Tuesday, March 13, 2012

Happy 3.14!


Albert Einstein's birthday is easy for mathematicians to remember, March 14 or 3.14! The Einstein-de Sitter Universe has a "critical" density of (6 $\pi$G t^2)^(-1), a density that keeps it from collapsing or expanding without bounds. Scientists long wondered why the density is exactly this, invoking strange ideas like "inflation" to explain it.

A cosmology where GM=tc^3 actually predicts that the "critical" density is in fact the stable density. If the Universe were less than this density, matter would be created via pair production until this density were reached. For a 4-dimensional spherical Universe of mass M, initial density is just (2 $\pi$^2 G t^2)^(-1). Difference between initial and final density is the difference between 3 and $\pi$ or 4.507034%. The density of baryonic matter that has been measured by the WMAP spacecraft may be precisely predicted from pure math.

Ancient Greek mathematicians first thought that the ratio between a circle's circumference and diameter was exactly 3, then later wondered why it should be an irrational number like 3.14. Fortunately $\pi$ does not equal 3, or the matter that we are made of would not exist!

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