientific validation? Sure, I like that, but that isn’t all that important really I think. It also is a taste of the unreasonable ability of math (that was very, very sophisticated math back there) to reflect reality.

And more importantly, as before, it reminds us that what we see, what we can determine about the nature of reality using our senses, is dependent on our limitations, our projections, our assumptions. The concepts, words and intuitions we have developed in the 4 dimensional world of space and time are mere approximations. Don’t get too attached to them. That is what this aspect of the quantum world says to me.


There is no fixed place.


Heisenberg’s Uncertainty Principle!



Circle Triangle Square



Sengai Gibon (1750-1838) was a Japanese Zen master who was an artist. There are many stories about Sengai. One I particularly like shows his courage and compassion. The Daimyo, the high ranking Samurai who was the local ruler, loved chrysanthemums. The gardener’s dog destroyed some of his prized blooms and so naturally the gardener needed to die. Sengai leveled the rest of the flowers with is trusty scythe, presenting himself to the Daimyo the next day. Sengai asked to be killed for certainly in this area the farmers, who were dying of hunger were ignored by the Daimyo while pretty plants were valued above a human life.

The Daimyo got the message.

One of Sengai’s most famous works is “Circle Triangle Square.”


There are many interpretations of what this painting is about.

The inscription at the left alludes to Sengai’s temple, an ancient temple already at that time about 700 years old. it was the first Zen temple in Japan. So maybe the shapes refer to the temple. Or the pagoda at the temple.

Or maybe the inscription is not about the subject of the painting and it is just in effect his signature. Zen masters in Japan and before that China were often identified with and named after their monasteries or the mountains they lived on.

Or maybe the circle is the cushion (zafu) the meditator sits on, the triangle is the mediator with the top point as the head, the solid base the butt and crossed legs. The triangle and meditator as mountain (the triangle/meditator/mountain idea was suggested in conversations with sensai Maezen at Hazy Moon). The square might be the zabuton, the square pad the zafu sits on.

I would like to interpret it geometrically.

I have no idea how much geometry Sengai knew. Clearly basic geometric shapes interested him enough to paint them.

I already discussed the perfect symmetry of a circle in the post “Circle and Wave.” Recall that a circle is an idea, defined as that object that is equally distant at every point from a central point. This distance is the radius of the circle. So all it takes is a distance to make a circle. The circle has perfect rotational symmetry, without beginning or end. When that symmetry is broken, we saw that we can produce waves, and these waves define particles, the basis of form.


We saw that breaking the symmetry of the ideal circle led to waves, particles and the manifest universe of form and is embodied in the yin yang symbol.

So circles are amazing but where do squares and triangles come in? How do they relate to the unwinding of the perfect circle as a wave.

Lets start with how a square and triangle relate. A square is two triangles.


Next, how do circles and squares relate? There are many ways, but here is one I like: Every circle precisely defines two squares, each of which intersects with the circle at four points. One square is inside, the other outside the circle. In a perfect symmetry, every square likewise defines two circles, each circle intersecting with the square at four points:


Those then define two larger and smaller squares and circles ad infinitum.



Now, how do circles and triangles relate and how do they make waves?

We can think of the circle as a clock face. This time we will think of the radius as a minute hand, here pictured as arrows, but this minute hand will go counter-clockwise starting at the 3 o’clock position. We will see how high the tip of the arrow is above or below the horizontal line bisecting the circle at several points.



Next lets put each vertical line along a horizontal line with each clock hour marked, starting at 3 o’clock and going counterclockwise. Even with just a few straight lines we see a wave emerging:


In this figure we placed the vertical lines above and below the horizontal line at their clock positions in the circle as marked on the horizontal line and connected the tips of the arrows with lines and got a rough wave.

If we were to add so many arrows and vertical lines so that the circle is filled with lines and arrows we would get a perfectly smooth wave.

What is this wave?

Each arrow in the circle goes from the center to the circle itself and so all are the same length. Each is a radius of that circle. It also is the hypotenuse (the longest side) of a right triangle.

Here is one triangle isolated to show that they are each indeed a triangle:


This figure Shows one of the triangles defined by the arrow and the line from the tip of the arrow to the horizontal bisecting line.

Now, lets say the arrow/radius/hypotenuse is one unit. It doesn’t matter one unit of what. A unit could be one inch, one mile, one light year, one unit of 6.753 mm, one diameter of an oxygen atom, it doesn’t matter; are all kosher as long as all other measurements that relate to that unit length, say of the other sides of the triangle, are measured in a way that is related to that basic unit (in inches, miles light years, multiples of 6.753 mm, diameters of oxygen atoms).

Remember basic trigonometry: the sine of a right triangle is defined as the length of the side opposite an angle (other than the right angle) over (that is, divided by) the hypotenuse. It is a ratio, no units, they cancel, just a relationship that always holds. Since the arrow/hypotenuse/radius here is 1, we defined it as one unit, the far side across from the angle is the length of that side over one, so that side IS the sine for that triangle. Slide4Here we see a right triangle, the thick line is the side across from the marked angle. The diagonal line is the hypotenuse (the arrow in our circle). The thick line is the side across from the angle. To the right of the triangle we have the thick line divided by the diagonal hypotenuse (which is 1) = the thick line. The length of that line is the sine.

So we collected these sides of the triangles, these sines, which were the length of the tip of the arrow above or below the horizontal bisecting line, and we created a sine wave!

Circle, square.  Every square defines two circles, every circle defines two squares, without beginning or end.  Square triangle. Two triangles make a square. Circle triangle. Every circle is made up of the hypotenuse of triangle after triangles, and these define waves (we just looked at one such wave, the sine wave).

So do you think Sengai had any of this in mind? Did he know trigonometry? Did he intuit that these basic forms could describe all form? These objects that have no physical existence but are abstractions, the product of mind, empty of substance?


“Nevertheless this great ocean is neither a circle nor has directions. The wondrous features of this ocean that remain beyond our vision are inexhaustible…. It is just that as far as my vision reaches for the time being, it appears to be a circle.”



photo courtesy of Susan Levinson

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”. Continue reading