Posts tagged with manifolds
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.
Beware of the pursuit of the Superhuman: it leads to an indiscriminate contempt for the Human. —George Bernard Shaw, Man and Superman
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.)
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.
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.
M. C. Escher’s painting Ascending and Descending illustrates a non-conservative vector field, impossibly made…. In reality, the height above the ground is a scalar potential field [the scalar (single number attached to a point) being the height above the ground]. If one returns to the same horizontal place, one has gone up exactly as much as one goes down.
So that’s that picture related to
Conservative vector fields obey the product rule:
conservative scalar field is also the output of a derivative operation…just a different dimensionality)