Chapbook

[sasha]

Nothin' Up My Sleeve...

There's a well-known series of numbers called the Fibonacci sequence, named for Leonardo Bonacci, aka "Fibonacci", a 13th century mathematician who studied it. A sequence is simply a list of numbers generated from some arithmetic rule - in this case, each member of the list is the sum of the previous two, starting with 0 and 1. Adding these together gives the third: 0 + 1 = 1. Adding this to the 1 just before it gives us the fourth, 2. Add this to the 1 preceding it and we get 3... which when added to the 2 gives 5; and so on.

You get the idea. Skipping the intermediate details, the sequence goes

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... ad infinitum.

This sequence is full of unexpected patterns & surprises, and has been delighting numerophiles for centuries. You'll find it in just about every mathematician's toy box. I was playing with it one rainy afternoon several years ago and discovered something that left me so baffled & amazed I thought I'd made a mistake. But I hadn't.

These numbers make a few surprise appearances in nature - the arrangement of leaves in plants, & the structure of chambered nautilus shells - but their greatest appeal lies in what happens when we calculate the ratio between adjacent pairs. The first pair is the one we started with: 0 and 1. The ratio of 0 to 1 is just 0 (0/1 = 0). The next pair is 1,1; 1/1 = 1. Now that we're past the starting gate, things start to get interesting:

third pair: 1/2 = 0.5
fourth pair: 2/3 = 0.66666
fifth pair: 3/5 = 0.6

followed by

5/8 = 0.625
8/13 = 0.61538...
13/21 = 0.61904...
21/34 = 0.61764...
34/55 = 0.61818...
55/89 = 0.61797...
...

It almost looks like the ratio is closing in on a value somewhere around 0.617 or 0.618 - and in fact, that's exactly what it's doing. That number is called the Golden Ratio (or sometimes the Golden Section) - rounded to 6 decimal places, its value is 0.618034, and was so revered by the ancient Greeks that they gave it a name, Φ (phi). They considered it - among other things - the most visually pleasing aspect ratio for a rectangular shape - the most ideal, most beautifully rectangly rectangle (something I've kept in mind when building shelves, as has my daughter when designing garden layouts.) The aesthetic appeal we associate with this number can't be overstated.

On that long-ago rainy day, I was curious about the path of convergence these ratios follow on their way to Φ, and had coded a spreadsheet to do the calculations. Progressive ratios seemed alternately too large and too small, but each time closer than before - a typical alternating path to convergence, like a bouncing ball gradually coming to a stop. Interesting.

This is where I stepped off the trail & found myself in the woods. It occurred to me that the Fibonacci sequence was only one of an infinite number of such sequences that could be generated by the same rule - all I had to do was start things off with a different pair of numbers to generate an entirely different sequence, whose pairs would converge to a different ratio - if they converged at all. So I replaced the two starting values with something else - I don't remember what. Let's just say 37 and -5.

As I'd known I would, I got an entirely different string of numbers: 32, 27, 54, 83... but I was more interested in where their succesive ratios would lead. So I ran off a few dozen, and saw that, as before, their successive ratios rapidly converged...

... to 0.618034 - the Golden Ratio.

This was so unexpected I was sure I'd made a mistake somewhere. I double-checked my coding, and found no errors - unsurprising given the calculations' simplicity. I tried doing a few by hand - and got the same result as the spreadsheet. I tried changing the starting values again, this time to non-integers, like π and √2. But no matter where these sequences started, the ratio of their consecutive values always converged to 0.618034. I was as baffled and awed as when I was 13 or 14 and quite accidentally discovered the parabola, hiding in the zig-zag path between the tiles of the shower stall in our house.

The story should end here - at an amazingly counterintuitive truth about numerical sequences I'd accidentally stumbled across on a rainy day. But a mystery without resolution is like an unscratched itch, & I had to go scratch it. If seeing how a magic trick is done spoils it for you, consider this The End. But if you're not averse to a little off-roading, buckle up & I'll try to explain just why all Fibonaccies lead to the Golden Ratio. It's going to involve some algebra and a bit of pre-calculus hand-waving, but I hope it doesn't diminish the magic and beauty of the result.



*********************************************************************

Still here? OK the question is: Why does the ratio between consecutive elements of ANY Fibonacci sequence - not just THE Fibonacci sequence - converge to the same value, regardless of where they start? To answer this, we need to cast the sequence in purely symbolic terms. We need to translate the statement "each number in the sequence is the sum of the previous two" into math-speak:

To do this, we need to give names to each of the components in this definition. So let's call the nth term in the sequence Fn - where n can mean ANY counting number. We'll call the one just before it - the (n-1)th - Fn1 (1st previous), and the one before that Fn2 (2nd previous). Now we can express the rule for generating the sequence in math-speak as

1) Fn= Fn1 + Fn2. But for this to be true, there have to be at least two members of the series already in existence before we can calculate a third. It doesn't tell us anything more about them, so they can be whatever we like. Fibonacci just chose 0 and 1 because - well, why not.


We want to find out why the ratio of Fn1 to Fn converges to 0.618034, as n gets very large. So let's call that ratio R, recognizing that R will be different for each pair of numbers. We'll also assume that n is fairly large, putting us some distance downstream from the starting pair.

2) R = Fn1/Fn.

Returning to equation 1): let's divide Fn by Fn1. We can do that, as long as we do the same to what it's equal to, namely (Fn1 + Fn2). (If I have the same number of coins in each pocket and remove half of them from each, or add three to each, the number in my left pocket still equals the number in my right - though not the same number as before.) So let's divide both sides by Fn1:

3) Fn/Fn1 = (Fn1 + Fn2)/Fn1.

This is almost equation 2), which tells is what Fn1/Fn is equal to. In fact, Fn/Fn1 is just ____1____    or 1/R.
                                                                                                                                      (Fn/Fn1)

Dividing the sum of two numbers by a third number is the same as dividing each number individually by the 3rd, and adding the two results. [ Try it yourself with (6 + 4)/2. ] Doing this to the right-hand side turns equation 3) into

4) 1/R = (Fn1/Fn1) + (Fn2/Fn1).

Any number divided by itself is just 1; so 4) becomes

5) 1/R = 1 + Fn2/Fn1.

But Fn2 and Fn1 are consecutive "Fibonacci" numbers too, the pair just before Fn; so as long as we're far enough along in the sequence, their R-ratios won't be too different. In other words, when n is very large,  Fn2/ Fn1 is ALMOST the same as Fn1/Fn - they're "close enough" to consider the same R. (This is the hand-wavy part.)

6) 1/R = 1 + R.

We can get rid of that fraction by multiplying it by R, as long as we do the same to 1+R:

7) R(1/R) = R*(1 + R)

The R's on the left cancel each other out; and the R outside the parentheses on the right multiplies each term inside them for the same reason equation 3) became equation 4):

8] 1 = R + R*R.

Finally, we can subtract 1 from both sides and rearrange the terms a bit to give us

9) R² + R -1 = 0.

This is a particular class of equation known as a quadratic, whose solutions are well-known; so without taking that side trail, I'll just cut to the chase and say that there are two values of R that satisfy it: (√5 - 1)/2, and -(√5 + 1)/2... equal to 0.618034 and -1.618034 respectively.

This is true only because each term in the series is the sum of the two preceding it - it says nothing about where such a series needs to start. Though each string of numbers is different, they ALL have the property that the further out you go, each one gets closer and closer to 0.618034 times the next.

I never saw this one coming - after all these years, Nature can still pull a fast one on me!
 

[sasha]

Here There Be Dragons...
 

 
I've discovered a trail nearby that until a week or two ago I never knew existed!



Well, not exactly - I knew it was there, and had followed it a few times as far as an impenetrable tangle of windfall & blowdowns left by a downburst last year.
 

 
I went back last week - and the impenetrable stretch has been made penetrable!

So penetrate it I did.


The trail runs along the eastern shore of Scott Brook, starting somewhere north of its confluence with Towne Brook. The conjoined waters then continue south as Priest Brook until they meet the Millers River. I'm intimately familiar with the trails running along the western banks, but points south of the storm damage on the east side were new territory for me. I set out with the eagerness of a child...
 

 
A soft gray day - warm but not too, humid but only a bit...


marsh birds I knew only by their calls singing from the reeds near shore...
 

 


the greenery in boisterous pagan abundance...
 

the mountain laurels in full bloom...
 

 


The jangling clamor of the outside world was now little more than a distant background hum, less distracting than the deerflies circling my head. Here I was occupied with more pressing matters - like where to place my feet, or how best to frame the surroundings in the viewfinder.

I'd assumed that the trail originated from a campground situated beside the confluence - but it doesn't - it can't. The site lies west of the watercourse, and without a bridge there's no direct access for an ATV. I came within sight of a few RVs and a beach, but never closer than the slowly gliding waters of Priest Brook would allow. Here the trail seems to veer off to the east - heading upslope into mixed hardwood, to... where? The map shows only a large undeveloped tract.

After a few "one more bend, then I'll turn around" 's, I reluctantly conceded that I'd gone far enough... for today. I'll be back. Next time, I'll park closer to the trailhead. And I'll get an earlier start.
 

.

 
 

[winddance]

how fun to be an adventurer. great pictures too. Is that white flower mock orange?

[sasha]

mountain laurel - cousins to rhododendrons. They're all over the place around here.

[saw]

appealing recount.....photos, story.....I sure miss those explorations.....who knows maybe I'll get another crack at it.....one must be hopeful........................................especially in the new world

[sasha]

thanks for joining me on the walk, last one for a bit. I've got an all-day kiilobuck car repair (just in time to celebrate my final credit card payment to the water fiasco back in March) - and a routine (I hope) maintenance stop with my physician. Weather's going to suck for the rest of the week anyway, might as well spend it in waiting rooms.

[mnaz]

Wow ... Phi sort of reminds me of Pi. As a (still employed) urban denizen myself, I have little chance to get deep into the woods-- but I do have a pretty cool 600 acre (mostly) forested park nearby to explore. I found a few narrow passageways between main trails a couple of years ago-- navigating over and through downed trees in some cases. Some good adventures, albeit on a smaller scale.

[sasha]

Pi at least has a geometric interpretation, a connection to the physical world you can intuitively grasp: the distance around a circle divided by the distance across. Phi emerges from a shadowy realm of infinite series, polynomials, converging limits, all that good stuff. Yet they both crop up in expected corners of mathematics everywhere that have nothing to do with circles or arithmetic sequences. They're hard-coded into the DNA of the universe, somehow. Euler's number, e, is another - it has no straightforward geometric interpretation (though you can gin one up), but shows up in applications from compound interest to vector analysis - and, of all places, the bell probability curve. There's no apparent reason for these unlikely connections - but there they are anyway...

Went back to the trail today, and completed a loop traverse I suspected was there but didn't verify until today! And discovered yet another in the process. I think I know where it goes... but there's only one way to find out for sure....

[sasha]

...Yet they both crop up in expected corners of mathematics...

UNexpected.... UNexpected corners... sheesh

I shouldn't post when 'm tired....

[sasha]

 

Zuihitsu for Bernie

April 7, 2020 - Winter's grip on our throats is weakening, Covid-19's is tightening. The gross ignorance & incompetence of those in charge is painfully obvious as they blunder about denying that anything's amiss, even as the death toll skyrockets. Meanwhile, "social distancing" and "shelter in place" compete for Newest Entry In The Lexicon, and life before the virus struck is wistfully referred to as "the Before times".

As a solitary beast, the isolation mandate has little effect on my lifestyle - if anything, it's an enhancement, freeing me from the task of coming up with excuses not to engage. (And I know I'm far from being the only one who felt that way.) It was like an official sanctioning of my fondness for solo tramping along woodland trails, and on this day I was eager to conduct such an expedition to one I'd only just discovered.

Right where Priest Brook crosses Royalston Road on the border between Royalston & Winchendon MA, there's a gauging station I've passed so many times I no longer see it. Some months earlier I'd parked across the road from it and idly strolled over for a closer look - and discovered it to sit at the beginning of what looked like a footpath winding south along the western bank of the brook. I followed this for a short distance, reluctantly turning back because of the hour, and catalogued the place as one worthy of a second look.

And today seemed as good as any for that look.

In the vicinity of the gauging station, the surroundings are quite open & accessible, but before long the brush presses in as the land levels off, and the brook slows as it broadens into a leisurely meander through the floodplain.

 

 

Where it's prone to springtime flooding, the trail veers off slightly upslope into the woods. At the edge of one of these stretches, I came upon a clearing beside the brook. It looked like an impromptu campsite: a well-worn track to the water, a firepit filled with charred branches.... a pair of cross-country skis propped upon a log, its poles strapped to a tree nearer the water.

The same strap supported a gaudy arrangement of artificial flowers; just above it there appeared to be a small photograph. Curious, I clambered down the embankment for a closer look.

 

The plot was open and had clearly seen a lot of foot traffic, but was remarkably free of trash. I paused to regard the skis, then continued to the tree. A laminated photograph was pinned to the trunk just above the floral arrangement, depicting a man maybe in his 50s seated in a kayak, holding the paddles feathered across his lap. He smiled back at the camera over his shoulder. I unpinned the photo and turned it over. On the back was a simple handwritten farewell to "Bernie". I assumed Bernie was the kayaker, the message from friends & family, and that they'd chosen this spot to erect a little memorial. The skis were probably his, too.

I was moved by this, and carefully returned the photo to the tree. I felt a bond with this stranger in the photograph who'd apparently loved this place enough for his tribemates to enshrine him here. I supposed that he'd fallen under the spell of the water and the woods as I had, and had regarded this as a holy place, a temple to be visited in quiet, reverential meditation - and his people were acknowledging that devotion. That made him a kindred spirit. I wanted to let them know that I, too, worshipped here - that I was a fellow parishioner who'd chanced by, and wanted to wish Bernie well.

So I dug the notepad I always carry out of my camera bag & scribbled : "My deepest condolences"... and after a moment's thought (and a few false starts) appended a little haiku-ish triplet: "in eternal sunlight / he now kayaks / with the angels". I signed it simply: " - a passing hiker".

I tore out the page & tucked it as best I could behind the photograph, with enough sticking out to alert any future visitors to its presence. Feeling a bit subdued, as one does leaving a church after a funeral, I returned to the trail.

I pressed on. The landscape continued unfolding before me as it always does in the wild, like in the pages of those magical picture books I used to pore over as a kid - now marshland, now upland slopes... deep woods, forest glens, open fields - whispering to me that such magic is real, that one only has to look to see it. It's a siren song I've heard all my life, and even after all this time I'm captivated by her voice.

 

But I couldn't quite shake the quiet melancholy of the shrine - and I began to wonder if my own offering would ever be seen by those it was meant for. One rain, I thought, and it's ruined. Even if the ink doesn't run, the paper will turn to mush. In the days before digital photography, I'd likely have been carrying an extra roll or two of film, and could have pressed an empty can into service as a vault for my greeting. But it wasn't - so I didn't, & I couldn't.

It was getting late enough in the day to consider turning back, so at the top of a particularly steep little overlook, I decided this would be far enough for today. After a lingering look around, I turned and descended the slope, watching my footing especially carefully - I'm not 35 any more.

Hell, I'm not even 60 any more.

It was on this descent, intent on my footing & watching the ground ahead, still contemplating the discovery of the shrine and a wee regretful that my halloo was open to the elements when something nearly buried caught my eye, and a handful of synapses snapped shut as they made a connection.

It was a spent shotgun shell, big enough to roll up a note scrawled in the woods for the family of a departed member - less durable than a PE film can, but better than leaving it out in the open. So I picked it up, cleaned it out best I could, & when I passed by the shrine again, stuffed my poem into it and tucked it upside-down in the strapping holding the poles. I don't know if it'll ever see the light of day again, and I never will - but at least the woods bore witness to my attempt to reach out, and if Bernie's out there tramping or paddling down the river or just sitting beside it in contemplative worship, he'll know - and that's good enough for me.
 
 
 

[sasha]

Ethereal Light

Sometimes when walking through the woods with Camera draped around my neck and Cannabis luring my subconscious out to play, something about the ambient light will catch my eye & stop me in my tracks. At the right time of day, and the right time of year, it can assume a certain aura, a luminosity, an almost imperceptible mistiness that invites a lingering look - quietly insists upon it, like a kindly old schoolmaster or well-seasoned nun gently shushing a roomful of spirited young children, calling their attention to something wondrous.

I suppose this effect is due primarily to dust and the sunlight's angle of incidence - it seems most prominent when the trees are backlit, the sides facing me awash in light reflecting off the forest floor. But this is just mechanics. A quiet mind can see beyond the brush strokes, and for just a moment glimpse the whole painting. It's as if the air itself is being illuminated, the way you sometimes see in classical renderings of churches or grand cathedrals. And in the absence of understory - as in a pine wood - the trees become a colonnade supporting the vaulted roof, the spaces between them filled with that mysterious essence, imbuing the grove with an august serenity, a stillness that can't help but inspire a hushed reverence far beyond that of optical scattering by airborne particulates.

How many times have I hoisted Camera and tried to capture that mysterious essence? How many times have I hunted for the proper framing, for the right exposure that reproduces the full dynamic range as a string of 1's and 0's? And how many times have I failed?

I don't even try anymore. I've never been able to capture it - that ethereal something, that trace of sanctity hovering between the trees in a little pocket of wilderness, that faint hint of Beyond that always tugs at my soul and draws me into a long, contemplative look. And maybe that's how it was meant to be. Maybe that whisper from eternity isn't meant to be caged. Maybe it's enough just to see it - to really see it, and to experience it in a kind of pagan beatitude. Maybe that's all it was meant to do - to remind us that magnificence doesn't have to be huge - that the ordinary can be extraordinary, and that the smallest things often turn out to be a lot bigger than they seemed. We can hear shouts whether we want to or not - but secrets are whispered, and can only be heard when we listen. Maybe it's assuring us that the magic we used to feel as children didn't go away when we grew up - that it's still there as long as we don't shout it down. Maybe it's just insisting that we honor Silence by listening more.

Works for me........

[winddance]

sometimes you can feel it but the camera just doesn't see it. know the feeling.

[sasha]

 
The camera doesn't always see the same scene that I do - it's more democratic in how it distributes the total incoming light among the individual pixels. I was struck on a walk one day by the interplay of light & shade in a particular sunlit clump of roadside weedage, & tried to capture it. This is what the camera saw - a perfectly exposed, perfectly boring photograph:
 

 
 

This was the scene as I saw it - the picture I' thought I was taking: 
 

 
Hardly museum quality, but at least truer to my subjective experience.

It took playing with settings after the fact that I knew nothing about - the gamma factor, in this case. Apparently it adjusts midrange brightness while leaving the highlights (and lows) unchanged. By reducing it to 30-35% of its default setting, I got the picture I'd seen at the time. I've since been able to rescue some shots that came out "muddy" by increasing it. It's fussy work, and I don't know enough about what each knob does to fix every broken picture - so I fiddle and "try stuff", sometimes with success, sometimes not.

Add this to your list of things us retired folk finally have the time for!

[mnaz]

I know the feeling of the camera not catching what I saw, the light play and ambiance. Plus, in the desert, moving across such great open distances, horizons I chase after are often unrecognizable when I get there.

[sasha]

A long time ago I tried my hand at underwater photography. Besides the logistics of just being there, the optical properties of water pose a whole new set of challenges. For one thing, New England waters are colder & more biologically productive than warmer tropical waters (that's why whales migrate north to feed). It's also why the best underwater scenics are taken in transparent southern waters - ours are green with algae that limits visibility to 10-15 ft at best. So to get a decent picture of anything, you've got to get right on top of it. And the water itself exerts a slight telephoto effect meaning you've got to back up to get the whole subject in frame. It's why I bought a wide-angle lens - underwater, a 28mm has about the same visual field as a 50 does on land. I also tried a magenta filter to counteract the green, but it increased exposure time to compensate for the light it absorbed. (This was in the days of film - digital post-processing wasn't an option for the hobbyist.) And because of the facemask and camera housing, I couldn't get my eyeball much closer than 4-5" to the viewfinder. I had fun exploring the lake I live near, but didn't give National Geographic much to worry about.