Wikipedia:Reference desk/Archives/Mathematics/2008 March 14

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March 14

Happy Pi Day

Just wishing all of you a happy Pi Day! Please celebrate responsibly. --Kinu ^t/_c 04:12, 14 March 2008 (UTC)

What should I do at 1:59:26? I've only got an hour! HYENASTE 04:22, 14 March 2008 (UTC)

Um... run around in a circle? —Keenan Pepper 05:54, 14 March 2008 (UTC)

Just returned from my circle! Happy Pi Day!!! HYENASTE 05:59, 14 March 2008 (UTC)

getting dizzy! this is so exciting87.102.83.204 (talk) 10:05, 14 March 2008 (UTC)

Damn, I've just fallen over.--88.109.224.4 (talk) 13:41, 14 March 2008 (UTC)

Happy 3.14... Day! --Mayfare (talk) 15:19, 14 March 2008 (UTC)

I've just drank 3.1 pina coladas oh goddd.... Damien Karras (talk) 15:23, 14 March 2008 (UTC)

I've never liked the idea of having a celebration based on an arbitrary and unnatural number system. What say we move pi day to the third of July (or the seventh of March for transatlantic types)? Algebraist 15:48, 14 March 2008 (UTC)

Let's just make every day pi-day see Big yellow disc in sky87.102.83.204 (talk) 16:49, 14 March 2008 (UTC)

You may be an algebraist, but is that an excuse for taking a rational approach to pi day?  --Lambiam 16:56, 14 March 2008 (UTC)

How are continued fractions less transcendental than decimals? And that username's out of date: I'm currently applying for a DPhil in logic. Algebraist 20:54, 15 March 2008 (UTC)

That explains a lot.  --Lambiam 07:27, 16 March 2008 (UTC)

"Give three point one four one six cheers for the Science Officer with pointed ears." —Tamfang (talk) 00:24, 19 March 2008 (UTC)

Percentages

Here I am at a very advanced age totally ashamed to say that I cannot work out percentages! Please advise me how to work out the percentage of one sum against the other; for example what percentage has my pension increased from last year's income to this year's. How to I do that ? Thanks in anticipation.--88.109.224.4 (talk) 08:47, 14 March 2008 (UTC)

Simple a percentage is one over another times 100.

So if income was 110 this year, and 95 last year this gives (110/95)*100 = 115.8%

So the increase is 15.8% (since 100% equals 1:1 ratio or no change I subtract 100%)

Or if my wage (77) increases by 2% my new wage is 77 + ( 2/100 x 77 ) = 77+1.54 = 78.54 —Preceding unsigned comment added by 87.102.83.204 (talk) 10:05, 14 March 2008 (UTC)

Did you check out the article Percentage?  --Lambiam 16:11, 14 March 2008 (UTC)

Thank you both, esp. Lambiam. The article is excellent and brings me up to speed. Thanks again.--88.109.224.4 (talk) 16:57, 14 March 2008 (UTC)

Also if you're struggling to do it 'mentally' (i.e. without a calculator/spreadsheet) I find it easiest to break things int0 10% and 1% then just do simple addition...E.g. If I want to know what 22% of 721 is I just go.... ok 10% is 72.1 and 1% is 7.21...I have 2 x 72.1 = 144.2 and then 2 x 7.21 so that's 14.42 add the two together and I get 158.62 which a quick google search (http://www.google.co.uk/search?hl=en&q=22%25+of+721&btnG=Search&meta= suggest is correct. This is particularly useful as a technique when you just want to get a ball-park amount really quite rapidly. ny156uk (talk) 17:50, 14 March 2008 (UTC)

Another thing. A lot of people get confused or thrown off by the percent symbol (%). Whenever you see that symbol, you can erase it altogether ... and simply replace it with the fraction 1/100 or, if you prefer, the decimal 0.01. That is, the % symbol simply means to take whatever number you are dealing with and to multiply it by the fraction 1/100 or (equivalently) the decimal 0.01 -- whichever you prefer or whichever is easier in a given situation. Thus:

• 7% means ( 7 * 1/100 ) = 7/100 --- if you need / want the fraction version of the answer
• 7% means ( 7 * 0.01 ) = 0.07 --- if you need / want the decimal version of the answer

Thanks. (Joseph A. Spadaro (talk) 22:25, 14 March 2008 (UTC))

Group Theory

The theory of groups interest me. I have a knowledge of mathematics up to:

a) Complex number arithmetic

b) Basic differentiation / integration

c) Basic knowledge of matrices (inverse, determinants)

d) Basic Logic and set theory

I understand the brief introductions and historical accounts of its development; and I'm tempted to pick up a book- although I don't know if I'm qualified to handle even the simplest introductory text.

It's basically a crude engineering mathematics background. Can you suggest a path of study to lead me up to investigations into group theory? Can I step into it with the basic knowledge I have or would you suggest deep study into which particular areas of mathematics? —Preceding unsigned comment added by 81.187.252.174 (talk) 14:12, 14 March 2008 (UTC)

Basic set theory is all you need, really. Find a book called something like "A first course in groups" and see how you find it. At a basic level, groups are pretty easy to understand. They are just sets with a way of combining 2 elements to get another element in a well-behaved way (for example, adding 2 integers to get a 3rd integer, or reflecting a polygon and then rotating it and realising that it's the same as having just reflected it in a different axis). --Tango (talk) 14:26, 14 March 2008 (UTC)

A textbook that is not expensive (used) is An introduction to the theory of groups by George W. Polites.[1] I don't know the book, so I can't vouch for it, but it is thin (80 pages), and if it is a mismatch to your needs, at least you did not waste lots of money.  --Lambiam 16:05, 14 March 2008 (UTC)

To hell with "not expensive"; here are three free online books for you to choose from: [2] [3] [4] —Keenan Pepper 21:59, 14 March 2008 (UTC)

Origin of asymmetry in conformation of symmetrical shape in a higher dimension (+ origami!)

Ok, this one has me totally stumped, and bear with me, as I'm not sure I'll explain it wonderfully well. There is a certain "mathematical" origami model, called a hyperbolic parabola (or paraboloid). Which is folded using a crease-pattern that is four-way symmetric under both reflectional and rotational transformations. (See: http://www.math.lsu.edu/~verrill/origami/parabola/ )

Now, here's the weirdness: when the crease-pattern is concertina-ed (fanned-up), the model assumes the three-dimensional shape for which it is named -- which is not completely symmetrical! The shape once conformed possesses chirality, or handedness, which is not possessed by the tetrahedron into which it will fit perfectly. Where does this rotational asymmetry come from? It is not there in the two-dimensional crease-pattern, but somehow emerges from the conformation into three dimensions, despite there being no way to distinguish between what is done to any four-way division of the model.

Can someone explain this puzzling phenomenon? (Preferably without using arcane technical terms). Many thanks, 85.194.245.82 (talk) 20:47, 14 March 2008 (UTC)

(original poster): Still be not-overly-confident of how well I explained my confusion, here's an addendum I just thought of: from the perspective of the paper as it is twisting into three-dimensions, both of its (indistinguishable) diagonals curve into circular arc-segments, one "up", the other "down" -- the puzzle is what logic does the paper use in conspiring with three-dimensional space to decide which of these diagonals goes up and which goes down?! 85.194.245.82 (talk) 20:53, 14 March 2008 (UTC)

I must admit I had no idea what you were talking about until I actually made one of these. Do you have one on hand? If so, try this: grab two opposite sides of the square piece of paper (which are two opposite edges of the tetrahedron) and twist in such a way as to flatten out the square. Then keep twisting in the same direction far past that point. Try to do it in one smooth motion. If you do it right, the thing will flip inside out and become an identical shape, but this time the diagonals go the opposite directions. So really it's not that a symmetrical pattern leads to an asymmetrical shape, it's that a symmetrical pattern leads to two possible stable shapes which are opposites, and they're symmetrical if you consider them as a pair. In between them is an unstable equilibrium which is a corrugated flat square (it's unstable because you effectively give the paper a negative Gaussian curvature, so it doesn't like to be flat anymore). Neat. —Keenan Pepper 21:41, 14 March 2008 (UTC)

Technical nitpick with one of your statements: The shape it makes is not chiral. It has point group $D_{2d}$, which is a proper subgroup of the point group of the flat square you start out with ($D_{4h}$), so it does have "less symmetry" in a specific sense, but it still has mirror symmetry, so it's not chiral. —Keenan Pepper 21:51, 14 March 2008 (UTC)

Hah, just realized I used many "arcane technical terms". Sorry about that. I'll be happy to explain them to you. —Keenan Pepper 21:53, 14 March 2008 (UTC)

Aye, empirical experimentation shortly after framing the question lead me to the "the paper does it" conclusion you point to, which at first made me think, "awww, reality's boring", as I was hoping for some kind of crazy metaphysical implication about the non-independence of spacial dimensions relative to one another (I have an overactive imagination that way). But then, I thought about it some more and had a little discussion on IRC, and decided it was still a very interesting phenomenon despite the "choice" of axis-curving resulting from environmental factors. Interesting because a purely geometrical consideration lead to an amplification of information. That is to say, although really only one bit of information is added to the system in choosing the subset of the point group, the effect is to cause a large difference in eventual topology. Someone said this was an example, or at least analogous, to the theoretical-physical concept of spontaneous symmetry breaking (I personally think the term "spontaneous" is a terrible misnomer, as it implies that the information arrives randomly, ex nihilo, from nowhere, which is not how mathematics or the universe works, no matter how hard we try to pretend otherwise). 85.194.245.82 (talk) 22:09, 14 March 2008 (UTC)

(thanks for the terminology info, btw. I had a feeling chirality wasn't right, but couldn't think of any other asymmetry. Is there a specific term for it? Will look at the article you linked to.) 85.194.245.82 (talk) 22:09, 14 March 2008 (UTC)

The phenomenon is an instance of spontaneous symmetry breaking. An even much simpler example is when you put a small ball (as from a ball bearing) exactly in the middle on top of a larger ball. Perfect symmetry. Turn around and the little ball will have dropped off, in an asymmetric way (since all ways of dropping off are asymmetric). If it does not drop off instantaneously, you may have to wait for a butterfly or Heisenberg to assist the process.  --Lambiam 23:43, 14 March 2008 (UTC)