### 5

#### Posted by John Baez

I take off for Glasgow tomorrow, so it’s about time to finish preparing my talks…

This one is the most elementary of the lot, about the number 5 and its rascally properties. As usual, you can click on the title to see the slides.

Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the ‘most irrational’ of irrational numbers. The plane cannot be tiled with regular pentagons, but there exist quasiperiodic planar patterns with pentagonal symmetry of a statistical nature, first discovered by Islamic artists in the 1600s, later rediscovered by the mathematician Roger Penrose in the 1970s, and found in nature in 1984.

The Greek fascination with the golden ratio is probably tied to the dodecahedron. Much later, the symmetry group of the dodecahedron was found to give rise to a 4-dimensional regular polytope, the 120-cell, which in turn gives rise to the and the root system of the exceptional Lie group $E_8$. So, a wealth of exceptional objects arise from the quirky nature of 5-fold symmetry.

This is the abstract in the announcements of my talk. I won’t actually talk about the Poincaré homology sphere — there’s a brief pop introduction to that in my closely related Tales of the Dodecahedron. When I turn this talk into a paper, I’ll include that stuff.

I don’t get to $E_8$, either — but I sketch two routes from the dodecahedron to $E_8$ in the second appendix of my talk on the number 8.

Note added later: you can now see a streaming video of this talk.

## Re: 5

I forgot to say it, but: please let me know about any mistakes you find!

Also: does anyone have an intuitive explanation — or even a non-intuitive one — for the appearance of the number 5 in those Ramanujan continued fractions? I don’t really understand this at all… This paper:

should explain everything. You can see the Dedekind $\eta$ function and also the number

$e^{ \pi i /24}.$

These must be related to the factor of

$e^{2 \pi i / 24}$

which plays such a key role in my talk on the number 24.

I want to see how the numbers 24, 5, and the golden ratio start talking to each other!

What facts about elliptic curves underlie these calculations? Apparently Hardy forbid everyone in his circle to say the phrase ‘elliptic curve’, for some reason. That’s a pity.