Thor’s Day Morning Mathematical Musings

Have you had your caffeine injection yet? Well, then, here are three puzzles (with answers, but the answers are not helpful!):

  1. Can you completely mix a mug of coffee, such that, at every point inside the mug, the coffee at that point is different after stirring from before stirring? Go get a cup of joe (or tea), stir it, and see what you think.
    Answer: no. There will always be at least one point that is the same after the liquid has settled, no matter how vigorously you stir it. It is mathematically impossible for there to be no such points inside the mug.
  2. Do there exist on the surface of the Earth, at any given time, two antipodal points that have exactly the same surface temperature?

    Answer: yes. What about two antipodal points that have exactly the same barometric pressure? Also yes. Two antipodal points that have exactly the same surface temperature and exactly the same barometric pressure? Yet again, yes. This is mathematically inescapable.‌

    antipodal points on a sphereAt any time there exists a continuous curve on the Earth’s surface on which every point has an antipodal point that also lies on the curve and that has the same temperature. There is a different continuous curve on which antipodal points have the same pressure. And the two curves must intersect, since both encircle the globe, each separating it into two pieces. So that means there must be, at any time, at least one pair of antipodal points somewhere on the surface of the Earth that have the same temperature and the same pressure.

    You’ve probably surmised by now—you drank that cup of joe, right?—that this is true not just for temperature and pressure but for any two continuously variable parameters (such as temperature, pressure, humidity, wind speed, solar and terrestrial radiation, cloud ceiling, particulate density, atmospheric composition, and so on). You would be correct.
  3. Think of a multi-digit positive integer. Any such number will do—for example, $76.$ Now add up its digits and subtract that sum from the original number. $76\,- (7+6) = 63.$ Now apply this algorithm to the new number: $63\,- (6+3) = 54.$ Keep doing this until the resulting number has shrunk to just one digit. $54\,- (5+4) = 45$, $\dots, 18\,- (1+8) = 9.$

    Ta da! (Yes, really.) No matter your starting number (as long as it has more than one digit), you will always end up at $9$.

    Here is a quick and dirty python program that performs this task for any positive integer, returning the end result (which had better be nine!) and the number of iterations it took to get there:

    def digi9(n):
        count = 0
        while True:
            k = sum(list(map(int,','.join(str(n)).split(','))))
            m = n - k
            if len(str(m)) == 1:
                return m, count+1
            n = m
            count += 1

    Let’s consider an example:

    >>> digi9(72459075)

    Starting with the randomly chosen number $72,459,075$, over two million iterations later we indeed end at $\dots, 27\,-(2+7) = 18,$ $18\,- (1+8) = 9.$

How are the answers to these little puzzles so? Welcome to the world of fixed point theorems! In mathematics, a fixed point is a member of a set such that an operation on the set at that point maps back to the point. The set can be anything—the set of integers, a Euclidean line, surface, or volume, etc. This concept has wide application and profound consequences in many branches of mathematics. The above puzzles are examples of fixed points in their respective sets. Put that in your mug and stir it!

Now go get some more coffee.

Show Me!

Suppose we have a function $f(x)$ such that $f(x) \in [a,b]~~\forall~x \in [a,b]$. That is, the function maps back to its domain. Then $f(x)$ has a fixed point $f(c) = c$ somewhere in the closed interval $a \le c \le b$.

Why? Well, it must be true that

\begin{equation}f(a) \ge a~~~ \mathrm{and} ~~~f(b) \le b \label{condition}\end{equation}

The intermediate value theorem says that if a function $f(x)$ is continuous on a closed interval $[a,b]$, then, for a given $c$ such that $f(a) \le c \le f(b)$, there must exist at least one value $x_0 \in [a,b]$ such that $f(x_0) = c.$

Since the range of our function is restricted to its domain, $f([a,b]) \in [a,b]$, we have from eq. \eqref{condition} that $f(a)-a \ge 0$ and $f(b)-b \le 0.$ If we define $g(x) \equiv f(x)-x$, this is $g(a) \ge 0 \ge g(b).$ By the intermediate value theorem there must then exist a value $c \in [a,b]$ such that $g(c) = 0$. Hence, there must exist at least one fixed point, $f(c) = c.$

This—or, rather, its generalization to any Euclidean space—is essentially a statement of the Brouwer fixed point theorem:

Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

Legend has it that Brouwer was lead to his theorem by pondering the surface of a cup of coffee upon stirring in a lump of sugar. (That someone would debase a good cup of coffee with sugar is a wholly different issue.)

Vsauce has an interesting video about fixed points, from which I stole the three examples above:



Polish immigrant tobacco farmers, 1940 (Getty Museum
Polish immigrant tobacco farmers, 1940 (Getty Museum)

Because, now and then, for the well being of your soul, you have to evict the empty diversions, addictive distractions, the noisome bile, and ponder, in the brief space exhumed by an image, a note of music, a spiraling leaf, a stranger’s touch, a kindness, a child’s wonder, Earthshine married to sliverous Moon, in this volume of relief, this anomalous bliss, this sudden expanse of silence—how is it that we, somehow, have willingly mongered purposeful calm for mindless glitter, mere noise?—and reflect on the inverse of nothing.

Comet C/2011 L4 (Pan-STARRS) and crescent Moon, 2013-03-12 19:25 MST
Comet C/2011 L4 (Pan-STARRS) and crescent Moon, 2013-03-12 19:25 MST



Rest in peace, Suki.

Littlest Kitty, Squirt, faithful companion.

Shoulder rider, lap sitter, mouse catcher, jumper.

Lover of chin scritches and forehead scritches and string cheese and pizza sauce.

Head butter, snuggler, blinds cord chewer, prancer.

Algebra proofreader, bookbinding assistant.

Fearless imperious disdainer of the Dread Great Vacuum Monster.

Heater hog.

Conqueror of Puffs.

The sweetest kitty.

April 2002 – 28 April 2016.


Suki 2015-10-27
Suki 2015-10-27 (click to enlarge)
Nap time (Suki, Frank, and Spike) 2015-01-04
Nap time (Suki, Frank, and Spike) 2015-01-04
Suki 2014-07-19
Suki 2014-07-19
Suki and Hat 2014-06-21
Suki and Hat 2014-06-21
Suki 2005-12-18
Suki the Bookbinder’s Assistant 2005-12-18
Suki watches the room 2005-06-02
Suki watches the room 2005-06-02
Suki and Puff 2003-02-05
Suki and Puff 2003-02-05
Suki and Cord 2002-12-17
Suki and Cord 2002-12-17


When I am at my desk, preparing for tonight’s observing. And it is evening.

Notes to self, part 437.

  1. When I am at my desk, preparing for tonight’s observing.
    1. And it is evening.
  2. If an email arrives from the satellite tracking app, you could open it.
    1. Be aware that this alert is for tonight.
    2. You did bring clothing for the weather, right?
      1. Not that it matters. You don’t pay attention to these things.
      2. Maybe you should.
      3. Come to think of it, you do recall thinking, this morning, that you could get away with not paying attention today, since you figured you’d be inside anyway.
        1. Running late, you were in a hurry.
        2. And you are lazy, when possible: it makes life more efficient.
  3. You bring your digital camera with you to the Observatory, because you never know what will demand photos on any given day.
    1. Or night.
    2. Mountain weather dances, flits, pirouettes.
      1. Cloud formations tend to be awesome.
      2. Atmospheric effects abound.
      3. Evanescent.
      4. Most, even in such a wondrous, sky-dance land, never look up.
        1. Is the mundaneness of our daily routines so important? That we must concentrate our gaze, glazed, on the mud of our feet?
        2. This is a great sadness.
  4. According to the alert, the International Space Station is due to pass overhead.
    1. Tonight.
    2. It is an especially good pass:
      1. For once, its path will track straight overhead.
      2. For once, it will largely miss the Earth’s shadow.
        1. This means the ISS will be a bright beacon from nearly horizon to horizon.
        2. This means it must be nearly either a north-to-south or a south-to-north pass. Ah, spatial geometry.
      3. For once, this good fortune is not tied to a predawn pass.
        1. You do not function well in the predawn hours.
    3. To compensate, Murphy’s Law will demand its due.
      1. It always does.
        1. This is consistent with observation.
      2. You hypothesize that this is a conservation law.
      3. Murison’s Corollary: When fortuitous good things happen, the balance of the Universe must be restored.
        1. Count on it.
  5. Fire up the satellite ephemeris program you wrote.
    1. Fetch the latest orbital elements from
    2. Create plots of azimuth and height above the horizon.
    3. Check that your observing window matches the alert’s prediction.
  6. Glance at the outside temperature: +12°F.
    1. You are surprised.
    2. But then you remember this morning, and your decision to leave the coat, the scarf, the gloves, behind.
    3. Tell yourself: that’s okay, this should be quick, it’s not that cold.
  7. Grab the camera and head outside ten minutes early.
    1. Always start early. Things go wrong.
    2. Rats: you didn’t bring a tripod.
      1. Hand-held video recording it is, then.
      2. You are secretly a little relieved at not being able to try anything fancy.
        1. Even though nobody else is here, it feels like a secret.
        2. Can we really keep secrets from ourselves?
  8. The door locks behind you: click.
    1. Memory trigger.
    2. Check your pocket for keys. After it locks behind you.
    3. This strikes you as humorous.
  9. Find a good spot: the middle of the small parking lot.
    1. Unobstructed view north, west, and south.
    2. The main telescope dome, three stories high, with a halo of Flagstaff light pollution, swallows the eastern sky.
    3. The satellite is on a south-to-north path tonight.
    4. Yes, this is perfect.
  10. The southwest wind is brisk.
  11. Unpack and check your camera.
    1. Breathe. Go slow. Be methodical. Think.
    2. Everything functions as expected.
    3. You don’t expect this. What will be the yin to this yang?
  12. +12°F is cold.
  13. Bare hands in +12°F will quickly go numb.
    1. Forty-five seconds to a minute, tops.
    2. You will marvel at the pain, though you cannot feel anything.
    3. Configure and start your camera before this happens.
  14. Check your watch: seven minutes to go.
    1. This, too, is unexpected.
    2. Try not to think about your body heat rapidly fleeing with the wind, that thief.
      1. Your warm, warm, cozy, comfortable body heat.
      2. Via your hands, and neck, and head, and feet.
      3. When did these jeans become so thin?
    3. Seven minutes is an eternity.
      1. When there is nothing to do but not think about how uncomfortable it is.
      2. When standing exposed in the wind.
      3. When it is +12°F.
  15. Keep your eyes on the view through the camera.
    1. Is that it, there, low in the southwest?
    2. Look up, blink-flick distorting tears, and verify with your eyes: yes, there it is.
      1. Right on time.
      2. In the right place.
      3. Glorious.
  16. Follow it slowly up, and over, and down to the northeast, where it softly slips into shadow before reaching the treeline. The five-minute pass passes quickly.
    1. Now you cannot feel your feet.
  17. It is done.
    1. Note the satisfaction in your gut: good data acquired, it says.
    2. Bask in that warmth as you lean down to pack up.
    3. And then your circumstances impinge.
  18. Fifteen minutes is a surprisingly long time when it’s +12°F out.
    1. And you’re wearing only a t-shirt and light jacket.
    2. And Birkies.
  19. If you can’t feel anything with the stumps at the ends of your arms, there will be consequences.
    1. You won’t be able to turn off or stow your camera.
    2. It will be surprisingly hard, and hence take a surprisingly large number of tries, and hence take a surprisingly long time, to get your key into the door lock and scurry back inside, to your office.
    3. Where it is not +12°F.


Camera: Canon G3 X. Video processed using kdenlive.

On Dirt

Star guts. Ground mountains. Seething motion unseen. Organism detritus. Feculence. Bug poop.


Knees pop as I bend down and pinch a gram of soil between my fingers. I bring it up to my face: grains, and the filler between the grains. I am looking at 1013 bacteria, each a tiny furnace eating chemical energy.

nutrients: carbon, nitrogen, phosphorous, potassium, calcium, magnesium, sunlight.

The critter universe in this quarter of a thimble of soil is more numerous than the stars swirling around the spiral galaxy we call home. By a factor of roughly a hundred. Or 1,500 times more numerous than the people on Earth. I work with such numbers every day. After thirty years, I still cannot fathom their import. Grains escape my clumsy dermal trap and sift back down, to the ground.

soil: earth, terra, qaḏāra, drytt. (The word dirt , from Middle English drytt, annoys soil scientists; it is an epithet.) Clay, silt, sand. Browns and tans, flecks of blacks and reds and whites. Crystalline facets sparkling in bright sunlight.


Soil is Earth’s largest reservoir of carbon, the basis of all known life. Too much carbon in the air, and we die, we voracious eaters and stirrers of dirt. Too little, and we die, we profligate disturbers of Nature. We live, we stumble, we contemplate, among a balance of energies that flow in overlapping cycles, large and small, short and long—a balance that seems ever more fickle, precarious, as the world grows warmer.


I look up and squint. The Sun, a middling, middle-aged star in an unremarkable part of the galaxy, warms my face, the skin on my arms. I know this sensation to be my brain, some still poorly understood interconnected agglomeration of neurons, synapses, and neurotransmitters, making sense (or such is my perceived reality) of neurochemical signals instigated by infrared quanta, packets of energy that were born of violent subatomic interactions and that fled the core of our star a thousand millennia ago. It takes that long for light to wend its random way from the Sun’s core to its surface.

macrofauna: woodlice, worms, beetles, ants.

A roundworm has 300 neurons and several thousand synapses. My cat has three-quarters of a billion neurons, and about 1013 synapses. She is gray and feisty and adorable, and getting on in years, like me, but she is not the sharpest knife in drawer, perhaps also like me. I rub a larger number of bacteria between my fingers than she has synapses, the connections between her brain cells.

mesofauna: mites, nematodes, roundworms, coneheads, blind and heartless pauropods, indestructible tardigrades.


Digital elevation map of the San Francisco Volcanic Field in Northern Arizona. (source: AZGS)
Digital elevation model of the San Francisco Volcanic Field in Northern Arizona. (source: AZGS, click to enlarge)

I wonder, where was this microcosm of mineralogy, pinched between my fingers, a million years ago? Flagstaff sits atop the San Francisco Volcanic Field, a complex of around six hundred volcanic cinder cones that have been active over the past six million years. The San Francisco Peaks, named in the 17th century for St. Francis of Assisi, themselves are the weathered remains of a stratovolcano that erupted between 0.4 and 1 million years ago. This bit of soil, at least its silica grains, may very well have been in the upper mantle, squeezing towards a volcanic hole in the Earth’s crust, when the photons warming my skin began their arduous journey.


We know about the bacteria in this pinch of soil, at least their rough numbers if surprisingly little else, because our optical instruments, microscopes, allow us that determination, given enough persistence.

microfauna: bacteria and fungi, thousands of species, mostly unknown to science; yeast; protozoa with their pseudopods, their flagella, their cilia; disintegrators of organics.

I don’t see stars in the daytime sky, other than our Sun, but I know they, too, are there, each of them a furnace converting matter in their cores to energy. At night, our telescopes show us their rough numbers, given enough time and persistence. Our microscopes also enable us a rough count of our neurons, and our synapses, these tangled little engines of thought. We don’t yet understand consciousness, our self-awareness that causes us to ask questions, stir the soil, and build tools so we can determine these incomprehensible numbers, and to ponder—if not grasp, for that seems a long way off still—their significance.



Navigating the Teratism, Or How I Came to Vote for Hillary Clinton

Given the widespread unreason in this mind-numbing political season, how can one cut through the din to make a good decision on who should be our Democratic presidential candidate? It is still not that hard to go about it at least somewhat rationally. Google is your friend—or can be, if you use it in the right ways. Here is a brief tale of the strategy I adopted in my quest to decide my Arizona primary—er, “presidential preference”—vote rationally.

From the start, I pointedly refused, both in public and, importantly, to myself, to take a position until just before election day here in Arizona. I’ve learned from past elections that the intense pressure of primary season can reveal facets of a candidate’s personality and experience that are important not to miss. So, I figured, the longer I stake out uncommitted territory, the more useful things I will learn, and the better my chances of making a sound decision. Furthermore—I did not realize this until later—being firmly uncommitted meant that I had no emotional investment in any candidate. Given our well-proven human tendency to defend our own tribe no matter the context, evidence, or consequences, this was a brilliant strategy for maintaining a certain amount of level-headedness and a boon to intellectual freedom. Alas, if only I could claim this brilliance was anything but an accident! Nevertheless, this lesson turned out to be the most valuable one for me in this experience.

I am now glad I did choose this course and hold off. It both allowed and forced me to check the substance behind the things people parrot, and the things people uncritically pass around as “memes”. I found that, primarily, these things are bunk—either untrue, or twisted to say or imply something untrue, or cherry-picked out of context to represent something untrue. It is little other than collective mental garbage going in and out, in and out of flaccid brains. Motivated reasoning and confirmation bias metastasized and run amok. This cannot be healthy.

The completeness of the logic FAIL (especially of very nearly every “meme” I’ve seen) coming from a certain segment of liberaldom astonishes, when you look into it. I did not expect this degree of unreason coming from liberals. But I suppose I should have: the psychologists tell us (and have rigorously shown) that people are people, whatever their ideological leanings. We all are surprisingly susceptible to the same biases, the same cognitive foibles—left, right, maybe not so much the mythical middle. Still, it has been disappointing to learn that critical thinking is not relevant to the very people who, at least occasionally, proudly pay it lip service: the educated liberal.


After a couple of months of observing the back-and-forth on Hillary Clinton and Bernie Sanders, and of chasing down the origins of some of the various topics that I care about, and of pushing back as gently but firmly as I know how against the tide of blatant unreason issuing from some of my friends, I arrived at the following conclusions (observations, really):

  1. Bumper-sticker thinking is not helpful. But it is oh, so seductive. This is a serious problem.
    1. Political “memes” mostly are dishonest, even false, and only encourage both lazy thinking and unproductive discord. They are like that cough that you just cannot shake, the one that at 2am you fear could be tuberculosis. Or cancer. It is easy to prove the dishonesty of most of them, but people almost never bother. Political memes are just too delicious, yet they are uniformly counterproductive.
    2. There is no breaking through the ideological barriers of many liberals. That is, the cognitively barricaded have little interest in facts, the truth, or, especially, complex contexts and shades of meaning, if it threatens the warped world view they’ve adopted nearly wholesale from two and a half decades of incessant drum beating from the right. Rational discussion is as hopeless among them as among rabid conservatives. Neither even notices the pounding drums.
    3. The right has largely succeeded in its long, sustained campaign of propaganda and negativity, even among liberals. Liberals, too, now unthinkingly assume the GOP’s disingenuous message framing as a matter of course, without ever questioning those false assumptions. Frank Luntz is unquestionably evil, but he is just as unquestionably brilliant.
  2. Sanders’s consistent message is THE progressive, liberal message…and has been, since approximately forever. This is good. This is excellent. Ours is a wondrously wholesome and healthy message. We care about people, and society, and the planet we live on. Further, Sanders does not do a bad job of framing our message effectively. This is unusual for a liberal. If only we had more who can do this.
  3. However, that’s pretty much it. To borrow a phrase, there’s little “there” there. Tastes great, less filling. Hardly a thought (although a larger amount of afterthought) is given to how we might usefully set about accomplishing any of the things Sanders drones on about—how to take plausible, substantive steps towards our shared liberal goals—given our current reality.
    1. This reality consists of past, present, and promised intransigence, nastiness, belligerent ignorance, blatant lies and cheating, unrelenting callousness, narcissism, and frequent infantilism among conservatives, as well as the sad fact that conservatives continue to control most of our country—Senate, House, Supreme Court, state legislatures, and state governorships. And it is unlikely any of this context is going to change much.
    2. So, bzzzt. This glaring yet persistent absence of substance, of a realistic plan moving forward, has been a deal-breaker for me.
    3. Clinton, however, plausibly claims to be about realistic (well, in large part based on realistic) solutions for making substantive progress toward the same goals, but taking into account our current reality (ding ding ding ding ding!), as I think any thinking person must. Short on pizazz, but pragmatic. This matters.
  4. Hillary Rodham Clinton has baggage. Big baggage. But the overwhelming majority of it ranges from mostly to completely bogus—nonsense, lies, and disingenuous exaggeration. That’s what over two decades of asinine, rapid-fire Republican attacks, ignorance, and dishonest agendas will pile on a person, especially if that person is competent (not to mention a woman), and especially when the media doesn’t do its journalism job (which abilities it willingly allowed to atrophy several decades ago). Follow up on any so-called “criticism” of HRC (as proclaimed by either Republicans or Sanders supporters), and you discover that—surprise!—95% of it is bullshit.
  5. HRC is likely more conservative than I am comfortable with on several important matters: many areas of foreign policy, a few areas of economic policy. (But no problems in her domestic policies that strike me as worrisome.)
  6. However, HRC also:
    1. has a buttload of experience in combat politics (Sanders has none); she is thoroughly battle tested,
    2. has an extra crap-ton of experience in dealing with and circumventing Republican assholery (Sanders has none),
    3. knows, and can adroitly handle, most if not all of the main players in the DC machinery (sorry, Bernie fans, but this matters),
    4. recognizes, readily acknowledges, and thinks strategically about the complex real world in which every policy decision resides (Sanders does not seem to understand—or at least acknowledge—that the real world is hugely complex),
    5. has a well-proven titanium spine (Sanders: indeterminate, as he’s never been tested), and
    6. appears to be, mostly out of public view, a genuinely warm human, despite all she’s been through.
  7. Hillary Clinton cannot frame a message effectively to save her life. This is unfortunate, for us all, and it unfairly hurts her in the polls. But to my mind this is not a valid deal-breaker since it does not affect the indomitable substance of what she brings to the table. It does, however, make it more difficult to uncover that substance from among all the bogus dreck. That’s on us: our failure has been and continues to be intellectual laziness.
  8. Sanders appears to be inflexible, unable to adapt much to a shifting, changing context. I suspect HRC is no yoga master, but Sanders is mineralized through and through. Has he substantively changed, in any way, in forty years? Perhaps he’s never been forced by circumstances to look at things differently to achieve a longer goal. This is a problem for me.

    I recently tried to explain this to a good psychologist friend (we went to their wedding in Thailand, even). He stopped me midstream and said that in psychology research circles this is a quantitatively well-studied thing and has a label: cognitive rigidity. He was pretty pleased to teach me something new. He also agreed that he, too, sees Sanders as notably rigid (his words: Sanders would likely score high on the scale of cognitive rigidity). That dawning in my head the week before our voting day I think is what clinched my evolving decision. If you can’t adapt, you won’t be effective.


So I voted for Clinton.


Idolatry is not my thing. I must admit that I am sick and tired of pervasive Bernie Sanders cult worship. Few if any in the Sanders crowd (at least any more) seem to actually think, and do research, with a serious eye toward considering the evidence as a dispassionate, unbiased observer. It seems to be mostly about seeking and sharing only those superficial fragments and tidbits that agree with predetermined opinions. This is not thinking.

political ideology circle (from reddit)
The left-right divide disappears not only in the moderate middle but also in the mental rigidity of fascism (from reddit, click to enlarge).

Further, you can trace the origins of most of the negativity and slime thrown at Hillary Clinton—regardless of who is flinging it today—directly back to Republicans (they play dirty, remember?). But, still…liberals? Misinformation and willful ignorance have been running rampant, even among us, and especially among Sanders zealots, who, in terms of blind ideology, are little different from conservatives (see graphic). The content is opposite, but the cognitive rigidity on display is the same. Further, voting primarily with your gonads (many conservatives do this) or your adrenal glands (Berniebots, that’s you) is not just unwise, irrational, counterproductive, intellectually dishonest—all true—but also unethical, in that abandoning your responsibility as a citizen to your fellow citizens, and to our shared society, is unethical.

This continues to surprise me, our abandonment of critical thinking; I just can’t seem to wrap my head around it yet. A sustained, twenty-five year barrage of shameless negative falsehoods and bullshit from the Republican machine, faithfully parroted by the mainstream media, must inevitably bias all of our perceptions and assumptions. Maybe that is the explanation. GIGO. But shouldn’t at least we, the educated liberal, be well aware of this bias?

It has been a disheartening several months.

excessive or blind adoration, reverence, devotion.


Here are a few resources I found helpful while pondering (mostly alphabetical by title). I will update this list sporadically, as I come across useful new articles. Latest update: 21 April 2016.


But where’s the butter?

“Hon, where’s the butter?”—a cartoon on our fridge.

A cartoon hung on our fridge for a couple of years. Its caption became a humorous private code, let’s call it The Code, between my wife and me. The Code is about assumptions, although, as with most frequently revisited negotiations, however small in the wider tapestry of life, a richness to it has developed over time.

honey wheres the butter
(click to enlarge)

Several years ago, I was searching the house for something (I forget what). I am thorough, and, like all pack-rat tendencied people, I am intimately familiar with all my Usual Locations for this or that object, the particular hierarchy of locations depending on how I categorize the object. They may be varied, but they are specific. I’ve noticed people tend to not believe this of us, because, probably, they are not like that. This is mistaken. I had searched the Usual Locations for where this thing, whatever it was, would have been, had I been the last one to move it. Having exhausted all known possibilities, I finally asked if she knew where it was. The thought of inefficiently looking in random places did not appeal. She interpreted my question as an accusation. I don’t remember it that way. Yet, in light of modern research on the fallibility of our recollections, which apparently are not the recalling of information so much as gap-ridden reconstructions of it, I cannot now be sure. Our brains seem to have uncommunicative minds of their own. Or maybe we still haven’t learned, most of us, to listen very well.

My demeanor can seem stern to others. But I am just being plainspoken. I don’t do sugar-coat very well. My wife excels at social diplomacy. Sometimes, I secretly wish I could do this. I marvel at people for whom this seems effortless (a sign of artistry). This societal expectation of largely inane filler pre- and post-pended to the message content—at times I view it as an affliction—seems in turns silly and exasperating to me much of the time, but even so I just don’t have the ability. (Or is it a skill? I would rather it be an ability, an innate talent, because then I would not be responsible for my learning difficulties with this chore.) Now I wonder: do our individual abilities, or lack of abilities, cause our value judgments? Do I think this inane because I’m not good at it? If true, this is disconcerting: is it even possible, then—especially in light of the spackled-over patchwork that is our memories—for any of us to assess objectively? Anyway, if the proffered filler accompanying a communication is (usually inaccurately) perceived as insufficient or insufficiently deferential to fragile egos, why do most people assume the intent is to attack? Perhaps in addition to physics I should have pursued psychiatry.

“Did you check here? There? . . .” She rattled off the most likely Usual Locations, surprising me with a couple I didn’t know she knew—an indication of how well we know each other, which in hindsight pleases me no end but at the time felt a little invasive, like a private area had been found out. “Yes, it’s not there. Yes, not there, either.” She assumed I had not thoroughly checked the Usual Locations. Or could it be my assumption of her assumption is inaccurate, and I don’t really know where this was coming from? She marched off to physically visit these places and check them herself, fully assuming it’d be sitting somewhere in plain view. As she passed through the kitchen I trailed along, a bit sullen. Try and imagine, if you will, a gait that broadcasts, “Rolling my eyes now”. Telling her that checking the same spots that I had just checked would be pointless would have been pointless. We can both be stubborn. And wrong.

She brought up short, arm outstretched, triumphal finger aimed at the counter. Imagine, I’m sure you can, a posture that fairly shouts, “See, I told you so!” Saying the words out loud would have been redundant. There it was. In the Wrong Location. But right in front of me this whole time, easily seen had I been less narrowly focused on only the Usual Locations. I had passed by it at least three times. I think we both felt a certain amount of smugness in being right, each in our way, neither of us really yielding to the other’s frame of reference, but also a small but significant awakening of awareness, a crack, inside our respective sectarian mental domains. Thus was born The Code, which showed up on the fridge I think a day or two later.

The Code is now a shortcut for reminding that one of us may not see a disagreement about something the same way as the other. “Honey, where’s the butter?” is a not infrequent non sequitur, which is not a non sequitur, in our back-and-forth. Couples and close friends tend to develop such language and (therefore) cognitive shortcuts, which often have their roots in a humorous—hence easily-remembered—episode which they happened to have navigated successfully. We remember these episodes, not always with the greatest fidelity to all the details, but for their lessons and the insights they give us into our personal relationships. And that is the important part.