It's Elemental

Tag: Shape

  • Waldorf Educators: A Light on the Path

    The world has problems; LOTS of them.  But as long as there are people like my friend Nancy, who devote their lives and love to one of the most important and practical callings on the planet, education of our young, I have hope for the future.

    Let me give you a picture, in her own words, of what I mean, as an example of how Waldorf education tries to meet the challenging tasks of our time.  This is but a tiny picture of what is possible–today– for education:

    My class went to our auditorium today and entered in the dark.  The room was covered in pine boughs in the shape of a spiral path.  In the center of the path was one large glowing candle that sat upon a tree trunk covered by a blue cloth with gold stars on it.  It smelled like a pine forest.  The students entered in silence and proceeded to a semi-circular row of chairs to sit down.  Two women, playing penny whistle, violin, and cello began playing various musical selections.

    The first child approached me.  I was standing by a table filled with red apples, each holding a slim candle in a hole carved into it.  Then, one by one, each child took the spiral walk into the center where the light glowed.  Lighting their own small candle from that big central light, they each turned and began walking back out from the center.  At some place of their own choosing along the path, they each picked a spot to place their little light.

    When the last child had gone, I took my candle filled apple and walked the path among the little lights, to the center.  After lighting my own candle, I found a place along the way to place my own light among the others.  It was an extraordinary experience for us all.  The metaphors were flooding my mind as I watched each of these students proceed on this path.  When they were little, their kindergarten teacher or parent held their hand on the way.  Now, they seemed to have grown so tall, only third graders, yet I marveled at how they had gotten so big.

    As I looked at them, a veil was drawn back for me and I could see much more than I could usually see.  Everything was revealed to me in a moment.  It seemed to me, as they walked, that I could sense them as younger children, as who they were now, and even who they were going to one day become.  In the darkness, I really felt I could see their future before my physical eyes.

    When my turn came, I realized that in the last two years of first and second grade, I always had walked the advent spiral first – before they did – so that they could see how to do it again.  This time, they walked confidently alone without my example.  This time, I followed their example.  As I passed by each one of the lights on my way into the middle, I felt as if I were floating along in the starry heavens, each star one of these students of mine that I hold so dear.  Each star lighting my way into the center.  I was so humbled to place my small light, no bigger or smaller than the others, down on the path to join in lighting the room.  I could also feel how profound it was for the students to see their teacher walk this inward path.

    With each eye intently and silently upon me, I felt the connection, the gravity, the grace, the sacredness of the responsibility I carry on their behalf, but also I felt, “I am the so-called teacher this time, but I am really only one of you.  You, who have been my teachers, who will be my teachers in the future.”  What a privilege to live in such light.  Blessings on us all.

  • Chaos theory and fractals: 1/4

    Chaos theory and fractals: 1/4

    The following four questions (one per post) were posed in a recent class. My edited responses follow.

    1. So I understand that any shape in nature can be converted to a mathematical formula, right? Then you take that formula and plug in the variable related to that shape and feed the answer into the variable spot in the next equation, continuing that as many times as one wants (with potential infinite number of times).   Here is where I get stuck.  Does each answer produce a smaller version of the original shape or does the answer produce the snowflake-like shape of the whole, complex, product?

    We do not know how to find mathematical functions for every shape in nature; indeed this is still an area of active research.  Usually we can model (i.e. approximate) natural shapes with mathematical functions, but nature is a little more slippery and variable than the stark purity of mathematics proper, so we don’t get exact results.  This isn’t usually a problem, however, because the whole point is that we can occasionally find mathematical functions that provide something of the… essence of the form.

    For example your lungs are fractal in nature, branching and branching and branching… but they don’t do it in only one way.  Rather, they have different ‘stages’ of branching, each with a different ‘fractal dimension’, depending on how many previous branches lie before it (I can explain what a fractal dimension is if you wish in another post).  (Read a single page about this from the Cellular and Molecular Life Sciences journal – slightly technical but you can probably get the basic idea through a different lens, which is always useful: http://www.springerlink.com/content/q103206u70402617/)

    But to answer the question directly re: “each answer…”,  most fractals both occur and are described by the taking of a single set of relations (it may be a shape), and repeating that relation/shape on different scales, with rotations, translations (moving without rotating), etc.  In other words, you start with a thing and then put it through some sort of process, then take the new whole that results and do the same process to it again, and so forth.  What is interesting is that a pattern emerges out of the process… and the process is primary, NOT THE THING.  In fact, you can take ANY shape, and subject it to a specific process over and over, and you will get the SAME fractal form, regardless of the original shape.  So each iteration (“each answer”) is a further unfolding of the process, a more ‘detailed’ rendering of the pattern inherent in the unfolding of the process.

    So it’s not just a question of “smaller versions” or of reproductions of the “whole”; the pattern that defines the particular fractal is really the expression of an infinite process (usually based on some pretty simple and limited rules, like “rotate 36 degrees counter-clockwise and scale by a factor of 1/2”), which can be “reversed” (rotate 36 degrees clockwise and scale by a factor of 2) – so size has nothing to do with it, and you can start anywhere in the process with whatever you have, because the thing isn’t ‘the thing’ if you know what I mean: ‘the thing’ is the process.  This is very good for nature because nature can take whatever is there and do some pretty simple things and get some very complicated results.  It doesn’t have to start with a ‘grand plan’ that requires everything to be exact and fit in just the right way in order for it to ‘work’, but can just take whatever is there and MESS AROUND.  This isn’t the whole story but I’m just trying to relate this to fractals specifically.

  • on distinction

    Trying to summarize:

    The drawing of a distinction is a formative act.  Drawing a distinction FORMS the space, constellates the infinitely possible unknown into spaces which are shaped by the particular WAY in which the distinction is drawn.  The resulting spaces can not help but take the complementary shape of the distinction that defines them.


    The drawing of a distinction is an INformative act.  Drawing a distinction INFORMS the observer to distinguish the space like this.  The resulting shape of any possible knowledge within the space cannot help but be informed (formed-in) by the particular way in which the distinction is drawn.


    The distinction is an epistemological act that creates itself, and in so doing, is also the creation of the space in which the distinction occurs, as well as of the observer who makes the distinction.


    There are levels of distinction; distinctions can be made about distinctions, and distinctions can be made about those distinctions…


    Distinctions are atemporal and aspatial.  They are the primitive act.  The distinction between the distinction, the space, and the observer are all formally equivalent, just as the three interior angles of a triangle are all formally equivalent.  In other words, they are interchangeable in any process that specifies ANGLE. 


    The only process that specifies distinction, space, and observer is DISTINCTION.  The form (which is the same thing as the LAW of form) is an ouroboros.


    Consciousness is the consciousness of distinction.  That is to say, consciousness is distinction; it cannot be other than consciousness of distinction. Therefore consciousness is co-extensive with all distinction.  What is distinguished is conscious of its distinction; this is why there is distinction; this is why there is consciousness.


    Time is a higher order distinction that requires recursivity.  It is a distinction about a distinction.  Time is dependent upon the oscillation of distinction between two states made by that distinction.  Time is the injunction to cross the boundary between the states, and once crossed, to cross again.  The crossing is atemporal.  Only the distinction of the crossing is temporal.  The distinction of the crossing of the distinction is time.


    <—————-distinction—————->

    Now, I have no idea if any of the above is sensible or insensible <crossing boundary: its sensibility or insensibility is only possible on the basis of the distinction as such.  In itself it is neither sensible nor insensible, but what it would be if it could> but it has raised some questions for me:


    Is distinction ONLY binary?  Is distinction always between this state and that state, that is, between one and another?  Is it possible to distinguish not just between one:one, but between one:many, between one and another+another+another?  In other words is this a binary reductionism? <crossing boundary: yes, if this binary distinction is made.  now make another distinction.>


    Okay, but what about the, um, distinction in quantum mechanics of the possible eigenstates of the wave function?  There are an infinite number of possible eigenstates for any given system (even the simplest), and they must ALL be taken into account (their probabilities must be worked in to the equation). Then, out of the whole field of possibilities, ONE is observed.  In the observation, all the other possible values have a probability of zero, while the observed value has a probability of one.   This sounds an awful lot like the idea that the fundamental act of distinction creates the space and thus the values of any distinction in the space thus created.  Does this have any validity, either as an analogue or at least a metaphor? <X: Do you distinguish it as such?>


    …


    Okay I have another question.  What happens if we DON’T DISTINGUISH?  I mean, we don’t un-distinguish and we don’t distinguish.  Without distinction… what happens?  Is this (GSB language) a marked empty mark? 


    Crap!  And can making a distinction of a distinction actually be the same as not making a distinction in the first place?


    Please, anyone, help?!

    <X:                 >

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