Search found 198 matches

by HenryS
Wed May 21, 2008 7:54 am UTC
Forum: Mathematics
Topic: The golden ratio, spirals, in 3d
Replies: 3
Views: 1394

The golden ratio, spirals, in 3d

There is a well known pattern that crops up all over the place in nature involving the golden ratio, \phi = \frac{\sqrt{5}-1}{2} . http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#seeds is a good description. Essentially, to get a sunflower head pattern, add the k^{th} new node at ...
by HenryS
Wed May 21, 2008 7:25 am UTC
Forum: Mathematics
Topic: Is the second uninteresting number interesting?
Replies: 17
Views: 2420

Re: Is the second uninteresting number interesting?

schmiggen wrote:How many common paradoxes are really just, to some extent, a neglect of the rules/language/meanings relevant to their expression (or to the topic to which they refer)?
If you believe Wittgenstein, all of them.
by HenryS
Thu May 01, 2008 5:59 pm UTC
Forum: Logic Puzzles
Topic: Where in the world is Matt Lauer?
Replies: 6
Views: 2784

Re: Where in the world is Matt Lauer?

Or "eight"/"ate".
by HenryS
Tue Apr 29, 2008 9:13 pm UTC
Forum: Serious Business
Topic: Anthropomorphization: how deep does it run?
Replies: 18
Views: 2745

Re: Anthropomorphization: how deep does it run?

There are limits to what evolution has given us, sure. But it does get you somewhere. You can be reasonably sure that there isn't a sabre-toothed tiger in the middle of the room, ready to pounce on you, if you cannot see such a tiger. Whether or not such a tiger exists is a matter of objective reali...
by HenryS
Tue Apr 29, 2008 6:49 pm UTC
Forum: Serious Business
Topic: Anthropomorphization: how deep does it run?
Replies: 18
Views: 2745

Re: Anthropomorphization: how deep does it run?

I think we can get somewhere in talking about objective reality if we believe that we evolved into who we are. Evolution tends to look unfavourably on replicating systems that misinterpret their sensory data in ways that lead it to courses of action based on beliefs that do not correspond to objecti...
by HenryS
Mon Apr 07, 2008 3:41 am UTC
Forum: Mathematics
Topic: Geometry on a torus
Replies: 18
Views: 1954

Re: Geometry on a torus

The flat metric on a torus is of course Euclidean, so it's really just as special as spherical or hyperbolic geometry.

You might also look into affine geometry for something a bit more exotic that will work on a torus.
by HenryS
Mon Feb 25, 2008 10:34 pm UTC
Forum: Logic Puzzles
Topic: Cycles in an infinite plane of arrows
Replies: 8
Views: 2500

Cycles in an infinite plane of arrows

This puzzle is from this: http://www.amazon.com/Mathematical-Mosaic-Patterns-Problem-Solving/dp/1895997283/ book by Ravi Vakil, and you can see an example on the background of the cover. Suppose you have an infinite grid of squares, and in each square there is an arrow, pointing in one of the 8 card...
by HenryS
Fri Jan 25, 2008 6:29 pm UTC
Forum: Mathematics
Topic: How do you end your proofs? QED, box, ...
Replies: 107
Views: 21312

Re: How do you end your proofs? QED, box, ...

mrbaggins wrote:I am now starting any decently long proofs on the board with "Behold!"

But I want more ideas for endings people :P
"As was prophesied!"?

Or perhaps it should really be "In accordance with the prophesy!"
by HenryS
Thu Jan 24, 2008 10:39 pm UTC
Forum: Mathematics
Topic: Why do people dislike maths so much?
Replies: 258
Views: 28557

Re: Why do people dislike maths so much?

Yakk wrote:One day the math-knowing humans will gather up and abandon the parasites to their fate.
Or better yet, send them off on Ark B.
by HenryS
Wed Jan 23, 2008 10:34 pm UTC
Forum: Logic Puzzles
Topic: Dissection of a SQUARE.
Replies: 10
Views: 2658

Re: Dissection of a SQUARE.

Nice.

Trying to think up some generalisations... other regular polygons are not interesting, and it looks like the exact same arguments (and solution) work for rectangles. Oh well.
by HenryS
Tue Jan 22, 2008 7:26 pm UTC
Forum: Logic Puzzles
Topic: Dissection of a SQUARE.
Replies: 10
Views: 2658

Re: Dissection of a SQUARE.

I'm pretty sure that the minimum is 8. A lower bound of 5 is easy to prove, maybe 6. I don't see a way to get that lower bound higher without some better ideas.
by HenryS
Mon Jan 14, 2008 11:12 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

In other words, I want a manifold that not only has no special points, but no special directions either. While a torus has no special points, it does have special directions. Right, true. However, any strategy that works for the plane will also work for the torus (ditto any covering space), so we c...
by HenryS
Sun Jan 13, 2008 9:22 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

Yes, there's one orientation reversing isometry that doesn't work, but we could ask which ones do and which don't (clearly some do). I'm not sure what they're called either, "isotropic" maybe? What's wrong with the 2-torus? Its universal cover is R^2 with Euclidean geometry, which does hav...
by HenryS
Sun Jan 13, 2008 8:38 pm UTC
Forum: Logic Puzzles
Topic: Bouncy balls
Replies: 9
Views: 2502

Re: Bouncy balls

I think the answer to 2) (and therefore to 1) is almost certainly yes, but I don't know enough about the relevant fields (ergodic flow and billiard table problems) to reel off a proof. One good first step is to replace the one square with a plane tiled with squares, by refle...
by HenryS
Sun Jan 13, 2008 7:28 pm UTC
Forum: Mathematics
Topic: A finite plane set with two points on every middle line?
Replies: 17
Views: 1903

Re: A finite plane set with two points on every middle line?

Cool solution. This problem reminded me of a similar kindof famous problem: Consider a finite set of points in the plane with the property that on the line through any two points there is at least one other point. A set consisting of at least 3 points all lying on a single line is clearly such a set...
by HenryS
Sun Jan 13, 2008 7:19 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

if you throw out orientation reversing isometries Ok, new problem. A robot and an identical-in-every-way-other-than-orientation anti-robot are parachuted down onto [insert space of your choice, start with the surface of a planet], and in order to destroy the planet in a blaze of photons, they need ...
by HenryS
Sun Jan 13, 2008 6:38 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

The answer is kindof nice in H^2: the only isometries with a fixed point inside H^2 are rotations. Everything else (parabolic and hyperbolic isometries) only have fixed points on the circle at infinity. In terms of moving forward in a straight line a distance d, then turning by an angle theta, if I'...
by HenryS
Sun Jan 13, 2008 6:50 am UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

S^3 doesn't work since the map x -> -x is orientation preserving in S^3. Oh neat. What about S^4 ;) And my guess for H^3 is the same as yours: being even bigger than R^3, it should be that much easier to find starting pairs of positions/orientations so that if the robots follow the same path (any p...
by HenryS
Sat Jan 12, 2008 11:08 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

The problem can be reduced to asking if there is a fixed point in the map from R^3 to R^3 taking the initial point of view of one robot to the initial point of view of the other. If there is then the robots just need to get to that point, and they can do that by visiting every point in space, start...
by HenryS
Sat Jan 12, 2008 6:15 pm UTC
Forum: Logic Puzzles
Topic: Deluxe Circle Game
Replies: 8
Views: 2633

Re: Deluxe Circle Game

You missed the condition that there are an even number of players and that half of them are red and half blue. Also that the players alternate colours, so there is in fact no choice as to the layout of the players other than how many there are.

I'd have to think about how to solve it though.
by HenryS
Sat Jan 12, 2008 5:48 pm UTC
Forum: Logic Puzzles
Topic: Robots in Space
Replies: 24
Views: 2759

Re: Robots in Space

So they have no input whatsoever and they don't know where they are? Then their paths will be the same (assuming they're oriented the same initially) and they will be the same distance from eachother, always. We could fix the problem by putting them in a different 3 dimensional geometry, say in S^3...
by HenryS
Tue Jan 01, 2008 8:39 pm UTC
Forum: Mathematics
Topic: Is the Koch Snowflake compact?
Replies: 17
Views: 3178

Re: Is the Koch Snowflake compact?

xprisoner wrote:...the important lesson about fractals being that although Sn is closed for each n, the limiting object is clearly not closed...
An easy non-fractal example to see this is Sn = [1/n,1].
by HenryS
Thu Dec 20, 2007 1:28 am UTC
Forum: Mathematics
Topic: Does it really qualify as math?
Replies: 17
Views: 2801

Re: Does it really qualify as math?

Believe it or not, if you go to college, you'll be exposed to a lot of math like this. This is the start of the series portion of math, in which you'll not only find these patterns, but see in which direction these patterns head the series as a whole (to where the series converges). I think there's...
by HenryS
Wed Dec 19, 2007 8:53 am UTC
Forum: Mathematics
Topic: Creating a model for the total number of human souls
Replies: 9
Views: 1928

Re: Creating a model for the total number of human souls

Haffi112 wrote:what if there is just one soul?
Then, a la Feynman, you would need to figure out what the entities with antisouls (souls travelling backwards in time) are.
by HenryS
Thu Dec 06, 2007 8:25 am UTC
Forum: Logic Puzzles
Topic: Small values of 20
Replies: 42
Views: 5599

Re: Small values of 20

Strilanc wrote:
Spoiler:
(mod 19)

Spoiler:
Doesn't work, you lose the ordering if you're working in modulo arithmetic.
by HenryS
Wed Dec 05, 2007 10:35 pm UTC
Forum: Mathematics
Topic: How do you end your proofs? QED, box, ...
Replies: 107
Views: 21312

Re: How do you end your proofs? QED, box, ...

Box.

At least in my area, you don't see "QED" used in journal articles and so on any more. The story I heard is that "QED" has too much of an air of finality, which would discourage readers from identifying any mistakes with the proof. So we moved to something less confident.
by HenryS
Mon Dec 03, 2007 6:43 pm UTC
Forum: Mathematics
Topic: Division by Zero (Please, no new threads about this)
Replies: 367
Views: 84077

Re: n/0?

I think it can occasionally make sense to add a new point, ∞, particularly when talking about topological properties more than algebraic properties. ∞ = -∞ if you compactify the real line in the right way. You can go ahead and try to define algebraic properties, and you can get some places, such as ...
by HenryS
Fri Nov 30, 2007 6:42 am UTC
Forum: Mathematics
Topic: Random thing that's bugging me: f(n) = 1, 3, 2, 4, 3, 5...
Replies: 24
Views: 3012

Re: Random thing that's bugging me: f(n) = 1, 3, 2, 4, 3, 5...

(-1) x isn't that far away from the trig... (-1) x = (e pi i ) x = (e (pi x)i ) = cos(pi x) + i sin(pi x) So...the (-1)^n solution: 3/4 -(-1^n)*3/4 +n/2 = 3/4 + n/2 -3/4[cos(pi n) + i sin(pi n)] ...has as real part the other solution (well, it would if the constant weren't wrong, but presuma...
by HenryS
Mon Nov 26, 2007 9:54 pm UTC
Forum: Serious Business
Topic: Men's brains are different!
Replies: 164
Views: 18328

Re: Men's brains are different!

It sounds like the question could be taken in a couple of different ways, and so calling one answer the "correct" one should come with scare quotes. My thought process went along the lines of: Well, presumably I have to find the center of the cross-section of the cup and draw a horizontal ...
by HenryS
Thu Nov 22, 2007 7:31 pm UTC
Forum: Mathematics
Topic: Uh oh. My math teacher says .999... does not equal 1.
Replies: 9
Views: 3917

Re: Uh oh. My math teacher says .999... does not equal 1.

ikerous wrote:If theres an infinite set of numbers between .999r and 1, certainly he can tell you what one of them is.
That isn't entirely fair. There are sets of real numbers that we know exist by cardinality arguments but that we cannot name a member of.
by HenryS
Tue Nov 20, 2007 4:23 pm UTC
Forum: Mathematics
Topic: Fast-tracking mathematics.
Replies: 55
Views: 4685

Re: Fast-tracking mathematics.

There are a number of summer camp style things for gifted high school students, in various subjects. The one I'm most familiar with is: http://math.stanford.edu/sumac/ It says there that financial aid is available, I don't know how much or what the availability is. Something to look into though. The...
by HenryS
Sun Nov 11, 2007 5:28 pm UTC
Forum: Serious Business
Topic: Artificial Creativity
Replies: 33
Views: 3581

Re: Artificial Creativity

I thought this thread was going to be about something else, but I think that something else is a relevant discussion anyway: Suppose I write a computer program to generate art. Suppose it uses some emergent stuff, so that I really, truly, have no idea what things that it will put out. This seems to ...
by HenryS
Fri Nov 09, 2007 6:38 pm UTC
Forum: Serious Business
Topic: Religion
Replies: 2000
Views: 184310

Re: Religion

we made up the morality first, and the supernatural entity second. Saying "made up" sounds like it was done with intent. I think it might be more accurate to say that our ancestors developed belief in the supernatural, that certain aspects of morality arose as a consequence of that belief...
by HenryS
Thu Nov 08, 2007 3:54 pm UTC
Forum: Mathematics
Topic: Proof by contradiction is never needed!
Replies: 22
Views: 2775

Re: Proof by contradiction is never needed!

What about the proof that sqrt(2) is irrational?

The direct statement would presumably be:

(x^2 = 2) => (x is irrational)

So the contrapositive would be that:

(x is rational) => (x^2 =/= 2)

This doesn't seem (to me) easy to do without contradiction somewhere.
by HenryS
Thu Nov 08, 2007 2:50 am UTC
Forum: Serious Business
Topic: Religion
Replies: 2000
Views: 184310

Re: Religion

Well, I personally would ask you to define what you mean by that first statement; I hear of a "biological drive" to do something but not of a "genetic drive", and in this case that refers to having sex, which is what the condom enables one to do with little thought or fear of re...
by HenryS
Thu Nov 08, 2007 1:20 am UTC
Forum: Serious Business
Topic: Religion
Replies: 2000
Views: 184310

Re: Religion

The fact that people use condoms is an example that we have broken free from obeying what our genes want us to do.

Our desires may have been formed by evolution, but there are many other effects in play now.
by HenryS
Sun Nov 04, 2007 12:21 am UTC
Forum: Mathematics
Topic: 2^x = x^3
Replies: 12
Views: 3205

Re: 2^x = x^3

Right, but if you use less well-known functions, it can be expressed concisely. http://en.wikipedia.org/wiki/Lambert_W_function I plugged the equation into Mathematica to check that I wasn't missing something obvious, and it solved it using a function it calls "ProductLog" which looks lik...
by HenryS
Sat Nov 03, 2007 4:18 pm UTC
Forum: Mathematics
Topic: 2^x = x^3
Replies: 12
Views: 3205

Re: 2^x = x^3

There's no reason that the answer should be something you can express nicely in terms of well known functions. You can show that a solution has to exist using the intermediate value theorem, and you can get good approximations using Newton's method or similar, but that's about it.
by HenryS
Sat Nov 03, 2007 4:12 am UTC
Forum: Mathematics
Topic: Geodesics: Donuts are Delicious
Replies: 10
Views: 1776

Re: Geodesics: Donuts are Delicious

(Also, general practice when someone asks what looks like an obvious homework question, especially when they don't admit it as such and haven't even deigned to come to the forum in months, is to overly complicate things. See, for example, the "mass of a feather" thread over in the Science...

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