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*To see some ordering and spatial concepts that might be involved in a
newer conceptual world than Picasso and Einstein lived in the 1920s,
track down lattice theory and its applications.
Here's a link to a wiki article:
http://en.wikipedia.org/wiki/Lattice_(group)
``In mathematics, especially in geometry and group theory, a lattice in R^n is a discrete subgroup of R^n which spans the real vector space R^n. Every lattice in R^n can be generated from a basis for the vector space by forming all linear combinations with integral coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell....''
People have been tiling space with primative cells or regular patterns even before tile was invented. If you've ever done tile work, you'll realize all sorts of math problems (spatial thinking) come up in fitting a particular sized and shaped tile system onto a particular sized floor or surface. You can play with regular patterned tiles and work out how to change the orientation of the tile to create extremely intricate patterns.
[Then there are the patterns of weaving and discovered limitations on the kinds of patterns that can be woven. This takes space concepts in lattices back to the neolithic. Well, and across the world's cultures in textiles and nitting.]
Since the ancient Islamic world pretty much banned iconography (except Persia for awhile), they developed the most spectacular of all art applications of these modularized spatical constructions or tiling principles or also called tessilations. Go here:
http://farm1.static.flickr.com/100/287190887_53900c4d13.jpg
And here is the text that goes with the image:
``Science 23 February 2007:Vol. 315. no. 5815, pp. 1106 - 1110DOI: 10.1126/science.1135491
Reports
Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture Peter J. Lu1* and Paul J. Steinhardt2
The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons (`girih tiles') decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West...''
Here is a link that merges computer art, high end quantum physics, engineering, and of course the underlying mathematics required to do all them at the same:
http://www.quantum.physik.uni-mainz.de/de/bec/gallery/index.html
The way I got into lattices was through tiling a big drawing with tetrahedrons (their 2-d version).
I did it the tedious way with a drafting machine back in the 70s. It was during the layout phase that I discovered some of the space concepts (like limited ordering structures) involved with lattice theory. For example, the angles of the unit cell determine whether or not the tiling edges will fit on a rectangular piece of paper. In other words, whether or not, the border of the tiling or tessalation will form a continuous straight line. Or put differenting, whether or not you'll have to cut some of the tiles. (You can see the problem, when you notice that all the illustrations of the Penrose tiling, seen below, are cut off by their rectangular picture frames.)
The usual rectangular shape for drawing paper is the wrong shaped `manifold.' Then remember too, the rectangular frame was invented to be especially well suited to the use of Eucidean geometry and perspective. The simple problem of tiling the rectangular with shapes other than the square or rectangle in 2-d leads into some really wild problems in topology where the packing orders and angles in point set are used to generate the manifold. Or something like that...
Part of my inspiration for a lot of this abstract work came from Sol Lewitt, especially this drawing:
http://www.diabeacon.org/exhibs_b/lewitt/lewitt-exhibs_b-top.jpg
Google `Sol Lewitt', then click `images' and you will see the direct connection with lattices. He made modular sculpture, mostly with cubes. He also played around with set theory ideas as part of his conceptual and minimalist art. Here is a link to Tony Smith, another big influence on me at the time:
http://seattletimes.nwsource.com/ABPub/2008/04/11/2004343238.jpg
...or do the wiki article on him. And of course there was Buckminister Fuller.
I found that the more I could understand about lattices and groups, the more I could understand in math and physics. In the above quoted Science Note, Penrose Tiling was the topic. Penrose is Roger Penrose and his main topic of research is general relativity and cosmology. Below is a link to a short article on Penrose tiling. The article also gives a method of construction, using straight edge and compass, so you can recreate the tiles and fiddle with them. This is essentially legos for grown-ups:
http://intendo.net/penrose/info.html
(BTW, I became a lego fiend [late-70s] because my kid was just getting old enough to play with legos rather than eat them. There was also some expensive Danish modular toy system I got him too. But I can remember the name of it now.)
The point is that these concepts from mathematics form our ultimate spatial constructions that are both part of our more popular concepts of space and plasticity (as I view these in the arts), and also part of the concepts of matter, space and time as physics views them in its ultimate reductions. In turn, the latter conceptual tools are used to create our cosmology. That is to say, our best efforts at a picture of the grand totality of being.
I think it's great that so far the quantum world and the relativistic world of gravity don't seem to fit together. That might mean there is something wrong with how we are forming our space-time concepts.
In the cultural history view (with as much Marxism as I can work in there), I think it is not co-incidental that this building controversy in cosmology (that has shattered the consensus on big bang) has been matched step by step with growth of the supposedly post-modern world dominated by neoliberal based economies. Reading and thinking about the big bang, and post-modernity, seems to constantly take me back to Europe in the 1920s and the explosions of math, physics, literature, arts, and of course philosophers like Cassirer and various and sundry others.
By turns, these conflicts return me to metaphysical levels and also back to Egyptian cosmology and the pre-Socratics, when the essential unity of the world was seen in terms water and sand, number and shape, metaphors and analogs to the flowing continuity of conscieousness, flow of time and the grainy modularity of some tiny indivisible thing. For me, this becomes a kind of marvelous mental circus, a primordial metaphysics where arts, conscieousness, and the most advanced and technical thought in mathematics and physics sometimes converge and sometimes clash in the abstractions of their respective symbolic forms.
In looking around for stuff on Penrose, I came across a set of three lectures. Go here:
http://www.princeton.edu/WebMedia/lectures/
You have to go down a very long list. Look for ``Fashion, Faith, and Fantasy in the New Physics of the Universe.''
In the one lecture I just saw, it was interesting he brought up the Platonic Solids and had a picture of them. He was going to talk about them, but didn't. Toward the end, he used three different versions of an Escher's tiling to illustrate the different versions of the geometries used in cosmology.
CG
