In praise of symmetry

The chief forms of beauty are order and symmetry and definiteness, and these are especially manifest in the mathematical sciences” (τοῦ δὲ καλοῦ μέγιστα εἴδη τάξις καὶ συμμετρία καὶ τὸ ὡρισμένον, ἃ μάλιστα δεικνύουσιν αἱ μαθηματικαὶ ἐπιστῆμαι) – Aristotle, Metaphysics, Book 13 (Mu), Section 3, my translation.

Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection” – Hermann Weyl, Symmetry, 1952, Princeton University Press, p. 5.

Regularity is successive symmetry; there is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular” – Ernst Mach, The Science Of Mechanics, 1919 edition, p. 395.

Bottom left image derived from a public domain photo by Vinoo202.

Advertisements

Curved networks as art

This Dagstuhl workshop included a wonderful art exhibit, titled “Bending Reality: Where arc and science meet.” Exhibits included this image by David Eppstein, and also this one:

There were also several metro maps, including ones based on concentric circles and freeform Béziers, as well as some curved annotations of the world and images of curved relationships, like this poster:

See also this blog post, and my previous post about CIRCOS diagrams.

—And He Built a Crooked House

One of the great science fiction short classics is Heinlein’s “—And He Built a Crooked House,” in which a character builds a house in the shape of an unfolded four-dimensional hypercube or tesseract (I’ve always wanted one like it):

Just as a cube folds out into six squares, a tesseract folds out into eight cubes. Now this makes for an interesting house design, but even more interesting is that, during an earthquake, the house “folds up” in the fourth dimension. Which is hard to picture. This is one view, which gets the connections between the eight cubes right, but ignores the fact that they all have the same size and shape:

In Heinlein’s words:

‘That’s a tesseract, eight cubes forming the sides of a hypercube in four dimensions.’
‘It looks more like a cat’s cradle to me. You’ve only got two cubes there anyhow. Where are the other six?’
‘Use your imagination, man. Consider the top of the first cube in relation to the top of the second; that’s cube number three. Then the two bottom squares, then the front faces of each cube, the back faces, the right hand, the left hand—eight cubes.’ He pointed them out.
‘Yeah, I see ‘em. But they still aren’t cubes; they’re whatchamucallems—prisms. They are not square, they slant.’
‘That’s just the way you look at it, in perspective. If you drew a picture of a cube on a piece of paper, the side squares would be slaunchwise, wouldn’t they? That’s perspective. When you look at a four-dimensional figure in three dimensions, naturally it looks crooked. But those are all cubes just the same.’

Here is another view (on second thoughts, maybe living in a house like that would make me sea-sick):

Nobels per dollar

I was looking at the list of scientific Nobels on Wikipedia. A good predictor (correlation 0.84) of the number of Nobels for a country is the size of the economy (the GDP). Population size has, surprisingly, a small negative effect.

Some countries do significantly better than predicted by GDP: Luxembourg (2), Hungary (8), Denmark (10), Austria (17), Switzerland (20), and Great Britain (94). What did they do right?

Meteoric blades and beads


Meteoric iron dagger of Emperor Jahangir, Mughal dynasty, 1621, India (photo by “Daderot,” public domain)

Many of the earliest objects made from iron were forged from iron meteorites. The dagger above and the hatchet below are later examples.


An iron meteorite and a hatchet forged from meteoric iron (photo by “Zaereth,” public domain)

Working with meteoric iron could be done in the absence of iron ore, or in the absence of smelting technology. A recent paper in the Journal of Archaeological Science reports that the oldest known iron artefacts (iron beads from Egypt, dated to 3200 BC) were made from meteoritic iron, rather than being smelted from ore (the presence of germanium is the strongest evidence). This indicates that the age of iron began with meteorites from the sky, and that the smelting of ore followed centuries of experience with meteoric iron.

Iron smelting may be technologically more sophisticated, but making iron blades from star-stones is far cooler!


Widmanstätten pattern in a meteorite (my photo)

Network Science: a new journal

From the journals.cambridge.org page: “Network Science is a new journal for a new discipline – one using the network paradigm, focusing on actors and relational linkages, to inform research, methodology, and applications from many fields across the natural, social, engineering and informational sciences. Given growing understanding of the interconnectedness and globalization of the world, network methods are an increasingly recognized way to research aspects of modern society along with the individuals, organizations, and other actors within it.

A very welcome step! Volume 1, issue 01 was April 2013, and is still online.