Re: [math-fun] Tan[Sin[x]]-Sin[Tan[x]] puzzles
Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg
On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote:
Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg
Oops, and here's A. Goucher's missing antecedent: When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions); however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/ Sincerely, Adam P. Goucher
Check out images of this fungus: clathrus pusillus by typing the name into Google and clicking on Images. --Dan
Looks a bit like one of them fancy bicycle helmets ... WFL On 4/12/13, Dan Asimov <dasimov@earthlink.net> wrote:
Check out images of this fungus:
clathrus pusillus
by typing the name into Google and clicking on Images.
--Dan
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Whoa, those pits in the leaves <http://gosper.org/.flattop2.png> go deep. Roots? --rwg (I omitted the /2 in the exponent, since I wasn't fudging for √2√π.) (Trying to save selection as pdf crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.) On Wed, Apr 10, 2013 at 1:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote:
Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg
Oops, and here's A. Goucher's missing antecedent:
When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions);
however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/
Sincerely,
Adam P. Goucher
Wow! That's an amazing picture! Is there any way to estimate the height/depth of the pits? Are they analogous to Gibbs's ears on square waves? If so, perhaps they have a finite size. http://en.wikipedia.org/wiki/Gibbs_phenomenon Henry may be the only funster who figured out that the url should really be http://gosper.org/flattop2.png <http://gosper.org/.flattop2.png> . (No period before flattop. It looks right in this compose buffer, but still 404s with a phantom period when I click it!) I don't see how they could be Gibbs, which is an artifact of series truncation. And I can't imagine that the pits aren't roots. Apropos Gibbs, http://www.tweedledum.com/rwg/gibbs.htm --rwg At 06:20 PM 4/18/2013, Bill Gosper wrote: Whoa, those pits in the leaves <http://gosper.org/.flattop2.png> go deep. Roots? --rwg (I omitted the /2 in the exponent, since I wasn't fudging for √2√π.) (Trying to save selection as pdf crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.) On Wed, Apr 10, 2013 at 1:58 AM, Bill Gosper <billgosper@gmail.com> wrote: On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote: Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg Oops, and here's A. Goucher's missing antecedent: When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions); however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/ Sincerely, Adam P. Goucher
Oh foo, they're obvious roots with closed forms. E.g, Out[11]= (-1)^(1/4)/Sqrt[2 Pi] In[13]:= E^-%11^-2 Out[13]= 1 In fact, In[14]:= Solve[1==E^-z^-2,z] Out[14]= {{z->ConditionalExpression[-(1/(Sqrt[2 \[Pi]] Sqrt[I C[1]])),C[1]\[Element]Integers]}, {z->ConditionalExpression[1/(Sqrt[2 \[Pi]] Sqrt[I C[1]]),C[1]\[Element]Integers]}} But strangely, In[15]:= Solve[1 == E^-z^-2] Out[15]= {} --rwg Correction: (Trying to save selection as [png] crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.) On Thu, Apr 18, 2013 at 8:10 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow! That's an amazing picture!
Is there any way to estimate the height/depth of the pits? Are they analogous to Gibbs's ears on square waves? If so, perhaps they have a finite size. http://en.wikipedia.org/wiki/Gibbs_phenomenon
Henry may be the only funster who figured out that the url should really be http://gosper.org/flattop2.png <http://gosper.org/.flattop2.png> . (No period before flattop. It looks right
in this compose buffer, but still 404s with a phantom period when I click it!)
I don't see how they could be Gibbs, which is an artifact of series truncation. And I can't imagine that the pits aren't roots. Apropos Gibbs, http://www.tweedledum.com/rwg/gibbs.htm --rwg
At 06:20 PM 4/18/2013, Bill Gosper wrote:
Whoa, those pits in the leaves <http://gosper.org/.flattop2.png> go deep.
Roots?
--rwg
(I omitted the /2 in the exponent, since I wasn't fudging for √2√π.)
(Trying to save selection as pdf crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.)
On Wed, Apr 10, 2013 at 1:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote:
Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg
Oops, and here's A. Goucher's missing antecedent:
When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions);
however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/
Sincerely, Adam P. Goucher
On Thu, Apr 18, 2013 at 9:49 PM, Bill Gosper <billgosper@gmail.com> wrote:
Oh foo, they're obvious roots with closed forms. E.g, Out[11]= (-1)^(1/4)/Sqrt[2 Pi] In[13]:= E^-%11^-2 Out[13]= 1 In fact, In[14]:= Solve[1==E^-z^-2,z] Out[14]= {{z->ConditionalExpression[-(1/(Sqrt[2 \[Pi]] Sqrt[I C[1]])),C[1]\[Element]Integers]}, {z->ConditionalExpression[1/(Sqrt[2 \[Pi]] Sqrt[I C[1]]),C[1]\[Element]Integers]}} But strangely, In[15]:= Solve[1 == E^-z^-2]
Out[15]= {} --rwg Correction:
(Trying to save selection as [png] crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.)
On Thu, Apr 18, 2013 at 8:10 PM, Bill Gosper <billgosper@gmail.com> wrote:
Wow! That's an amazing picture!
And amazingly misleading. That "tentacled lemniscate" is shown clipped at height 1, but it's <1 in a (very) shallow valley surrounding the real axis, excluding 0,0. gosper.org/flattop3.png shows the plotware dissolving in tears while failing to show this. Neil B figured out that the little white "lemniscate" is the plotware so boggled by the exponential of a double pole that it can't even manage to clip.
DanA> P.S. Out of curiosity, I just plotted the surface z = (x^2 - y^2) / (x^2 + y^2) which is of course just z = cos(2 theta), and it gives a cute graph with a vertical interval as singular points at (x,y) = (0,0). This is well-known to some people, but it wasn't to me. You get essentially the identical graph for z = 2xy / (x^2 + y^2) = sin(2 theta). --Dan I was trying to plot for a kid the corresponding verticality at 0^0, but I absolutely could not cajole Mma out to the edge. Macsyma does plot3d(x^y,x,0,1,y,0,1) with no fuss: gosper.org/x^y.bmp --rwg Well, you do have to make up a value for 0^0. How do you drag this surface out of Mma?
Is there any way to estimate the height/depth of the pits? Are they analogous to Gibbs's ears on square waves? If so, perhaps they have a finite size. http://en.wikipedia.org/wiki/Gibbs_phenomenon
Henry may be the only funster who figured out that the url should really be http://gosper.org/flattop2.png <http://gosper.org/.flattop2.png> . (No period before flattop. It looks right
in this compose buffer, but still 404s with a phantom period when I click it!)
I don't see how they could be Gibbs, which is an artifact of series truncation. And I can't imagine that the pits aren't roots. Apropos Gibbs,
http://www.tweedledum.com/rwg/gibbs.htm --rwg
At 06:20 PM 4/18/2013, Bill Gosper wrote:
Whoa, those pits in the leaves <http://gosper.org/.flattop2.png> go deep.
Roots?
--rwg
(I omitted the /2 in the exponent, since I wasn't fudging for √2√π.)
(Trying to save selection as pdf crashes 9.0.1 for PlotPoints->666. And the plotting artifacts are worse.)
On Wed, Apr 10, 2013 at 1:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Wed, Apr 10, 2013 at 1:09 AM, Bill Gosper <billgosper@gmail.com> wrote:
Here <http://gosper.org/flattop.png>'s a more traditional plot near z=0. --rwg
Oops, and here's A. Goucher's missing antecedent:
When you said `ultraflat', I thought you were referring to the property of all derivatives being zero at that point. Obviously, complex-differentiable functions (such as yours) cannot have this property (except for constant functions);
however, there are infinitely differentiable examples over the reals such as f(x) = exp(-1/x^2). A function with this property is considered here: http://cp4space.wordpress.com/2013/02/28/radical-tauism/
Sincerely, Adam P. Goucher
I had the same problem until realizing that a parametric polar coordinate plot would do the trick. --Dan On 2013-04-26, at 11:41 PM, Bill Gosper wrote:
P.S. Out of curiosity, I just plotted the surface
z = (x^2 - y^2) / (x^2 + y^2)
which is of course just z = cos(2 theta),
and it gives a cute graph with a vertical interval as singular points at (x,y) = (0,0).
This is well-known to some people, but it wasn't to me.
You get essentially the identical graph for
z = 2xy / (x^2 + y^2) = sin(2 theta).
--Dan
I was trying to plot for a kid the corresponding verticality at 0^0, but I absolutely could not cajole Mma out to the edge. Macsyma does plot3d(x^y,x,0,1,y,0,1) with no fuss: gosper.org/x^y.bmp --rwg Well, you do have to make up a value for 0^0. How do you drag this surface out of Mma?
participants (3)
-
Bill Gosper -
Dan Asimov -
Fred lunnon