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September 12, 2008

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 8E_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 8E_8, either — but I sketch two routes from the dodecahedron to E 8E_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.

Posted at September 12, 2008 11:47 PM UTC

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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 η\eta function and also the number

e πi/24.e^{ \pi i /24}.

These must be related to the factor of

e 2πi/24e^{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.

Posted by: John Baez on September 13, 2008 5:32 PM | Permalink | Reply to this

Re: 5

These talks are great. I really liked the pictures of the 120-cell.

My impression was that the buckyball is a truncated icosahedron, but not a truncated dodecahedron, since it doesn’t have decagon sides.

Posted by: Scott Carnahan on September 14, 2008 5:11 AM | Permalink | Reply to this

Buckyball; Re: 5

Buckminsterfullerene (IUPAC name (C60-Ih)[5,6]fullerene) is the smallest fullerene molecule in which no two pentagons share an edge (which can be destabilizing; see pentalene). It is also the most common in terms of natural occurrence, as it can often be found in soot.

The structure of C60 is a truncated (T = 3) icosahedron, which resembles a soccer ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge.

The van der Waals diameter of a C60 molecule is about 1 nanometer (nm). The nucleus to nucleus diameter of a C60 molecule is about 0.7 nm.

The C60 molecule has two bond lengths. The 6:6 ring bonds (between two hexagons) can be considered “double bonds” and are shorter than the 6:5 bonds (between a hexagon and a pentagon).

[wikipedia]

Posted by: Jonathan Vos Post on September 14, 2008 3:16 PM | Permalink | Reply to this

Re: 5

Scott wrote:

My impression was that the buckyball is a truncated icosahedron, but not a truncated dodecahedron, since it doesn’t have decagon sides.

Whoops! You’re right. Unfortunately I noticed this in the middle of giving my talk — just when I was explaining how to obtain the buckyball as a truncated dodecahedron.

I probably was thinking of a buckyball as a ‘severely truncated’ dodecahedron. There are some pictures here showing how a red dodecahedron can morph into a yellow icosahedron by chopping off more and more of the corners. However, you only reach the buckyball rather far along in this process, after you’ve chopped off ‘more than half’ of the dodecahedron’s corners — so it’s rightly called a truncated icosahedron.

(On a wholly other note: thanks for pointing out that on my website, the link to my paper ‘Hochschild homology in a braided tensor category’ is broken. Annoyingly, I can’t find any trace of the TeX file of that paper anywhere on my computer anymore.)

Posted by: John Baez on September 16, 2008 6:24 PM | Permalink | Reply to this

No Quintics?

This looks like a really cool lecture series. To add to the strange properties of 5: Abel’s proof, using Galois theory, that there’s no general formula for solving quintic equations. (I was surprised not to see that one there!)

Posted by: Ari on September 15, 2008 5:08 AM | Permalink | Reply to this

Re: No Quintics?

or Klein’s lectures on the icosahedron!

Posted by: curious on September 15, 2008 11:44 PM | Permalink | Reply to this

Re: No Quintics?

There was no time for these things in the talk, but I certainly hope to talk about them in the paper based on the talk!

I love the ideas in Klein’s Lectures on the Icoshedron — in particular, how he solves the quintic using the ring of functions on the Riemann sphere that are invariant under the symmetries of the icosahedron. But, I find this modernized treatment a lot easier to read:

  • Jerry Shurman, Geometry of the Quintic, John Wiley and Sons, New York, 1997.

This is also interesting… it’s much more advanced, and harder to read, but it has the advantage of being free:

Posted by: John Baez on September 16, 2008 6:35 PM | Permalink | Reply to this

Re: No Quintics?

Geometry of the Quintic is available for free at my home page.

Posted by: Jerry Shurman on April 17, 2010 9:30 PM | Permalink | Reply to this

Re: No Quintics?

Thanks, Jerry! I love your book and I ordered it for the UCR library. Now I don’t know if they’ll be able to get it. But it’s really kind of you to make it publicly available. I took the liberty of adding a direct link from your comment to the relevant page.

Everyone who likes the icosahedron should read this book!

Posted by: John Baez on April 18, 2010 9:50 PM | Permalink | Reply to this

Re: 5

I attended your lecture in Glasgow yesterday, and thoroughly enjoyed it. I hope to get to the others later in the week.

A couple of related things that you might find interesting, if you don’t already know them.

1) 360-degree rotation is not the identity. It’s possible to demonstrate physically that after a 360-degree rotation, the object is in the same position but it is not embedded in the surrounding space in the same way. See here You can do this by holding an object (e.g. a cup; fill it with liquid for extra entertainment) and rotating it 360 degrees. Your arm will be all twisted up. Continuing to rotate in the same direction, after another 360 degrees the cup is back in the same position and your arm is back to normal.

2) Penrose tilings. The company Pentaplex markets a puzzle based on Penrose tiles. The two shapes of tile have been made into interlocking plastic pieces in the shape of fat and thin chickens. The puzzle is to make a tiling that covers a maximum area. There are also some jigsaw puzzles with pictures of Penrose tilings, again using animal shapes. The puzzle version is called Game Birds. Several years ago they used to sell a much larger version called Perplexing Poultry, which I have a set of. (Actually I could show you later this week if you are interested - email me).

Posted by: Simon Gay on September 16, 2008 11:28 AM | Permalink | Reply to this

Re: 5

I’m glad you enjoyed my talk, Simon!

A 360° rotation is the identity in SO(3) , but not in its double cover. Indeed, that’s what I was vaguely trying to allude to before my discussion of the 120 symmetries of the ‘spinor dodecahedron’. But alas, I forgot to demonstrate this fact using the famous cup trick until the reception after talk, when a persistent young girl came up and asked me how it was possible that an electron, when rotated 360°, does not come back to its original state. By then I was sipping a glass of wine, so the cup trick came naturally to mind. Luckily, I wasn’t so sozzled as to whip off my belt and attempt the belt trick.

I’d be delighted to see the Pentaplex puzzle.

Posted by: John Baez on September 17, 2008 11:10 AM | Permalink | Reply to this

Re: 5

Latest Notices have some interesting 5’s
from ancient Babylon - including a polygon
that can be assembled to an icosahedron.

Posted by: jim stasheff on September 20, 2008 1:52 PM | Permalink | Reply to this

Re: 5

Yes, Thomas Riepe already pointed out this article to me:

He quoted a bit:

After this explanation of all the computations in the Kassite text in Figure 19, it remains only to explain the meaning of the term “horn figure”: which figure can be constructed by use of 20 equilateral triangles, where 20 is computed as (61)4(6 - 1) \cdot 4? The only possible answer seems to be that “horn figure” was the name for a regular polyhedron with 20 faces, more precisely what we call an icosahedron, a term of Greek origin…

Apparently this Kassite text dates to sometime after the second half of the second millennium BC. I’m not sure how much later! But, since the Kassites fizzled out around 1155 BC, this text seems to vastly predate the supposed discovery of the icosahedron by the Greek mathematician Theaetetus sometime between 380 and 370 BC (see week236). It may come after the Scottish stone balls carved in the shape of Platonic solids (see week241).


Posted by: John Baez on September 20, 2008 2:32 PM | Permalink | Reply to this

Re: 5

I just downloaded the talk and watched. Very entertaining and nicely edited too.

You asked to know about any mistakes and maybe its a bit late to say now but, on slide 32 the sum giving the partial continued fraction for pi is the wrong one. You have to leave the 1/292 out to get the good approximation 355/113.

Posted by: PhilG on November 16, 2008 3:34 PM | Permalink | Reply to this

Re: 5

Allen Hatcher already pointed out this idiotic mistake, and I have corrected it in the slides.

I haven’t looked at the videos yet. I’m sort of scared to.

Posted by: John Baez on November 16, 2008 5:20 PM | Permalink | Reply to this

Re: 5

John, I didn’t think you’d be scared to see yourself on video! I assumed you’d looked at them weeks ago. I had no idea you were so shy.

Posted by: Tom Leinster on November 17, 2008 8:20 PM | Permalink | Reply to this

Re: 5

I love talking to big crowds of people. But what if it turns out I look silly when I’m doing it? I’m not sure I want to know.

(Of course you’ll say I don’t look silly. But what I mean is: how I look to me.)

Posted by: John Baez on November 17, 2008 9:52 PM | Permalink | Reply to this

Re: 5

Actually you do look silly. The part where you can’t figure out how to do the rotation group of a dodecahedron is pure slapstick!

Luckily for me I can say this in the safe and certain knowledge that I will never be asked to deliver a talk worth recording.

Posted by: PhilG on November 18, 2008 8:17 AM | Permalink | Reply to this

Re: 5

For some reason I still see the error in the version of the slides linked to.

Posted by: PhilG on November 17, 2008 7:21 AM | Permalink | Reply to this

Re: 5

Hmm! Thanks for catching that. For some reason I’d forgotten to put the new version on my website. It should be fixed now.

There was also a stupid mistake in the definition of the Leech lattice in my talk on the number 24, which is fixed in the slides now.

Posted by: John Baez on November 17, 2008 4:39 PM | Permalink | Reply to this

Videos available!

Streaming videos of all three of John’s talks are (at last) available. If your computer is set up right, you can just click the link and start watching. You no longer need to wait ages for the whole file to download.

To watch the full-screen version, click on the little icon at the bottom-right of the video screen. To choose which lecture to watch, use the menu to the right of the video screen. For further information, including copies of the slides, see here or here.

Posted by: Tom Leinster on February 28, 2009 10:16 PM | Permalink | Reply to this

Re: 5

“Ice can build an extended one dimensional chain structure entirely from pentagons and not hexagons”

Posted by: Thomas on March 8, 2009 6:58 PM | Permalink | Reply to this

Re: 5

“Chuan Xiao discovered the true structure of the mimivirus when he decided to try reconstructing the virus, assuming it had not the standard icosahedral symmetry but another configuration called five-fold symmetry.”

Posted by: Thomas on April 28, 2009 6:50 AM | Permalink | Reply to this

Re: 5

Just showed “5” at our undergraduate math club. Great talk. It’s really cool we can download them for free, thanks Glasgow Mathematical Journal Trust! The video editing was quite professional, just the right mix of John vs slides and even some zoom-ins.

Posted by: Bruce Bartlett on April 28, 2009 7:36 PM | Permalink | Reply to this

Re: 5

Cool! Maybe I’ll get up the nerve to watch it sometime. There was indeed a professional team of 3 involved in making this video.

Good to hear from you — you’ve been scarce around these parts. I hope you can spend a bit of time on that “journal club” discussing Ben-Zvi’s new paper… it should be good.

Posted by: John Baez on April 28, 2009 7:52 PM | Permalink | Reply to this

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