{"id":17805,"date":"2016-03-11T13:18:44","date_gmt":"2016-03-11T18:18:44","guid":{"rendered":"http:\/\/johnesimpson.com\/blog\/?p=17805"},"modified":"2016-03-11T13:19:11","modified_gmt":"2016-03-11T18:19:11","slug":"an-infinity-of-reflexive-trajectories","status":"publish","type":"post","link":"https:\/\/johnesimpson.com\/blog\/2016\/03\/an-infinity-of-reflexive-trajectories\/","title":{"rendered":"An Infinity of Reflexive Trajectories"},"content":{"rendered":"

\"one<\/a><\/p>\n

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

From whiskey river<\/em>:<\/p>\n

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.<\/p><\/blockquote>\n

(Iris Murdoch [source<\/em><\/a>])<\/p>\n

…and:<\/p>\n

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.<\/p><\/blockquote>\n

(Robert Fulghum [source<\/em><\/a>])<\/p>\n

…and:<\/p>\n

Poem to My Daughter<\/strong><\/p>\n

The sky has, is, one exit, one excuse,
\nand if I’m dead now that I’m saying this,
\nI can’t vouch for my transition from life
\nas having been rough or even evident.
\nHave I tried turning it off and then on again?
\nHave I tried throwing it against the wall?
\nGetting to know you, getting to know all
\nabout you getting the mirror to mean
\nnot only me, and thinking I must look
\ndumber than I look — dumber, then, than prose —
\nI walk through the laundry room regretting
\ngetting the weekend done this way, as if
\nbackstage, and say the name of your birthplace
\nas if I’d lost a hundred dollars there,
\nwhich I may have \u2026 Dear, when nowhere, don’t do
\nas those of us in nowhere do — just go.<\/p><\/blockquote>\n

(Graham Foust [source<\/em><\/a>])<\/p>\n

<\/p>\n

Not from whiskey river<\/em>:<\/p>\n

The Student<\/strong><\/p>\n

She never spoke, which made her obvious,
\nthe way death makes the air obvious
\nin an empty chair, the way sky compressed<\/p>\n

between bare branches is more gray or blue,
\nthe way a window is more apparent than a wall.
\nShe held her silence to her breast like a worn coat,
\nsmoke, an armful of roses. Her silence
\ncolored the smaller silences that came and went,
\nthat other students stood up and filled in.<\/p>\n

I leaned near the window in my office. She sat
\non the edge of a chair. Hips rigid, fidgeting
\nwhile I made my little speech. February<\/p>\n

light pressed its cold back against the glass,
\nsealing us in. She focused on my lips
\nas I spoke, as if to study how it’s done,
\nthe sheer mechanics of it: orchestration
\nof jaw and tongue, teeth shifting in tandem,
\nshaping the air. So I stopped, let her silence<\/p>\n

drift over us, let it sift in like smoke or snow,
\nlet its petals settle on my shoulders.
\nI looked outside to the branches<\/p>\n

of a stripped tree, winter starlings
\nfolded in their speckled wings, chilled flames
\nshuddering at the tips. Students wandered
\nacross campus as if under water, hands and hair
\nunfurling, their soundless mouths churning—
\nirate or ecstatic, I couldn’t tell—ready to burn<\/p>\n

it all down or break into song. When I looked back
\nher eyes had found the window: tree, students,
\nbirds swimming by, mute in their element.<\/p>\n

It was painful to hear the papery rasp
\nof her folding and unfolding hands, to watch
\ncolor smudging her neck and temple, branching
\nto mist the delicate rim of one ear. I listened
\nto the air sunder between us, the feverish hush
\ncollapse. I could hear her breath—smoke<\/p>\n

rising from ice. I could see what it cost her
\nto make that leap. What heat it takes
\nfor the body to blossom into speech.<\/p><\/blockquote>\n

(Dorianne Laux [source<\/em><\/a>])<\/p>\n

…and:<\/p>\n

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.)<\/p><\/blockquote>\n

(Wikipedia [source<\/em><\/a>])<\/p>\n

…and:<\/p>\n

#13:<\/strong> 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”…<\/p>\n

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<\/em> was expanding — space itself<\/em>: 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.<\/p>\n

What he did not<\/em> know before developing his blog post: the universe is believed to be expanding not in three dimensions, but in four<\/em>. (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<\/em> projection of a four-dimension object<\/em>. Think of a line drawing; think of a shadow. These two<\/em>-dimensional abstractions are projections of their three-dimensional counterpart<\/em>. 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-<\/em> or a triple <\/em>torus.)<\/p>\n

He included in the blog post an image of one<\/em> 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.<\/p>\n

\"a<\/a>But 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<\/em> of a creature’s (or a laser beam’s) wandering around a 3D projection<\/em> of a 4D object<\/em>.<\/p>\n

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<\/em> route shown…<\/p>\n

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.<\/p><\/blockquote>\n

(JES, Maxims for Nostalgists<\/em>)<\/p>\n","protected":false},"excerpt":{"rendered":"

[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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","h5ap_radio_sources":[],"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":3,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":"","jetpack_publicize_message":"Topology meets the mind: 'An Infinity of Reflexive Trajectories'","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[38,247,12,1393,95,5,251,324,4159],"tags":[61,2268,2813,2946,3285,3653,3708,4268,4269,4270],"class_list":{"0":"post-17805","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-backwards","7":"category-ruminations","8":"category-03_runningaftermyhat","9":"category-whiskey-river-runningaftermyhat","10":"category-science-medicine","11":"category-06_writing","12":"category-poetry-writing_cat","13":"category-researchresources","14":"category-essays","15":"tag-memory","16":"tag-dorianne-laux","17":"tag-the-universe","18":"tag-robert-fulghum","19":"tag-maxims-for-nostalgists","20":"tag-graham-linehan","21":"tag-wikipedia","22":"tag-iris-murdoch","23":"tag-3-torus","24":"tag-topology","25":"entry"},"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6kZSG-4Db","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/posts\/17805","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/comments?post=17805"}],"version-history":[{"count":14,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/posts\/17805\/revisions"}],"predecessor-version":[{"id":17822,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/posts\/17805\/revisions\/17822"}],"wp:attachment":[{"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/media?parent=17805"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/categories?post=17805"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/johnesimpson.com\/blog\/wp-json\/wp\/v2\/tags?post=17805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}