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)























[Karol] Borsuk&#8217;s geometric shape theory works well because &#8230; any compact metric space can be embedded into the &#8220;Hilbert cube&#8221; [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 &#8230;
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&#8230;. In Borsuk&#8217;s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

&#8212;-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&#8217;s cube → 0.

[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.

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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.

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

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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”.

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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.

Photo: Horse paintings in  Chauvet Cave

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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
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  • houses are square; efficient industrial production of the straight and right-angle-based construction materials
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    and work plans
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    means it would be relatively expensive to build otherwise.
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  • yards are square
  • parks are square
  • city blocks are square
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  • (…except older cities which resemble a CW complex more than a grid)
    ComplexCity: Moscow
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In general relativity flat Euclidean spaces are deformed by massive or quick-spinning objects. 

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Due to an uneven distribution of mass inside the Earth, its gravity field is not uniform, as indicated by the lumps in this illustration.

Still image of world gravity map

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

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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.

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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.

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EDIT: Totally forgot about tattoos. 






Stones from neolithic Scotland, for the story see Lieven Le Bruyn.

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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)




Tony Robbin

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A fun exercise/problem/puzzle introducing function space.

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