An Infinity of Reflexive Trajectories

one view of a triple torus

[Image (courtesy of Wikipedia): one of numerous graphic representations of a mathematical (and perhaps physical) space called a 3-torus (also three-torus, or triple torus). For more information, see below.]

From whiskey river:

We are such inward secret creatures, that inwardness is the most amazing thing about us, even more amazing than our reason. But we cannot just walk into the cavern and look around. Most of what we think we know about our minds is pseudo-knowledge. We are all such shocking poseurs, so good at inflating the importance of what we think we value.

(Iris Murdoch [source])


Every person passing through this life will unknowingly leave something and take something away. Most of this “something” cannot be seen or heard or numbered or scientifically detected or counted. It’s what we leave in the minds of other people and what they leave in ours.

(Robert Fulghum [source])


Poem to My Daughter

The sky has, is, one exit, one excuse,
and if I’m dead now that I’m saying this,
I can’t vouch for my transition from life
as having been rough or even evident.
Have I tried turning it off and then on again?
Have I tried throwing it against the wall?
Getting to know you, getting to know all
about you getting the mirror to mean
not only me, and thinking I must look
dumber than I look — dumber, then, than prose —
I walk through the laundry room regretting
getting the weekend done this way, as if
backstage, and say the name of your birthplace
as if I’d lost a hundred dollars there,
which I may have … Dear, when nowhere, don’t do
as those of us in nowhere do — just go.

(Graham Foust [source])

Not from whiskey river:

The Student

She never spoke, which made her obvious,
the way death makes the air obvious
in an empty chair, the way sky compressed

between bare branches is more gray or blue,
the way a window is more apparent than a wall.
She held her silence to her breast like a worn coat,
smoke, an armful of roses. Her silence
colored the smaller silences that came and went,
that other students stood up and filled in.

I leaned near the window in my office. She sat
on the edge of a chair. Hips rigid, fidgeting
while I made my little speech. February

light pressed its cold back against the glass,
sealing us in. She focused on my lips
as I spoke, as if to study how it’s done,
the sheer mechanics of it: orchestration
of jaw and tongue, teeth shifting in tandem,
shaping the air. So I stopped, let her silence

drift over us, let it sift in like smoke or snow,
let its petals settle on my shoulders.
I looked outside to the branches

of a stripped tree, winter starlings
folded in their speckled wings, chilled flames
shuddering at the tips. Students wandered
across campus as if under water, hands and hair
unfurling, their soundless mouths churning—
irate or ecstatic, I couldn’t tell—ready to burn

it all down or break into song. When I looked back
her eyes had found the window: tree, students,
birds swimming by, mute in their element.

It was painful to hear the papery rasp
of her folding and unfolding hands, to watch
color smudging her neck and temple, branching
to mist the delicate rim of one ear. I listened
to the air sunder between us, the feverish hush
collapse. I could hear her breath—smoke

rising from ice. I could see what it cost her
to make that leap. What heat it takes
for the body to blossom into speech.

(Dorianne Laux [source])


The triple torus… can be obtained by “gluing” the three pairs of opposite faces of a cube, where being “glued” can be intuitively understood to mean that when a particle moving by inertia in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face, in the same direction. (After gluing the first pair of opposite faces the cube looks like a thick washer, after gluing the second pair — the flat faces of the washer — it looks like a hollow torus, the last gluing — the inner surface of the hollow torus to the outer surface — is physically impossible in three-dimensional space so it has to happen in four dimensions.)

(Wikipedia [source])


#13: The blogger was preparing a post in which he considered the nature of reflexiveness: things, thoughts, creatures, and souls folding back on themselves. And he came across something called a “3-torus”…

As he knew, the nature of the universe is to expand, perhaps infinitely. He knew that an expanding universe didn’t imply everything becoming further distant from everything else; it meant that space itself was expanding — space itself: that which distance measures. He knew of the difficulty of picturing infinity, and he knew in general of the idea that “infinity” could be conceptualized as a circle or sphere: that the “infinite” merely refers to a shape with no beginning or end — a shape that folds back on itself.

What he did not know before developing his blog post: the universe is believed to be expanding not in three dimensions, but in four. (The fourth, of course, being time.) He had difficulty conceptualizing this, until learning that the fourth dimension is often depicted — mathematically, at least — as a three-dimensional projection of a four-dimension object. Think of a line drawing; think of a shadow. These two-dimensional abstractions are projections of their three-dimensional counterpart. The analogy, then, extends to the question: what does a four-dimensional universe even look like? One common answer: it looks like a 3-torus. (This shape is also referred to as a three- or a triple torus.)

He included in the blog post an image of one type of 3-torus. (It looks like a pretzel.) He imagined a creature wandering its surface; depending on the direction of movement, the creature might come to traverse the entire surface.

a 2D projection of a creature's wandering around a 3D projection of a 4D objectBut then he took a leap: imagine that the green torus is hollow, or filled with some other solid color — say, white. Imagine that the path of the creature’s wanderings is actually the path of a laser, slicing through the 3-torus. If you lopped off a complete portion of the 3-torus, at just the right angle, and then looked at what was left behind, you’d see a miracle. A simplified view of the miracle is shown at the right: a two-dimensional projection of the laser’s three-dimensional movement. More precisely, it is a 2D projection of a creature’s (or a laser beam’s) wandering around a 3D projection of a 4D object.

From this perspective, we might say: The creature wandered two entirely separate routes. But we’re blinded to the reality that there is but one route shown…

And then he realized: memory works in exactly this way. We fancy that events “happen” or “have happened,” as though they’re discrete entities. But it’s really just one long memory, with 99% of its reality lopped off for temporary viewing. Coincidences do not exist. And we can (most of us do) wander forever — through our lives, through our memories — and never see the same thing twice… or recognize it, if we do.

(JES, Maxims for Nostalgists)

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  1. Oh my! I was freaked out enough by the sight of a clover-shaped pretzel made of green magic putty at the top of this post (an oblique St. Paddy’s Day reference??) – and then I discovered it was about polydimensional geometry, which is the thing that does my head in more than anything. (I recall an “I give up” moment about 25 or 30 years ago where I was chatting to a doctoral physics student in a bar in Oxford and he told me that they were currently working on an model of the universe with 11 dimensions! It’s probably gone up since then….)

    Also, on a technical note: WordPress seems to have perpetrated one of its oddest glitches yet – comments here (at least, the more recent ones that I just checked) all seem to have had the lower-case ‘i’ removed throughout. It’s remarkable how little this affects understanding; indeed, you don’t even notice at first. I wonder if this is part of some sort of cognitive psychology experiment?

    • How… bizarre (the stripped lowercase i’s). I hadn’t noticed but will have to look into that. It could be a sign of some weird server (mis)behavior!

      I’ve got a Facebook friend — a former chemistry professor, now some sort of finance-sector computer guy — who eats up stuff like multidimensional geometry with a spoon. Alas, he’s not a blog reader.

      If I’d had more time, I think I’d have tightened up that little parable at the end there. But I was in a panic to get it posted soon (especially after Not Posting At All on a Friday right before this one!).

    • The i’s are back!

      (The culprit was a new version of a popular WordPress plugin, called Jetpack, which improves (as a rule, haha) commenting among many other WP features. They issued a fix a couple of days ago, which hadn’t yet installed — so thanks for letting me know about it.)

  2. Nice to know there’s a “rational explanation”. With oddities like these, my mind slides too easily into either paranoia (“Someone” is messing with the Internet – as it appears to ME only!) or hypochondria (Something has gone wrong with my brain!).

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