Posts tagged with geometry

`(x²−y<²−1) • (x²−z²−1) •  (y²−z²−1)   =   0`

(Source: imaginary.org)

Double integrals `∫∫ƒ(x)dA` are introduced as a “little teacher’s lie” in calculus. The “real story” requires “geometric algebra”, or “the logic of length-shape-volume relationships”. Keywords

• multilinear algebra
• Grassmann algebra / Grassmanian
• exterior calculus
• Élie Cartán’s differential-forms approach to tensors

These equivalence-classes of blobs explain how

• volumes (ahem—oriented volumes!)
• areas (ahem—oriented areas!)
• arrows (vectors)
• numbers (scalars)

"should" interface with each other. That is, Clifford algebra or Grassman algebra or "exterior algebra" or "geometrical algebra" encodes how physical quantities with these dimensionalities do interface with each other.

(First the volumes are abstracted from their original context—then they can be “attached” to something else.)

` `

EDIT:user mrfractal points out that Clifford algebras can only have dimensions of 2,4,8,16,… https://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension Yes, that’s right. This post is not totally correct. I let it fly out of the queue without editing it and it may contain other inaccuracies. I was trying to throw out a bunch of relevant keywords that go along with these motivating pictures, and relate it to equivalence-classing, one of my favourite themes within this blog. The text here is disjointed, unedited, and perhaps wrong in other ways. Mostly just wanted to share the pictures; I’ll try to fix up the text some other time. Grazie.

(Source: arxiv.org)

## Extrinsic vs Intrinsic

In The Geometry and Topology of 3-ManifoldsThurston speaks of the “extrinsic” vs “intrinsic” view of `{the 3-sphere − a knot}`.

I like learning words for something that’s been kicking the walls of my head trying to get out. Sometimes I can look at the world in an objective way, and sometimes everything centres on me.

• Extrinsic I and you and they are cerebral plains-apes, doing all of the things one would expect such to do. Chasing after money, status, sex, it’s all pretty simple when you see things this way.
• Extrinsic your exotic mind-expanding holiday is just a status signal.
• Extrinsic the hobby or job you hang your identity on tells me that you’re bourgeois and therefore define yourself through your activities or achievements.
• Intrinsic reinterprets monkeys’ “social grooming” as an activity that actually feels like something.
• Intrinsic is being in love.
• Extrinsic is love as a neurochemical sequence—dopamine, adrenaline, vasopressin, norepinephrine, oxytocin—generated by chance mutations and selective reproduction in response to an evolutionary problem.
• Extrinsic is 5,000 applicants for 5 jobs = 1‰ chance of getting it.
• Intrinsic my application was rejected so I’m a failure.
• Extrinsic is the pale blue dot, spaceship Earth, a wet rock bearing eukaryotes.
• Extrinsic I’m one of 10000 options s/he has on this dating site.
• Intrinsic is me; my likes, my interests; the homunculus behind the camera, 我, my life. Extrinsic this display is again an outgrowth of my bourgeois background.
• Intrinsic is exotic Peshawar, a world away and uncomfortably hot.

Extrinsic is more primates with a different culture-function applied to them. Going through the same life-stages, wearing different customs. Instead of going to a pub they do some sober, Pakistani alternative—but it’s all the same, all human stuff just expressed differently.
• Extrinsic when Parisians kiss each other on the cheek it’s not romantic, it’s just like a handshake.
• Intrinsic it’s gross that my parents have sex.
• Intrinsic I hate this town I hate the football games I hate the fakeness I hate the parochial small-mindedness there’s something better in store for me I need to get out of here I will get out of here
• Extrinsic is a visitor look how excited these people are about their quaint sports and amusements! That’s so nice.
• Intrinsic is being engrossed in reading.
• Extrinsic is seeing someone sitting still looking at paper with glyphs on it.
• Intrinsic is noticing how Thomas Piketty’s lower teeth are not-at-all straight like George Clooney’s, and being taken back a bit. Extrinsic is knowing that Piketty earns his stature from books and papers (arrayed behind him) whereas Clooney earns based on his looks.
• Intrinsic is waking up in the morning driven by the will to understand pseudo-Riemannian metrics and thereby, the Universe.

Chanting “sheaf cohomology" like a shamanic totem, carving it into my forehead my topknot my yarmulke my niqab, showing the world what’s inside my head and wishing they love me for it.
• Extrinsic I’m yet another plains-ape distracted by ego, status-signalling, and a scientific religion, cultured by stimuli that practically guaranteed I would behave this way.
• Intrinsic my new friend is so cool! She is friends with someone famous!
• Extrinsic of course she leads with her connection to the desirable; it’s all part of her personal branding scheme.
• Intrinsic is the many self-serious songs about my chain, my ding-a-ling, whatever.
• extrinsic is a boy in love with his homeostatic skin flap.
• Intrinsic is watching an ad targeted at you, and just reacting.
• Extrinsic is watching an ad that makes no sense to you, and thinking about the ad on a higher level.
• Intrinsic is appetite; craving; addiction.
• Extrinsic is flavourful sustenance.
• Extrinsic is asthexia; cachenia; syncope; renal failure.
• Intrinsic is the only death that matters (mine).
• Intrinsic is write drunk.
• Extrinsic is revise sober.

Beware of the pursuit of the Superhuman: it leads to an indiscriminate contempt for the Human. —George Bernard Shaw, Man and Superman

[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” `[0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …`

A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …

We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory

(I rearranged their words liberally but the substance is theirs.)

in `R` do: `prod( factorial( 1/ 1:10e4) )` to see the volume of Hilbert’s cube → 0.

It was the high zenith of autumn’s colour.

We drove her car out to the countryside, to an orchard. Whatever the opposite of monocropping is, that’s how the owners had arranged things.

The apple trees shared their slopey hillside with unproductive bushes, tall grasses, and ducks in a small pond in the land’s lazy bottom.

Barefoot I felt the trimmed grass with my toes. A mother pulled her daughter away from the milkweeds—teeming with milkweed nymphs—because “They’re dangerous”.

It was only walking along the uneven ground between orchard and forest that I realised that I almost never walk on surfaces that aren’t totally flat, level, hard, and constant.

` `

In the Chauvet cave paintings of 32 millennia before sidewalks, the creator — rather than being hampered by the painting surface — used its unevenness to their advantage.

But today

• sidewalks are completely flat in New York City; if you trip and hurt yourself because of their ill repair you can actually sue the City
• art (not all art but a lot of painting or screen-media) is conceived on a flat surface
• houses are square; efficient industrial production of the straight and right-angle-based construction materials

and work plans

means it would be relatively expensive to build otherwise.
• yards are square
• parks are square
• city blocks are square

• (…except older cities which resemble a CW complex more than a grid)

In general relativity flat Euclidean spaces are deformed by massive or quick-spinning objects.

and in sheaf theory things can be different around different localities.

The cave walls in Chauvet have been locally deformed even to the point that knobs protrude from them—and the 32,000-year-old artist utilised these as well.

Maybe when Robert Ghrist gets his message to the civil engineers, we too will have a bump-tolerant—even bump-loving—future ahead of us.

EDIT: Totally forgot about tattoos.

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The spirit of mathematics is not captured by spending 3 hours solving 20 look-alike homework problems. Mathematics is thinking, comparing, analyzing, inventing, and understanding.

The main point is not quantity or speed—the main point is quality of thought.
Geometry and the Imagination with Bill Thurston, John Conway, Peter Doyle, and Jane Gilman

(Source: geom.uiuc.edu)

hi-res