From the chaotic

Hidden away in Frank Oppenheimer’s glorious Exploratorium is an unnamed, little-noticed exhibit.

Thousands of tiny white spheres are enclosed between glass plates that form a plane, upon which the spheres are free to move. You give the whole thing a random shake or two, and wait for the little quanta to settle.

Patterns!

Here’s a simulation. Move your mouse over the window below to release the spheres. Moving the mouse outside the window will freeze the spheres. Click within the window to add new spheres.

(Uses verlet integration for the physics of collisions, with code I adapted from Florian Boesch. You may need a fast browser like Chrome to view the simulation)

So. Alternately enchanted and dismayed by this universe of order arising from and dissolving into chaos, I’m always heartened to find the occasional calm in the eddies:

Aztlan

Duncan West and Cesar Chavez, United Farm Workers; Delano, California, 1974.

It rains and blows monsoon style. I drift offramp, where a pale man in a gray windbreaker with PROBATION on the back peers under the hood at the silent aircraft engine in his brand new Ford Behemoth II. He and his truck block the right lane.

Roll down the window. “Hey buddy, want me to help push it off the road?”

He won’t meet my eyes. No tools in hand – just staring at the Big Engine That Couldn’t. His NRA sticker and gun rack support a working differential diagnosis: 1) too-much-anxiety disorder.

He hesitates; yells something and nods, so I park in the dirt and hop out.

Good God. He’s jumped back into his truck to steer, closing and locking the door. Not sure I can push this heap myself, although the road is flat here. I’m having a hard time even budging her. Knock on window: “We’ll probably both have to push, you know?”

Traffic all around now – people getting off work, holiday shoppers safe and dry in cars we’re now blocking. And me, wearing my sharply creased slacks and carefully ironed soul, crouching behind a squat, shiny truck I now realize I’m going to be attempting to push up a freeway offramp in about 15 yards. Bending to get some leverage, shoulder to bumper level, I form a hypotenuse which rain flows down.

Rolling a little. Yeah. Problem is, he’s driving out into traffic instead of pulling over to the right curb. Angry honks. My best diplomatic tone: “Lot of room over on the right, buddy!”

We slow, still well out in the right hand lane. Sounds like America’s Silent Majority has lined up behind us, unwilling to push but happy to lean heavily on their Ess-You-Vee horns.

Then the loud laughing Spanish-speaking voices suddenly all around. A half dozen men join in, leaning into the rear of truck. One fellow smiles at PROBATION jacket, then bends his knees, braced against tailgate.

The truck rolls easily, and bumps the curb. Success. Perfect ending: driver turns key, Ford Behemoth starts right up. Flooded.

We the Movers grin wide. Team Aztlan runs back to their own old truck idling in the midst of the furious Christmas shoppers and employees on work-release from the endless gray buildings.

Each pusher has another laugh and an honest-to-God living smile for me as I walk past. I drive on.

And then I remember. We somehow forgot to check their “citizenship” papers before they pushed…”citizenship” papers for these Natives whose ancestors are dust beneath the endless California freeways.

And they didn’t care a bit about checking ours.

Imagining the Death of Nalagiri

Once a queen, who though possessing great beauty or perhaps because of it, desired the rare things of the world. One night in her castle of gray stones and carefully laid lines she dreamt of a magnificent elephant with glowing eyes. Her dream elephant strolled calmly and quietly through jade forests, moving trees gently to one side with pure white tusks that shone in the moonlight.

Here was rarity. The queen awoke and could not forget her dream nor her desire. She wanted the pure white tusks. She asked her husband the king to find the calm elephant and the pure ivory.

Although it seemed to him an impossible task, the king loved this beautiful queen very much. He sent word throughout the kingdom, asking for anyone who knew of this great elephant to report to him. He offered a bountiful reward to any hunter who could bring the tusks to him.

Now it happened that there was indeed a great elephant living in the endless, deep green forests nearby – an elephant whose heart was as great as his very being. This elephant wandered the forests and highlands, searching for a purity and a grace. He roamed the beloved land, gently touching what he found there with his long, delicate trunk, searching and learning.

It turned out that during this elephant’s long meanderings, he had once saved a hunter who had been lost in the forest, guiding him from the dangerous parts of the forest back onto familiar paths leading to safety. The hunter returned home. But as coincidence would have it, this same hunter lived in the kingdom where a queen dreamt of an elephant with glowing eyes.

Hearing about the great reward the king offered for the tusks, the hunter forgot the elephant’s great kindness, and set out on a journey back into the magical forest. From his previous contact with the elephant, he knew the elephant was a seeker of truth, so he disguised himself as a wise sage.

At dawn the elephant emerged from the forest and walked quietly into a clearing, where the hunter waited. Seeing what looked to him like a kindly sage, the elephant bowed, his beautiful white tusks gently scraping the ground. When he did this, the hunter quickly pulled and drew back his bow, and sent three poison arrows like lightning into the body of the great elephant.

The elephant felt the arrows pierce his body. He looked down and felt them burning and he knew they were poisoned. He felt his heart beat faster and his legs weaken. He knew his end was near. Other elephants nearby heard his cry, and came to investigate. When these other elephants saw what had happened, they were overcome with fury and attacked the hunter.

The great tusked elephant, however, knew the hunter had been overcome by desire.

He sheltered the hunter within the protection of his great limbs from the terrible anger of the other elephants.

Then when it was safe, the elephant quietly asked the hunter why he had done such a foolish thing. The hunter wept and looked at the ground in shame, and told of the reward the king had offered. The hunter confessed that he coveted the pure tusks.

Though weakening rapidly from the poison, the elephant immediately struck his great tusks again and again on a tree trunk. The tusks made sounds like thunder as they broke off. The elephant knelt, then turned slowly. His eyes shone, and as he turned his long trunk gently caressed the earth. He gave the broken tusks to the hunter, saying to him: “By this gift I have completed my search. My gift to you may vanquish the three poisonous arrows that have already pierced you – arrows of greed and anger and ignorance.”

– adapted from BUKKYŌ DENDŌ KYŌKAI

Wide Night Skies

(Wherein we try to quote Erich Marie Remarque from memory, for nights when such words will be needed):

A clear voice utters words that bring me peace, to me, in big boots, belt, and knapsack, taking the road that lies before me under the high heaven, quickly forgetting and seldom sorrowful, for ever pressing on under the wide night sky.

A clear voice, and if anyone were to caress him he would hardly understand, marching with the big boots and the shut heart, who marches because he is wearing big boots, and has forgotten all else but marching.

Beyond the skyline is a country with flowers lying so still he would like to weep.

Teachers

img_2631.jpg

The morning after the storm is just for returning north along the coast, but the next exit is east to Ojai and I remember I have someone else to thank. Coleus from the farmers’ market makes a small gift.

I’ve never been here, but do know this has been his home for more than fifty years. A low stone fence. Beyond the back gate a young woman walks the lush green toward me, and I ask after Mr. David Essel. She points to a man already walking with care, slowly and unafraid down a path from the house.

He is dressed in coat and knit hat on this sunny day in Southern California, although the warmth I remember across decades is still right there inside. He asks after me and my life and talks to me softly, some of science, some of India, and some of Krishnamurti and Christ and a certain Tibetan monk. Compassion. I tell him I remember his lessons.

In his mid-eighties he cares for his ill wife in their home. As we part, he tells me, “she will die at home.”

I thought I knew all roads there, but found myself inland driving through what felt like open doors. Soft light and emptiness, snow on far peaks, and a sky to fall into. Then a crossroads: east to the freeway and the fastest return. Or West. Back to the sea on a road I’d never traveled.

Effortless curves winding ever upward, then a plateau where horizons vanish and the day is full of sun. Hardly any other travelers in this stillness, but then just up ahead a car is swerving around something standing right in the middle of the road. Up close I see it’s a goose. Up closer, I see the long slender beak and know this is no goose. It looks to me like a brown pelican.

Even before curiosity, guilt flows in like a tide. What’s that about? Obviously something has gone wrong when a seabird is trying to hitch a ride along what amounts to a high desert highway, sixty miles inland. I’m tired. I want to go home. What am I supposed to do about it?

How comes a seabird here? I drive on and think about it. Along the coast I’d driven through the leading edge of the fierce storm two days before on my way south. Black, pelting rains and hail and winds lifting against the mountains, winds that made my truck shudder and slew near the summits.

Minutes and miles go by, and when I finally sigh and get turned around I’m no longer sure where I saw the bird. I slow. There is nothing in the road. No cars. Nothing. I drive back and forth, searching the shoulders.

Then I see, just out of the corner of my right eye, down a steep ravine to the right. A large bird has flown out just high enough I can see his head as he falls.

No shoulder. No place to park. A quarter mile down the road I pull into a turnout.

I can’t see anything. No movement. No flight. I watch the stillness. Nothing. Nothing.

Well I tried, I think to myself. The guilt recedes, and it’s an easy walk back to my truck. Out the window, there’s where I thought I saw the bird. Nothing. Accelerating, I start to think about where I might stop to eat.

I drive north. Twenty miles of high, soft grasslands, and a day like a dream. But also the thoughts of the seabird I’ve left there. Thoughts of storm and cloud canyons, and winds like walls in the sky.

When you fly in storm, lightning strikes are strobes that halt and hang raindrops before you, motionless, filling an infinitude.

Thirty miles on. I picture the separation of bird from flock and sea. I remember Terry Tempest Williams and wild geese passing in front of a full moon, their great hearts beating in flight; of the storm’s unexpected fury and the sudden parting from the flock, of the panic and calling out, of endless tumbling. I remember how she found the goose there by the lake and cleaned and arranged its broken body, covering the golden eyes with small dark stones.

And I thought of the times I have been alone and afraid, in the air, or at sea.

Turn back. Forty minutes later I find the little stretch of road with its small, lonely tree. Walk along the shoulder. At first the seabird is hidden from my sight, then all at once so easy to see. Wings drawn in tight. Head on chest. Eyes open. He seems to be waiting.

img_2630-1.jpg

Walk back to the truck. Drive it closer. Park. This may take a while. I put on boots and jacket. Sort gear out of a big black duffel.

He is calm at first at my approach, then stands and spreads his wings. He makes a little hop and glides down clumsily a few feet. There is bewilderment in his eyes at a sky which no longer buoys the graceful long wings. I move above him and he stumbles on feet made for ocean swells, struggling ever down through sand and sage to the bottom of the ravine, the small dark stones all about our feet.

Shimmering heat – we watch the other’s eyes: golden; blue.

At my first try at covering his head with my black bag, there is the clack of beak and the feeble cry. The long, lovely beak catches my finger, but I realize then how defenseless and weak he really is, for he can’t even break the skin. A few more tries as he, master of sky and wave, tries to walk away from this danger. I drop the bag and we stare deeper into each other. Quiet words of a longing for horizons that recede forever, for winds laden with sea mist, for the thrust of surf and birth. Calm, calmer, open. I carry him out of the dry ravine.

We drive together a few hours. When the cell phone signal appeared again, I called the Marine Mammal Center in Sausalito, and with a few more calls I reached their sister facility in Morro Bay. The saints there will wait for me late beyond their usual hours on this Sunday night.

Over the hills and out of dryness, now the sea mists makes soft cones of light beneath road lights, and then I hear surf behind the power plant at Morro Bay. The pelican stirs in the front seat with a low, liquid calling.

Morro Bay Marine Mammal Center

They are kind and gentle, as I knew they would be. I think they will let him be at rest at least for this night, here at the sea and the end of all things. I drive on again filled with a quiet elation. Maybe we both can return to a home.

Then back on the freeway near Cupertino, the call comes. They’ve taken him to a vet, who has found a shattered wing joint. It can’t be repaired. The talk on the phone is of what is best, and I am left to drive on with my quiet grief for this son of the sea, and the sky.

I think of another someone I want to thank sometime, for teaching – so much a Haiku, who was strong in life and who would have chosen all things be always in motion, just for the delight of it. I think of hope, and how meaning comes from dissolving into the world. And I wonder how many times I have given up too early, and too easily.

Integers in Euclidean Space Order

Consider the factored integers N=[(0), (1), (2), (3), (2,2), 5, (2,3), 7, (2,2,2)...], with each factor representing a point in Euclidean space. Each of these points is at a distance d from the origin, and can be ordered by d.

-- Ieso.hs
-- Integers in Euclidean space order

-- Uses the Haskell Number Theory functions, from the "Haskell for Maths"
-- site: http://www.polyomino.f2s.com/david/haskell/numbertheory.html
import Data.List
import Primes

-- Return distance between two points in Euclidean space. Points are lists 
-- of possibly unequal lengths, padded with zeros as needed.
dist_points x y = sqrt (realToFrac ((foldl (+) 0 (map sq (zipWith (-) x1 y1)))))
           where x1 = pad (lengthLongest [x,y]) x
                 y1 = pad (lengthLongest [x,y]) y
                 sq x = x * x
                 lengthLongest l = maximum (map length l)
                 pad len x = (x ++ zeroPadding) 
                     where zeroPadding = genericTake (len - genericLength x) (repeat 0)

-- Return list of all prime factors of n
primeFactors 1 = [1]
primeFactors n = flatten (map expandedFactors (primePowerFactors n))
            where
              expandedFactors x = genericTake (snd x) (repeat (fst x))
              flatten :: [[a]] -> [a]
              flatten = foldl (++) []

-- Return distance between two factored integers 
distance n m = dist_points (primeFactors n) (primeFactors m)

-- Return list of distances between origin and integers up to n,
-- sorted by distance d. Where ties occur, the lowest n comes first, i.e., 
-- where d=5 for both n=48 and n=5, the order is ... 5, 48 ....
-- Neil Sloane has suggested a different way of breaking ties is use of 
-- lexicographic order, ie for n=48 [2,2,2,3] comes before n=5 [5] - see
-- below.
distanceOrigin n = map snd (sort d)
                   where
                     d = [(distance 0 i, i) | i <- [0..n]]

-- In order to list the integers within a certain distance d from the origin,
-- we have to calculate distances up to 2^(d^2/4), e.g. to list ALL 
-- integers within a distance of 10, use range 2...2^25.

-- I believe a list is "complete" up to where a power of 2 appears, 
-- because numbers greater than that power of 2 can't be closer 
-- to the origin (except ties, depending on they are handled). However, 
-- I am curious about how to determine the  minimum number of 
-- integers you have to calculate in order to make a list complete, 
-- e.g., how high do we have to look to be sure we've found the
-- 20 integers closest to the origin? 

main = do
  -- Calculate integers within ~7 units of origin
  print "n ordering for ties"
  let d = [(distance 0 i, i) | i <- [0..8192]]
  print (map snd (sort d))

  -- Recalculate using lexicographic order to break ties
  print "lexicographical ordering for ties"  
  let l = [(distance 0 i, primeFactors i, i) | i <- [0..8192]]
  print (map lastx (sort l))
        where
          lastx (x, y, z) = z
 

For example, d=8.66 for N=125 at the point (5,5,5), while d=5.29 for N=128 at (2,2,2,2,2,2,2) :

*Ieso> distance 0 125
8.660254037844387
*Ieso> distance 0 128
5.291502622129181

Here is a simple view of several of these integers that lie in 3-space, with the distance between the origin and N=125 highlighted.
3d.gif
The sequence N=[0..8192] in Euclidean space ordered by d begins

 
*Ieso> distanceOrigin 8192
0,1,2,4,3,8,6,16,12,9,32,24,18,64,5,48,36,27,128,10,96,72,54,256,20,192,15,144,108,81,512,40,384,30,288,216,162,1024

(One way to think about this may be that integers with higher Ω(n) should be relatively clustered around the origin).

(The sequence is listed as OEIS A168521).

Who hath desired the Sea?

Who hath desired the Sea? — the sight of salt water unbounded –
The heave and the halt and the hurl and the crash of the comber wind-hounded?
The sleek-barrelled swell before storm, grey, foamless, enormous, and growing –

Who hath desired the Sea? — the immense and contemptuous surges?
The shudder, the stumble, the swerve, as the star-stabbing bow-sprit emerges?

Who hath desired the Sea? Her menaces swift as her mercies?
The in-rolling walls of the fog and the silver-winged breeze that disperses?

Who hath desired the Sea? Her excellent loneliness
Rather than forecourts of kings?

- Kipling

Black and White

“It can get bitter on these heights, and the winds, how they blow. There has been many a roaring night Hector and I have thought we would perish on this Hill, but, of course, how could we? What comfort is the hearth fire without raw weather howling outside the door, and what pleasure is there in rest unless the day’s toil was almost more than you could bear?”

“Which were your greatest storms?” I asked.

He furrowed his brow. “We’ve had so many,” he spoke slowly, “and each comes upon us afresh, as if it never had a predecessor. It is a curious truth that I cannot remember the snows here in any particular…at night, it is not these storms that I dream about, but always springtimes and storms below. In my dreams, I forget my century in heaven and recall only my few springtimes on earth.”

– Donald McCaig

Garrison, and Crystal Peak

Old journal entries, May 21, 2003, West Desert, Utah

Cottonwood, graves, and light.

The raw native sandstone, faintly inscribed.

Granite black, etched roses. Ruby Young Fowler, May 6, 1891 – April 1, 1918, “Beloved Mother”.

Dove in sandstone. “In memory of Mary E. Young”, born Dec 30, 1871, died Dec 17th 1880, “…farewell my dear…no longer must I stay.”

“In memory of Leroy Young, son of B and Mary E. Young, born Oct 15th 1867, died Nov 30th 1880. “Weep not for me.”

“Ray G, son of E & M Heckethorn, Dec 17, 1884 – May 15, 1911″ – “The angels called him.”

“Eugene L. Son of RC & L Heckethorn, April 15, 1909, June 6, 1911. Darling we miss thee.”

Then next day the journey through heat and light to Crystal Peak.

After years of seeing the mountain from afar, from over horizons, I am here.

Silence. Stillness, except my mind.

Warm yellow glow on startling white; improbable peak.

Oh how I wish I were free.