The Trouble With Five

We are all familiar with the simple ways of tiling the plane by equilateral triangles, squares, or hexagons. These are the three regular tilings: each is made up of identical copies of a regular polygon — a shape whose sides all have the same length and angles between them — and adjacent tiles share whole edges, that is, we never have part of a tile’s edge overlapping part of another tile’s edge.

In this collection of tilings by regular polygons the number five is conspicuously absent. Why did I not mention a regular tiling by pentagons? It turns out that no such tiling can exist, and it’s not too hard to see why: a regular pentagon has five interior angles of 108°. If we try to place pentagons around a point, we find that three must leave a gap — because 3 × 108 = 324, which is less than the 360° of the full circle — and four must overlap — because 4 × 108 = 432, which is more than the 360° of the circle (plus.maths.org)

David Byrne’s Survival Strategies for Emerging Artists — And Megastars

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The fact that Radiohead debuted its latest album online and Madonna defected from Warner Bros. to Live Nation, a concert promoter, is held to signal the end of the music business as we know it. Actually, these are just two examples of how musicians are increasingly able to work outside of the traditional label relationship. There is no one single way of doing business these days. There are, in fact, six viable models by my count. That variety is good for artists; it gives them more ways to get paid and make a living. And it’s good for audiences, too, who will have more — and more interesting — music to listen to. Let’s step back and get some perspective. (Wired)