Posts tagged with geometry

[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.

hi-res

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

hi-res

Transgressing boundaries, smashing binaries, and queering categories are important goals within certain schools of thought.

Reading such stuff the other week-end I noticed (a) a heap of geometrical metaphors and (b) limited geometrical vocabulary.

In my opinion functional analysis (as in, precision about mathematical functions—not practical deconstruction) points toward more appropriate geometries than just the `[0,1]` of fuzzy logic. If your goal is to escape “either/or” then I don’t think you’ve escaped very much if you just make room for an “in between”.

By contrast `ℝ→ℝ` functions (even continuous ones; even smooth ones!) can wiggle out of definitions you might naïvely try to impose on them. The space of functions naturally lends itself to different metrics that are appropriate for different purposes, rather than “one right answer”. And even trying to define a rational means of categorising things requires a lot—like, Terence Tao level—of hard thinking.

I’ll illustrate my point with the arbitrary function ƒ pictured at the top of this post. Suppose that ƒ∈𝒞². So it does make sense to talk about whether ƒ′′≷0.

But in the case I drew above, ƒ′′≹0. In fact “most” 𝒞² functions on that same interval wouldn’t fully fit into either “concave" or "convex”.

So “fits the binary” is rarer than “doesn’t fit the binary”. The “borderlands” are bigger than the staked-out lands. And it would be very strange to even think about trying to shoehorn generic 𝒞² functions into

• one type,
• the other,
• or “something in between”.

Beyond “false dichotomy”, ≶ in this space doesn’t even pass the scoff test. I wouldn’t want to call the ƒ I drew a “queer function”, but I wonder if a geometry like this isn’t more what queer theorists want than something as evanescent as “liminal”, something as thin as "boundary".

hi-res

[Alexander] Grothendieck expanded our … conception of geometry … by noticing that a geometric object 𝑿 can be … understood in terms of … maps from other objects into 𝑿.
Kevin Lin (@sqrtnegative1)

(Source: quora.com)

## Why is the slope of perpendicular lines flipped over and switched signs?

Oh! This one only took me 17 years or so to figure out. This was a “fact” I had committed to memory in school but never thought about why.

` `

From The Symplectization of Science by Mark Gotay and James Isenberg:

There are some connections to circles and homogeneous coordinates (`v/‖v‖`) but let’s leave those for another time.

Gotay & Isenberg’s exposition using the metric makes it clear that the
`/‖v‖` part of the definition of `cosine` isn’t where the right-angle concept comes from. It comes from the `v``₁ w``₁ + v``₂ w₂`.

$\large \dpi{150} \bg_white \!\!\!\! \text{Given two vectors named } {\color{Golden} \vec{v}},{\color{DarkBlue} \vec{w}} \text{ made up of } \\ \\ {\color{Golden} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} =\vec{v}} \text{ and } {\color{DarkBlue} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} =\vec{w}} \\ \\ \text{the metric \scriptsize{(which is going to encode geometric information)}\Large\ is:} \\ \\ \boxed{g = {\color{Golden} v_1} \cdot {\color{DarkBlue} w_1} + {\color{Golden} v_2} \cdot {\color{DarkBlue} w_2}}$

` `

So if the slope of my starting line is `m`, why is the slope of its perpendicular line `−1/m`?

First I could draw some examples.

I drew these with http://www.garrettbartley.com/graphpaper.html which is a good place to count out the “rise over run” and “negative run over rise” `Δx` & `Δy` distances to make sure they really do look perpendicular.

The length and the (affine or “shift”) positioning of perpendicular line segments doesn’t matter to their perpendicularity. So to make life easier on myself I’ll centre everything on zero and make the segments equal length.

The metric formula is going to work if let’s say my first vector `v` is `(+1,+1)` (one to the right and one up) and my second vector goes one down and one to the right. Then the metric would do:

`+1 • +1` (horizontal) `+ +1 • −1 `(vertical)

which cancels.

` `

What if it were a slope of 9.18723 or something I don’t want to think about inverting?

This is a case where it’s probably easier to think in terms of abstractions and deduce, rather than using imagination in the conventional way.

If I went over `+a` steps to the right and `+b` steps to the up (slope=`b/a`), then the metric would do:

`a•? + b•¿`

What is that missing? If I plugged in `(?←−b, ¿←a)` or `(?←b, ¿←−a)`, the metric would definitely always cancel.

And in either of those cases, the slope of the question marks (second line) would be `−a/b`.

So the multiplicative inverse (flipping) corresponds to swapping terms in the metric so that the two parts anti-match. And the additive inverse (sign change) means the anti-matched pairs will “fold in” to zero each other (rather than amplifying=doubling one another).