functions that give cool designs
Moderators: gmalivuk, Moderators General, Prelates
functions that give cool designs
Well lets try something. Post functions in here which when plotted give cool looking designs when plotted on the graph paper.
in the form y=f(x) where x is the independent variable.
If you dont have a graph plotting software use this
http://www.chromeexperiments.com/detail ... lotter/?f=
If you post cool looking 3d graphs.... do give a link to a inline 3d graph plotting software.
But i encourage the 2d stuff.
[Additonal details on the function are encouraged]
in the form y=f(x) where x is the independent variable.
If you dont have a graph plotting software use this
http://www.chromeexperiments.com/detail ... lotter/?f=
If you post cool looking 3d graphs.... do give a link to a inline 3d graph plotting software.
But i encourage the 2d stuff.
[Additonal details on the function are encouraged]
 gmalivuk
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Re: functions that gives cool designs.
Nah, all the most interesting stuff is implicit or parametric.Afif_D wrote:in the form y=f(x) where x is the independent variable.
Re: functions that gives cool designs.
I think you are correct. So you can post your implicit and parametric shit too. Lets see whose is the best...
Re: functions that gives cool designs.
The most interesting stuff?
f(x)=(net earnings by the highest 10% of US earners)/(net earnings of the lowest 10% of US earners)
x=quarter (or year if you want some smoothing)
The design is really cool if you're in the top income brackets.
f(x)=(net earnings by the highest 10% of US earners)/(net earnings of the lowest 10% of US earners)
x=quarter (or year if you want some smoothing)
The design is really cool if you're in the top income brackets.
Time flies like an arrow, fruit flies have nothing to lose but their chains Marx
Re: functions that gives cool designs.
Velifer wrote:The most interesting stuff?
f(x)=(net earnings by the highest 10% of US earners)/(net earnings of the lowest 10% of US earners)
x=quarter (or year if you want some smoothing)
The design is really cool if you're in the top income brackets.
I've seen that one before: it's the one that looks like puppies and rainbows and unicorns all being horribly murdered and impaled on stakes, yes?
 silverhammermba
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Re: functions that gives cool designs.
[imath]x^x[/imath] looks cool if you consider the real values for negative x.
Re: functions that gives cool designs.
silverhammermba wrote:[imath]x^x[/imath] looks cool if you consider the real values for negative x.
And even cooler if you treat it as a multivalued function from ℂ to itself!
wee free kings
Re: functions that gives cool designs.
I thought you all genius guys would pt some cool functions in here.... but i was wrong....
Re: functions that gives cool designs.
Try urm... [imath]y^2 = \sin(2 \pi x^n)[/imath] for any natural n, that's a fairly pleasant set of blobby things, I'm fairly sure that the area enclosed by the blobby things is always infinite but don't hold me to that.
What's wrong with z^z from C to itself? Looks nice to me. Honestly, after a while the pretty pictures that mathematics creates aren't considered nearly as cool as the actual maths behind them, even if you want a cool design then trying to force one out of a single valued function just seems a little contrived.Afif_D wrote:I thought you all genius guys would pt some cool functions in here.... but i was wrong....
Re: functions that gives cool designs.
Afif_D wrote:I thought you all genius guys would pt some cool functions in here.... but i was wrong....
I am posting this solely in an effort to reinforce your conception that sardonically insulting people is a good way to get what you want.
[math]\small x=\frac{4z\left( 1+\sin \left( 2\pi n \right) \right)+\left( \left y \right1 \right)\cos \left( 2\pi n \right)}{5}\max \left( 0,\; 2\left y \rightz^{2}y^{2}+\frac{5\max \left( 0,\; 0.03\left( z+0.1+0.2\sin \left( 2\pi n \right) \right)^{2}\left( \left y \right1+0.2\cos \left( 2\pi n \right) \right)^{2} \right)}{2} \right)[/math]
It is a 3D graph where x is an explicit function of y and z, and n is an animated parameter ranging from 1 to +1.
wee free kings

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Re: functions that gives cool designs.
Qaanol wrote:Afif_D wrote:I thought you all genius guys would pt some cool functions in here.... but i was wrong....
I am posting this solely in an effort to reinforce your conception that sardonically insulting people is a good way to get what you want.
[math]\small x=\frac{4z\left( 1+\sin \left( 2\pi n \right) \right)+\left( \left y \right1 \right)\cos \left( 2\pi n \right)}{5}\max \left( 0,\; 2\left y \rightz^{2}y^{2}+\frac{5\max \left( 0,\; 0.03\left( z+0.1+0.2\sin \left( 2\pi n \right) \right)^{2}\left( \left y \right1+0.2\cos \left( 2\pi n \right) \right)^{2} \right)}{2} \right)[/math]
It is a 3D graph where x is an explicit function of y and z, and n is an animated parameter ranging from 1 to +1.
Mathematica friendly (to the extent of my ability and knowledge):
Code: Select all
Manipulate[
Plot3D[((4  z (1 + Sin[2*Pi*n]) + (Abs[y]  1) (Cos[2*Pi*n]))/5)*
Max[0, 2*Abs[y]  z^2  y^2 +
5*Max[0, .03  (z + .01 + .02*Sin[2*Pi*n])^2  (Abs[y] 
1 + .2*Cos[2*Pi*n])^2]/2], {z, 2, 2}, {y, 2,
2}], {n, 1, 1}]
Don't know what syntax was killed via copy+paste, but... nice graph!
Re: functions that gives cool designs.
wasapiguy2 wrote: snip
Fixed:
Code: Select all
Manipulate[
Plot3D[((4  z (1 + Sin[2*Pi*n]) + (Abs[y]  1) (Cos[2*Pi*n]))/5)*
Max[0, 2*Abs[y]  z^2  y^2 +
5*Max[0, .03  (z + .01 + .2*Sin[2*Pi*n])^2  (Abs[y] 
1 + .2*Cos[2*Pi*n])^2]/2], {z, 2, 2}, {y, 2,
2}], {n, 1, 1}]
You had an extra zero in there (0.02 instead of 0.2). Try running that and see if it looks better.
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Re: functions that give cool designs
Can someone post a screenshot for those of us without Mathematica?
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Re: functions that give cool designs
I'm already fairly sure of what it'll turn out to look like, based upon Qaanol's post.

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 Joined: Mon Apr 20, 2009 3:43 am UTC
Re: functions that give cool designs
the tree wrote:I'm already fairly sure of what it'll turn out to look like, based upon Qaanol's post.
I'd be very surprised if your guess was correct.
Sagekilla wrote:wasapiguy2 wrote: snip
Fixed:Code: Select all
Manipulate[
Plot3D[((4  z (1 + Sin[2*Pi*n]) + (Abs[y]  1) (Cos[2*Pi*n]))/5)*
Max[0, 2*Abs[y]  z^2  y^2 +
5*Max[0, .03  (z + .01 + .2*Sin[2*Pi*n])^2  (Abs[y] 
1 + .2*Cos[2*Pi*n])^2]/2], {z, 2, 2}, {y, 2,
2}], {n, 1, 1}]
You had an extra zero in there (0.02 instead of 0.2). Try running that and see if it looks better.
D'oh!
Re: functions that give cool designs
x%100*tan(X) is pretty cool
 Hackfleischkannibale
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Re: functions that gives cool designs.
Afif_D wrote:I thought you all genius guys would pt some cool functions in here.... but I was wrong....
This is only a side note, but there is something that I learned* about math as I started to think about becoming a real mathematician:
Apparently you think that a genius would think like you, and care about the things you care about, only more so. Since it's very hard to actually imagine how a smarter person would think, most people think like you do.
But mathematics moves forward at an incredible pace, because no one cares about doing very well the stuff everybody can do, if they can do something new instead. So as a mathematician, when you learn something, you don't play around with it for longer than you need to get a good grasp of it. You just learn the theory, and then try to get to the next theory as fast as possible, until you are either so exhausted that you have to move at a slower pace, or alternatively until you reach the border of the map, the end of the road, where you have to chop free your own path.
Since there are many mathematicians who practically all start at the same point, many of them get short of breath before they reach the edge, and so the central parts of the map are extremely well known, not any more very surprising, and rather uninteresting for the intelligent and curious.
Now I've been rambling on for quite a bit, but how is this relevant? Well, if you're interested in something or when you think you found out about something new, and you haven't learned a hell lot of math or you aren't an extraordinary genius, chances are it's already wellknown and/or uninteresting to those who have learned and/or who are geniuses.
This is not meant to discourage you from being interested in your own discoveries and interesting bits within your reach, but if you want to talk about these things, don't think you can get the geniuses interested, expect to talk to people at your level.
This post probably sounds condescending, but I think that it is useful information, I don't know how to write it down differently, and also I'm not yet very knowledgeable myself.
*I am not entirely sure how true this is, and part of the reason why I post it is that I want to hear whether I'm being overly pessimistic or something.
If this sentence makes no sense to you, why don't you just change a pig?
Re: functions that gives cool designs.
Hackfleischkannibale wrote: ... a lot...
*I am not entirely sure how true this is, and part of the reason why I post it is that I want to hear whether I'm being overly pessimistic or something.
While I wouldn't really advise asking a mathematical genius for tips on Algebra homework (unless that's their job), I think it's more or less the duty of someone who is very good and thoroughly enjoys math (or any field) to tell people who ask 'why is this interesting?' It should be their inherent duty to make other people interested. Not only are they doing a good deed, they're probably helping out the field as a whole by getting more people interested. I'm nowhere near as gifted in the sciences as some people that frequent these fora, but people ask me questions about space and black holes and whatnot to the point where I have a script of cool things I can tell them about. But that could be because I'm nowhere near a genius and those things are just still interesting to me.
So in conclusion, you're probably right, most people probably won't care what you're asking. But some do, and that's why you see the same scientists' faces on TV all the time. Some do, and all should.
 Hackfleischkannibale
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Re: functions that gives cool designs.
Korrente wrote:While I wouldn't really advise asking a mathematical genius for tips on Algebra homework (unless that's their job), I think it's more or less the duty of someone who is very good and thoroughly enjoys math (or any field) to tell people who ask 'why is this interesting?' It should be their inherent duty to make other people interested.
That isn't really what I meant. If you don't know math at all, and/or aren't really interested, then the top guys could be well the ones to talk to (even though I think it takes a slightly different set of qualities that "top math researcher" to do that, like being able to communicate and "dumb down"); but if you are already interested, I think most of what you are interested in will be much more interesting for those of a level of skill close to you, and much less to those at the frontier of math research.
The reason I posted that was that I recognize the line of thought, to pose your questions to the best because they are the best and therefore seem much more interesting than your peers or those, say, a year above you. Many people do it. I did that, and I still do that, but I'm trying to stop it, because I don't want to be annoying.
If this sentence makes no sense to you, why don't you just change a pig?
Re: functions that gives cool designs.
Hackfleischkannibale wrote:Korrente wrote:While I wouldn't really advise asking a mathematical genius for tips on Algebra homework (unless that's their job), I think it's more or less the duty of someone who is very good and thoroughly enjoys math (or any field) to tell people who ask 'why is this interesting?' It should be their inherent duty to make other people interested.
That isn't really what I meant. If you don't know math at all, and/or aren't really interested, then the top guys could be well the ones to talk to (even though I think it takes a slightly different set of qualities that "top math researcher" to do that, like being able to communicate and "dumb down"); but if you are already interested, I think most of what you are interested in will be much more interesting for those of a level of skill close to you, and much less to those at the frontier of math research.
The reason I posted that was that I recognize the line of thought, to pose your questions to the best because they are the best and therefore seem much more interesting than your peers or those, say, a year above you. Many people do it. I did that, and I still do that, but I'm trying to stop it, because I don't want to be annoying.
I see what you're saying and you're probably right. But when you get into just what you're interested in, it's probably better to just find others who share the interest, regardless of skill level. I think that's what OP was trying to do but didn't really find anyone. Either way I don't think it can be generalized as much you did in your first post, I would still advocate asking the best if they'll listen to you. If they're wholly disinterested or bothered by you, then stop asking. Case in point, if OP had just asked his classmates what some cool linear equations were, he'd never have discovered that implicit or parametric equations are more interesting.
Re: functions that gives cool designs.
Okay then...Korrente wrote:Case in point, if OP had just asked his classmates what some cool linear equations were, he'd never have discovered that implicit or parametric equations are more interesting.
Afif_D:
 If you really want to carry on with explicit functions then spend some time working on the algebra and basic calculus behind them. Take a look at infinite series (power series, Taylor expansions, Fourier series...) and how they can be used to represent nearly every explicit function under the sun in terms of well known functions.
 If not then take a look at implicit functions of the form f(x,y)=0 and how they relate to parametric functions of the form (x,y)=(X(t),Y(t)).
 Then look at different coorindate systems, specifically polar coordinates.
 There are other ways to generate an interesting image than plotting functions. A plot of y=f(x) might be considered to be an image the set of ordered pairs (x,y) that satisfies the equation the y=f(x) but how else might you define an interesting set of ordered pairs (x,y)? That's just something to think about.
Re: functions that give cool designs
Honestly, I wouldn't get too bothered rationalizing about anything that Afif_D posts. He regularly comes in here, asking questions, and when his errors are pointed out (mind you, he's often asking about errors in reasoning/mechanics), he acts like he's too cool for math and the passiveaggressive doucheometer goes to 11. This is best evidenced by his favorite sentence format, "Look man, [insert poorly formatted jerkish statement here]." More than once I've caught him post something remarkably aggressive and quickly delete it. Now, I'll grant him that English is probably not his first language, but that's no excuse to be a dick.
Afif, either you want to learn math, or you want to try to show all the mathenthusiasts in this forum how cool you are. Honestly, we don't really care. You could be hanging out with Bono and doing lines of coke of Tila Tequila, and none of us really give a damn. If you want to learn math, there are dozens of people in this forum who willingly spend time answering your questions. But not getting the answer you want is not an excuse to go all John Fitzgerald Page.
Afif, either you want to learn math, or you want to try to show all the mathenthusiasts in this forum how cool you are. Honestly, we don't really care. You could be hanging out with Bono and doing lines of coke of Tila Tequila, and none of us really give a damn. If you want to learn math, there are dozens of people in this forum who willingly spend time answering your questions. But not getting the answer you want is not an excuse to go all John Fitzgerald Page.
 Eebster the Great
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Re: functions that give cool designs
If you want an f: ℝ → ℝ whose graph looks interesting when viewed as an image, and which furthermore can be expressed explicitly and compactly using a certain limited set of operations, we can still probably help. Unfortunately, you are leaving out all the most interesting ones.
By the way, does anybody else here find the Cantor function and the Question Mark function to be interesting?
By the way, does anybody else here find the Cantor function and the Question Mark function to be interesting?
 jestingrabbit
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Re: functions that give cool designs
Eebster the Great wrote:By the way, does anybody else here find the Cantor function and the Question Mark function to be interesting?
The question mark is pretty fun. It doesn't look like much though.
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Re: functions that give cool designs
Someone had this in their signature:
x^2+(3y/2(x^2+x6)/(x^2+x+2))^2 = 36
I put it into Wolfram Alpha and it turned out to be pretty cool:
http://www.wolframalpha.com/input/?i=x^2%2B%283y%2F2%28x^2%2Bx6%29%2F%28x^2%2Bx%2B2%29%29^2+%3D+36
Sorry it's not linked properly but phpbb didn't seem to like me trying to link it very much.
x^2+(3y/2(x^2+x6)/(x^2+x+2))^2 = 36
I put it into Wolfram Alpha and it turned out to be pretty cool:
http://www.wolframalpha.com/input/?i=x^2%2B%283y%2F2%28x^2%2Bx6%29%2F%28x^2%2Bx%2B2%29%29^2+%3D+36
Sorry it's not linked properly but phpbb didn't seem to like me trying to link it very much.
 Eebster the Great
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Re: functions that give cool designs
If we are allowing parametric plots, I'm a fan of this one for its simplicity and craziness: [math]x = \sin{(t \cos{t})}[/math][math]y = \cos{(t \sin{t})}[/math][math]2\pi \le t \le 2\pi[/math] Increasing the range of t just makes the curve more and more complicated, as it never repeats.
WA link: http://www.wolframalpha.com/input/?i=ParametricPlot[{Sin[t+Cos[t]]%2C+Cos[t+Sin[t]]}%2C+{t%2C+2pi%2C+2pi}].
Not sure why this forum won't accept this link.
WA link: http://www.wolframalpha.com/input/?i=ParametricPlot[{Sin[t+Cos[t]]%2C+Cos[t+Sin[t]]}%2C+{t%2C+2pi%2C+2pi}].
Not sure why this forum won't accept this link.
Re: functions that give cool designs
Eebster the Great wrote:If we are allowing parametric plots, I'm a fan of this one for its simplicity and craziness: [math]x = \sin{(t \cos{t})}[/math][math]y = \cos{(t \sin{t})}[/math][math]2\pi \le t \le 2\pi[/math] Increasing the range of t just makes the curve more and more complicated, as it never repeats.
WA link: http://www.wolframalpha.com/input/?i=ParametricPlot[{Sin[t+Cos[t]]%2C+Cos[t+Sin[t]]}%2C+{t%2C+2pi%2C+2pi}].
Not sure why this forum won't accept this link.
It's the brackets.
she/they
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Re: functions that give cool designs
Sizik wrote:Eebster the Great wrote:If we are allowing parametric plots, I'm a fan of this one for its simplicity and craziness: [math]x = \sin{(t \cos{t})}[/math][math]y = \cos{(t \sin{t})}[/math][math]2\pi \le t \le 2\pi[/math] Increasing the range of t just makes the curve more and more complicated, as it never repeats.
WA link: http://www.wolframalpha.com/input/?i=ParametricPlot[{Sin[t+Cos[t]]%2C+Cos[t+Sin[t]]}%2C+{t%2C+2pi%2C+2pi}].
Not sure why this forum won't accept this link.
It's the brackets.
Carets seem to break links as well.
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Re: functions that give cool designs
For those two graphs, copy the following lines into Wolfram Alpha (and I agree, they're both pretty neat):
x^2 + ((3 y)/2  (x^2 + Abs[x]  6)/(x^2 + Abs[x] + 2))^2 == 36
ParametricPlot[{Sin[t Cos[t]], Cos[t Sin[t]]}, {t, 2pi, 2pi}]
x^2 + ((3 y)/2  (x^2 + Abs[x]  6)/(x^2 + Abs[x] + 2))^2 == 36
ParametricPlot[{Sin[t Cos[t]], Cos[t Sin[t]]}, {t, 2pi, 2pi}]
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 Eebster the Great
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Re: functions that give cool designs
Sizik wrote:It's the brackets.
Replacing the square brackets with %whatever didn't seem to fix it, but maybe it doesn't like curly brackets either.
Re: functions that give cool designs
...You guys know you can just highlight and manually copypaste the entire link there to get the desired results, despite being unable able to just click on the link?
Although it that event, sticking it in code tags is apt to make the copypasting easier without inadvertently clicking the broken first part of the link...
e.g.
I suspect those various sites that provide short links (like bit.ly) would work too, although I don't care for them since you never know if it leads where you want it to lead, and besides, that takes slightly more effort on the behalf of the poster.
Although it that event, sticking it in code tags is apt to make the copypasting easier without inadvertently clicking the broken first part of the link...
e.g.
Code: Select all
http://www.wolframalpha.com/input/?i=x^2%2B%283y%2F2%28x^2%2Bx6%29%2F%28x^2%2Bx%2B2%29%29^2+%3D+36
I suspect those various sites that provide short links (like bit.ly) would work too, although I don't care for them since you never know if it leads where you want it to lead, and besides, that takes slightly more effort on the behalf of the poster.
Re: functions that give cool designs
Was gonna suggest the Peano curve, but Eebster the Great's suggestion has everything we need.
 Eebster the Great
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Re: functions that give cool designs
Jyrki wrote:Was gonna suggest the Peano curve, but Eebster the Great's suggestion has everything we need.
I don't think my curve hits every point in the box, though (i.e. I don't know if its image is all of [1,1]x[1,1]). I don't think it even come close.
 jestingrabbit
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Re: functions that give cool designs
Eebster the Great wrote:Jyrki wrote:Was gonna suggest the Peano curve, but Eebster the Great's suggestion has everything we need.
I don't think my curve hits every point in the box, though (i.e. I don't know if its image is all of [1,1]x[1,1]). I don't think it even come close.
Agreed. Consider points of the form (x,0): tsin(t) is of the form (k+1/2)pi for integer k only countable many times. You can further demonstrate that it only intersects any horizontal or vertical line through the square at only countable many points too. Its image also has Lebesgue measure 0.
It is very interesting though.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: functions that give cool designs
All Shadow priest spells that deal Fire damage now appear green.
Big freaky cereal boxes of death.
 Eebster the Great
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Re: functions that give cool designs
jestingrabbit wrote:Eebster the Great wrote:Jyrki wrote:Was gonna suggest the Peano curve, but Eebster the Great's suggestion has everything we need.
I don't think my curve hits every point in the box, though (i.e. I don't know if its image is all of [1,1]x[1,1]). I don't think it even come close.
Agreed. Consider points of the form (x,0): tsin(t) is of the form (k+1/2)pi for integer k only countable many times. You can further demonstrate that it only intersects any horizontal or vertical line through the square at only countable many points too. Its image also has Lebesgue measure 0.
It is very interesting though.
However, it would be interesting if one could demonstrate that graphing that function using a line of finite thickness, one hits every point. That is, is every point in [1,1]x[1,1] of less than distance d from some point on the image for any d?
 jestingrabbit
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Re: functions that give cool designs
Yeah "Is the image dense in the spuare?" is the right question.
The first simplification is to strip one level of trig off. Consider f(t) = (t*cos(t) (mod 2pi), t*sin(t) (mod 2pi)). If the image of f is dense in [0,2pi)x[0,2pi), then you're done. That isn't really a huge help though. Without the modulo stuff, you've got a pretty nice looking spiral ([imath]r=\theta[/imath]). I thought you might be able to get somewhere by just considering values of t near integer multiples of pi, but that just gets you density on two of the edges of the square [edit] and a parallel line through the centre. Likewise for integer and a half multiples.
Still, I suspect it is dense. But the proving will take a little work.
Edit: The image is indeed dense. Argument to follow after I've had some sleep.
The first simplification is to strip one level of trig off. Consider f(t) = (t*cos(t) (mod 2pi), t*sin(t) (mod 2pi)). If the image of f is dense in [0,2pi)x[0,2pi), then you're done. That isn't really a huge help though. Without the modulo stuff, you've got a pretty nice looking spiral ([imath]r=\theta[/imath]). I thought you might be able to get somewhere by just considering values of t near integer multiples of pi, but that just gets you density on two of the edges of the square [edit] and a parallel line through the centre. Likewise for integer and a half multiples.
Still, I suspect it is dense. But the proving will take a little work.
Edit: The image is indeed dense. Argument to follow after I've had some sleep.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: functions that give cool designs
Not at all what you've actually asked for, however if you're looking for some of the beauty in Mathematics, I can thoroughly recommend http://www.bathsheba.com/sculpt/
For example
For example
 jestingrabbit
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Re: functions that give cool designs
So, the argument that the image is dense.
Let g(t) = (t*cos(t), t*sin(t)) and f(t) = (t*cos(t) (mod 2pi), t*sin(t) (mod 2pi)). Notice that the square [0,2pi)x[0,2pi) is a topological group, with coordinatewise addition being the operation in question. My claim is that for any [imath]\alpha[/imath] with [imath]\tan(\alpha)[/imath] irrational, the set [imath]f(\alpha + 2k\pi)[/imath] is dense. Note that [imath]g(\alpha + 2k\pi) = g(\alpha) + 2k\pi(\cos(\alpha), \sin(\alpha)).[/imath] Therefore, my claim is equivalent to the set [math]A=\{(2k\pi \cos(\alpha) \pmod{2\pi}, 2k\pi \sin(\alpha) \pmod{2\pi}): k\in \mathbf{Z}\}[/math] being dense in the square. Its closure is a (topologically) closed subgroup of the square. The only closed, proper subgroups are subgroups of the images (under modding both coords by 2pi) of lines with rational gradients through the origin. You can check that all the straight lines through the point [imath]2\pi(\cos(\alpha), \sin(\alpha))[/imath] must have irrational gradients, so the only lines that could contain A have irrational gradients, so the closure of A isn't a proper subgroup, ie it is the whole square, ie A is dense in the square.
The argument I originally thought of, which needs only that that the images of lines with irrational gradients are dense (rather than a knowledge of all closed subgroups), was along the lines of finding ever larger parts of the spiral which were within epsilon of straight lines with irrational gradients. That works okay, but the above argument is a little cleaner imo. Plus, it demonstrates that there's a lot of redundancy in the density ie that just taking the image of reals separated by 2pi gives you enough points to get density.
Let g(t) = (t*cos(t), t*sin(t)) and f(t) = (t*cos(t) (mod 2pi), t*sin(t) (mod 2pi)). Notice that the square [0,2pi)x[0,2pi) is a topological group, with coordinatewise addition being the operation in question. My claim is that for any [imath]\alpha[/imath] with [imath]\tan(\alpha)[/imath] irrational, the set [imath]f(\alpha + 2k\pi)[/imath] is dense. Note that [imath]g(\alpha + 2k\pi) = g(\alpha) + 2k\pi(\cos(\alpha), \sin(\alpha)).[/imath] Therefore, my claim is equivalent to the set [math]A=\{(2k\pi \cos(\alpha) \pmod{2\pi}, 2k\pi \sin(\alpha) \pmod{2\pi}): k\in \mathbf{Z}\}[/math] being dense in the square. Its closure is a (topologically) closed subgroup of the square. The only closed, proper subgroups are subgroups of the images (under modding both coords by 2pi) of lines with rational gradients through the origin. You can check that all the straight lines through the point [imath]2\pi(\cos(\alpha), \sin(\alpha))[/imath] must have irrational gradients, so the only lines that could contain A have irrational gradients, so the closure of A isn't a proper subgroup, ie it is the whole square, ie A is dense in the square.
The argument I originally thought of, which needs only that that the images of lines with irrational gradients are dense (rather than a knowledge of all closed subgroups), was along the lines of finding ever larger parts of the spiral which were within epsilon of straight lines with irrational gradients. That works okay, but the above argument is a little cleaner imo. Plus, it demonstrates that there's a lot of redundancy in the density ie that just taking the image of reals separated by 2pi gives you enough points to get density.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
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