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Posts tagged with Gestalt

Imagine you were a wealthy writer — so wealthy that you could pay servants to look stuff up for you. Instead of drudging through tomes (or internet searches) to fact-check yourself, find original references, and so on. You just do the fun part: pontificate on paper.

Now let’s say after you have finished an essay, your servant / employee / virtual personal assistant comes back from his footnote research and tells you that statement #13 should be revised based on the best-known research on the topic. In fact, statement #13 is almost the reverse of the truth.

I can imagine things going one of two ways from here.

  1. In the less interesting case, statement #13 is an offhand remark upon which little else in the essay depends. You correct yourself, modify some text directly before and after statement #13, and move on. The only neighbours of the concepts in statement #13 are the transition sentences directly before & after it on a 1-dimensional topological line.

     
  2. In the more interesting case, what you said before statement #13 was meant to lead up to the exact statement you made. Perhaps #13 was a key point, or the thesis of the essay. And let’s further imagine that the text following statement #13 depended critically on the exact value of statement #13 being as you wrote it. When #13 is altered, the preceding text is no longer necessary and the succeeding text no longer works.

In an especially dire scenario, your PA’s research might overturn the worldview that led you to write the essay in the first place.

Like Holger Lippmann’s “Flower Circles 13”, changing one element renders the entire whole needful of alteration. Everything is so thoroughly enmeshed (see “complete graph” below for the neighbourhood relations) that no element of the text speaks in isolation.

That’s in distinction to the calculus, where smooth functions can be approximated by a differential.

“It’s easy to learn calculus and then forget what the point was.”
—Gilbert Strang

In physicists’ language, due to tightly, globally connected topology, perturbations cannot be localised. Rather, the opposite: local perturbations cause global changes in the object.

OK, someone dared me. I’ll say it: Gestalt.




"The whole is more than the sum of the parts"

Who says scientists are reductionistic? Any superadditive system—due to complexity, interaction terms, valuation by an Lₚ norm with 0<p<1, or some other reason—adds up to more in total than the pieces individually do.

(Such Lₚ norms are semimetrics but not seminorms.)




A lot of people think of &#8220;geometric&#8221; art as being math-y, in the same sense that the band Maps &amp; Atlases is math-y.
But I don&#8217;t think lines, circles, squares, tessellations, grids, and polygons are more mathematical than globs, leaves, aleatorics, colours, nets, or scribbles. In fact, I can link to a math post about each: lines, circles, hypersquares, polytopes, aleatorics, tessellations, blobs, grids, leaves, nets, scribbles, colours.
The mathematical thought that occurs to me when looking at this painting is how, in composition, every spot on the canvas influences every other spot. Holger Lippmann couldn&#8217;t have swapped a few of these circles because it would have ruined the effect.
Similarly in painting like this, if you added a splotch of yellow in the bottom right, that would affect the look of several other parts of the canvas.
Algebraically, the pieces of the composition are like a highly connected graph (in &#8220;how good it looks&#8221; space).

If you regressed compositional outcome against the content of each point in the painting (or just against the style of each circle), the relevant explanatory variables would be highly interactive terms. All the monomial, binomial, trinomial, &#8230; terms would be irrelevant.
 
The image is: 29417FlowerCircles_13_grid3 by holger lippmann, via wowgreat

A lot of people think of “geometric” art as being math-y, in the same sense that the band Maps & Atlases is math-y.

But I don’t think lines, circles, squares, tessellations, grids, and polygons are more mathematical than globsleaves, aleatorics, coloursnets, or scribbles. In fact, I can link to a math post about each: lines, circles, hypersquarespolytopes, aleatoricstessellations, blobs, gridsleaves, nets, scribbles, colours.

The mathematical thought that occurs to me when looking at this painting is how, in composition, every spot on the canvas influences every other spot. Holger Lippmann couldn’t have swapped a few of these circles because it would have ruined the effect.

Similarly in painting like this, if you added a splotch of yellow in the bottom right, that would affect the look of several other parts of the canvas.

Algebraically, the pieces of the composition are like a highly connected graph (in “how good it looks” space).

image

If you regressed compositional outcome against the content of each point in the painting (or just against the style of each circle), the relevant explanatory variables would be highly interactive terms. All the monomial, binomial, trinomial, … terms would be irrelevant.

 

The image is: 29417FlowerCircles_13_grid3 by holger lippmann, via wowgreat


hi-res