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If you try to get several regular polygons to meet snugly at a point in the plane, what's the most sides any of the polygons can have?

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. The reason this works is that

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

There are 17 solutions of

(1/2 - 1/p) + (1/2 - 1/q) + (1/2 - 1/r) = 1

with p ≤ q ≤ r, but this one features the biggest number of all!

But why is this so important? Well, you can take

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

and rewrite it like this:

1/2 + 1/3 + 1/7 + 1/42 = 1

So, 1/2 + 1/3 + 1/7 comes very close to 1. And in fact, if you look for natural numbers a, b, c that make

1/a + 1/b + 1/c

as close to 1 as possible, while still less than 1, the very best you can do is 1/2 + 1/3 + 1/7. It comes within

And why is

1/a + 1/b + 1/c < 1

for these triangles to fit on the hyperbolic plane. A clever trick then lets you get a Riemann surface with at most

2(g-1)/(1 - 1/a - 1/b - 1/c)

symmetries. So, you want to make 1 - 1/a - 1/b - 1/c be as small as possible! And thanks to what I said, the best you can do is

1 - 1/2 - 1/3 - 1/7 = 1/42

So, your surface can have at most

84(g-1)

symmetries. This is called

In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It's a really amazing thing! It has

84 × 2 = 168

symmetries. That number looks really mysterious when you see it. Of course it's the number of hours in a week, but the

But why is this stuff the answer to

For more details, see:

http://math.ucr.edu/home/baez/klein.html

and for a webpage version of this post with a few more pictures, see:

http://math.ucr.edu/home/baez/42.html

*The Hitchhiker's Guide to the Galaxy*by Douglas Adams, the number**42**is the "Answer to the Ultimate Question of Life, the Universe, and Everything". But he didn't say what the question was! Let me reveal that now.If you try to get several regular polygons to meet snugly at a point in the plane, what's the most sides any of the polygons can have?

**42**.The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. The reason this works is that

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

There are 17 solutions of

(1/2 - 1/p) + (1/2 - 1/q) + (1/2 - 1/r) = 1

with p ≤ q ≤ r, but this one features the biggest number of all!

But why is this so important? Well, you can take

(1/2 - 1/3) + (1/2 - 1/7) + (1/2 - 1/42) = 1

and rewrite it like this:

1/2 + 1/3 + 1/7 + 1/42 = 1

So, 1/2 + 1/3 + 1/7 comes very close to 1. And in fact, if you look for natural numbers a, b, c that make

1/a + 1/b + 1/c

as close to 1 as possible, while still less than 1, the very best you can do is 1/2 + 1/3 + 1/7. It comes within

**1/42**of equalling 1.And why is

*this*important? Well, suppose you're trying to make a doughnut with at least two holes that has the maximum number of symmetries. More precisely, suppose you're trying to make a Riemann surface with genus g ≥ 2 that has the maximum number of symmetries. Then you need to find a highly symmetrical tiling of the hyperbolic plane by triangles whose interior angles are π/a, π/b and π/c, and you need1/a + 1/b + 1/c < 1

for these triangles to fit on the hyperbolic plane. A clever trick then lets you get a Riemann surface with at most

2(g-1)/(1 - 1/a - 1/b - 1/c)

symmetries. So, you want to make 1 - 1/a - 1/b - 1/c be as small as possible! And thanks to what I said, the best you can do is

1 - 1/2 - 1/3 - 1/7 = 1/42

So, your surface can have at most

84(g-1)

symmetries. This is called

**Hurwitz's automorphism theorem**. The number 84 looks really mysterious when you first see it - but it's really there because it's twice 42.In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It's a really amazing thing! It has

84 × 2 = 168

symmetries. That number looks really mysterious when you see it. Of course it's the number of hours in a week, but the

*real*reason it's there is because it's four times 42.But why is this stuff the answer to

*the ultimate question of life, the universe, and everything?*I'm not sure, but I have a crazy theory. Maybe all matter and forces are made of tiny little strings! As they move around, they trace out Riemann surfaces in spacetime. And when these surface are as symmetrical as possible, the size of their symmetry group is a multiple of 42, thanks to the math I just described.For more details, see:

http://math.ucr.edu/home/baez/klein.html

and for a webpage version of this post with a few more pictures, see:

http://math.ucr.edu/home/baez/42.html

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blogged...

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A(1) resolutions as Art.

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**Tip: Create an empty Save circle to save posts for later sharing**

Great G+ circles tip from +Richard Kashdan in a comment on one of my posts: "You can mark posts you are interested in by creating a new circle, called "Save" for example, and leave it empty; i.e. don't put anyone in it. You can then save a post for later by Sharing it with the Save circle. No one but you will see that you did that, but you can later go to the Save stream on the left column to get the posts that you saved."

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The revolt has spread to the US Congress! The

More precisely: any federal agency that spends more than $100 million per year funding research must make that research available in a public online repository for free download no later than 6 months after the research has been published in a peer-reviewed journal. This is

**‘Federal Research Public Access Act**says**research paid for by our tax dollars should be freely visible to us**. We shouldn't have to pay for it twice - once to have it done, and again to see the results.More precisely: any federal agency that spends more than $100 million per year funding research must make that research available in a public online repository for free download no later than 6 months after the research has been published in a peer-reviewed journal. This is

**already done**by the National Institute of Health: we just need to expand it to the National Science Foundation and other agencies. Post has shared content

I wonder why I wasn't listed?

MIT Faculty boycott Elsevier journals

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Anyone who is thinking about ways to augment or change the current academic publishing model should have a read through +Scott Morrison's article, especially if you are contemplating trying to move to an entirely open platform.

In the comments we receive an anonymous quote from senior mathematician on the differences between editing for Advances and for a major nonprofit journal:

“Papers were submitted directly to me, I was responsible for logging them, assigning referees, keeping track of the time-table in terms of when to remind referees, writing all the correspondence to referees and authors, reminding referees again, corresponding with other editors, writing delicate rejections, etc. For Advances, I do only two thing: assign the referee and make decisions.”

It seems that employing good software might alleviate some of these problems (though definitely not all!). One suggestion from +Dmitri Pavlov is to use the OJS http://en.wikipedia.org/wiki/Open_Journal_Systems#External_links

created by the Public Knowledge Project (see also http://pkp.sfu.ca/?q=ojs).

Below you will find a list of mathematics journals using the OJS software.

Bulletin of the Australian Mathematical Society http://www.austms.org.au/Bulletin

Contributions to Discrete Mathematics, http://cdm.ucalgary.ca/index.php/cdm

Discrete Mathematics & Theoretical Computer Science, http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs

Internationsl Journal of Mathematics and Soft Computing, http://www.ijmsc.com/index.php/ijmsc

International Journal of Pure and Applied Mathematics, http://www.ijpam.eu/en/

Journal of Informatics and Mathematical Sciences, http://www.rgnpublications.com/jims/

Le Matematiche, http://www.dmi.unict.it/ojs/index.php/lematematiche

Studies in Mathematical Sciences, http://www.cscanada.net/index.php/sms

In the comments we receive an anonymous quote from senior mathematician on the differences between editing for Advances and for a major nonprofit journal:

“Papers were submitted directly to me, I was responsible for logging them, assigning referees, keeping track of the time-table in terms of when to remind referees, writing all the correspondence to referees and authors, reminding referees again, corresponding with other editors, writing delicate rejections, etc. For Advances, I do only two thing: assign the referee and make decisions.”

It seems that employing good software might alleviate some of these problems (though definitely not all!). One suggestion from +Dmitri Pavlov is to use the OJS http://en.wikipedia.org/wiki/Open_Journal_Systems#External_links

created by the Public Knowledge Project (see also http://pkp.sfu.ca/?q=ojs).

Below you will find a list of mathematics journals using the OJS software.

**What are people's experiences with this software? What does it do well and what not so well? and why aren't more journal using it!?**If we don't know what all the problems and issues are, how can we expect to create a system that will solve them all?Bulletin of the Australian Mathematical Society http://www.austms.org.au/Bulletin

Contributions to Discrete Mathematics, http://cdm.ucalgary.ca/index.php/cdm

Discrete Mathematics & Theoretical Computer Science, http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs

Internationsl Journal of Mathematics and Soft Computing, http://www.ijmsc.com/index.php/ijmsc

International Journal of Pure and Applied Mathematics, http://www.ijpam.eu/en/

Journal of Informatics and Mathematical Sciences, http://www.rgnpublications.com/jims/

Le Matematiche, http://www.dmi.unict.it/ojs/index.php/lematematiche

Studies in Mathematical Sciences, http://www.cscanada.net/index.php/sms

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It's nice that even the business magazine

I like Andrew Stacey's idea of putting papers on free archives like the arXiv and then having independent "review boards" evaluate those papers. He writes:

"My proposal would be to have “boards” that produce a list of “important papers” each time period (monthly, quarterly, annually – there’d be a place for each). The characteristics that I would consider important would be:

1. The papers themselves reside on the arXiv. A board certifies a particular version, so the author can update their paper if they wish.

2. A paper can be “certified” by any number of boards. This would mean that boards can have different but overlapping scopes. For example, the Edinburgh mathematical society might wish to produce a list of significant papers with Scottish authors. Some of these will be in topology, whereupon a topological journal might also wish to include them on their list.

3. A paper can be recommended to a board in one of several ways: an author can submit their paper, the board can simply decide to list a particular paper (without the author’s permission), an “interested party” can recommend a particular paper by someone else.

4. Refereeing can be more finely grained. The “added value” from the listing can be the amount of refereeing that happened, and (as with our

*Forbes*has noticed the Elsevier ban. But their business model will only go up in smoke if we develop a new model. It's easy to distribute information; the hard part is evaluating it. Bean-counters at universities and funding agencies need these evaluations to decide who to hire and promote, which departments to give more money to, and so on.I like Andrew Stacey's idea of putting papers on free archives like the arXiv and then having independent "review boards" evaluate those papers. He writes:

"My proposal would be to have “boards” that produce a list of “important papers” each time period (monthly, quarterly, annually – there’d be a place for each). The characteristics that I would consider important would be:

1. The papers themselves reside on the arXiv. A board certifies a particular version, so the author can update their paper if they wish.

2. A paper can be “certified” by any number of boards. This would mean that boards can have different but overlapping scopes. For example, the Edinburgh mathematical society might wish to produce a list of significant papers with Scottish authors. Some of these will be in topology, whereupon a topological journal might also wish to include them on their list.

3. A paper can be recommended to a board in one of several ways: an author can submit their paper, the board can simply decide to list a particular paper (without the author’s permission), an “interested party” can recommend a particular paper by someone else.

4. Refereeing can be more finely grained. The “added value” from the listing can be the amount of refereeing that happened, and (as with our

*nJournal*) the type of refereeing can be shown. In the case of a paper that the board has decided themselves to list, the letter to the author might say, “We’d like to list your paper in our yearly summary of advances in Topology. However, our referee has said that it needs the following polishing before we do that. Would you be willing to do this so that we can list it?”"**Someone needs to start one of these boards for math and/or physics. We can discuss it endlessly, but the time is ripe for action. If someone starts one, other people will start more, and natural selection will optimize the concept.** Post has shared content

From +Scott Aaronson

http://www.scottaaronson.com/writings/journal.pdf

"I have an ingenious idea for a company. My company will be in the business of selling computer games. But, unlike other computer game companies, mine will never have to hire a single programmer, game designer, or graphic artist. Instead I’ll simply ﬁnd people who know how to make games, and ask them to donate their games to me. Naturally, anyone generous enough to donate a game will immediately relinquish all further rights to it. From then on, I alone will be the copyright-holder, distributor, and collector of royalties. This is not to say, however, that I’ll provide no “value-added.” My company will be the one that packages the games in 25-cent cardboard boxes, then resells the boxes for up to $300 apiece"

http://www.scottaaronson.com/writings/journal.pdf

"I have an ingenious idea for a company. My company will be in the business of selling computer games. But, unlike other computer game companies, mine will never have to hire a single programmer, game designer, or graphic artist. Instead I’ll simply ﬁnd people who know how to make games, and ask them to donate their games to me. Naturally, anyone generous enough to donate a game will immediately relinquish all further rights to it. From then on, I alone will be the copyright-holder, distributor, and collector of royalties. This is not to say, however, that I’ll provide no “value-added.” My company will be the one that packages the games in 25-cent cardboard boxes, then resells the boxes for up to $300 apiece"

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