User talk:Sam Derbyshire/Archive2

Swastika curve

Thank you for the image at Swastika curve, it is much improved (I had planned to replace it too). I have two suggestions. First, I wonder if you could make the text on the axes a bit bigger (the curve now looks great, but the text is still too small). Second, if you can, it would be great if you could provide the picture source code (at the image page), just in case it may need modification in the future (and that's a bit more in the free documentation spirit). Wonder what you think. You can reply here. Thanks. Oleg Alexandrov (talk) 04:41, 10 January 2008 (UTC)

There, I uploaded a better version. Feel free to tell me what you would like to see changed again (if anything). I hope that's all right. -- Xedi (talk) 08:39, 10 January 2008 (UTC)

Oh, by the way, what is the recommended font to use on mathematical images ? I remember seeing a manual of style for mathematical graphs but can't seem to find it anymore. If you see any images that need improvement (mine or not) I'll gladly try to help. Thanks. Xedi (talk) 10:33, 10 January 2008 (UTC)

Thanks. I'd argue that the font is still a bit blurry and may need enlargement, but that's up to you (here's the size I used at Bernoulli's inequality). I don't know about font type, see maybe WP:MSM, or Wikipedia:WikiProject Mathematics/Graphics, or ask at WT:WPM).

Also, since you make a lot of images, a long term idea would be to use SVG instead of PNG, as SVG is editable and I think works better for curves and other line art. But this is again a comment in the long term. Oleg Alexandrov (talk) 16:25, 10 January 2008 (UTC)

The idea was to have a higher resolution when clicking on the image, not a full resoluion image in the article. If you think it's preferable that the full size image is in the article, fine, but I thought it would be to small as there would be no higher resolution. About the font, I didn't find any information of those article. I think I remember someone recommending Computer Modern (the font used in LaTeX), but I don't think it goes too well (or maybe I haven't got a good version, I didn't really understand what I was supposed to download). As for the SVG, I don't really know how vectorising works - would I have to manually go over each line and draw it (like with the pen tool), or is the process somewhat automatized ? I don't think I have the time to manually convert those images to SVG. (Also, many of my images are actually animations, and for the moment browser support for animations is low, and I really don't see how I could convert my images to animated SVG in a reasonable amount of time). What do you think ? -- Xedi (talk) 18:09, 10 January 2008 (UTC)

My own philosophy is that it is good for an image to look good both in full resolution and in the thumbnail (as most people are looking only at the thumbnail in the article anyway). About SVG, from what I've seen MuPad exports natively to SVG (there's some command from that, or from the file menu). But again, the SVG remark was more about creating future images. I do agree that it is not worth it converting old images to SVG, and that animations can't be converted. Cheers, Oleg Alexandrov (talk) 20:10, 10 January 2008 (UTC)

Well the MuPAD SVG export feature doesn't really work well - often the font doesn't come out correctly, and some other bugs frequently occur. And, having had a look at the SVG that were created, they honestly aren't really editable nor convenient to work with. I do agree however that using SVG would be a good idea, just that I'm not a graphist and don't want to be approximating the curves by drawing splines in some vector image editing software. Sorry.

Sure, I understand. Thank you for the many very nice pictures you contributed. Oleg Alexandrov (talk) 04:19, 11 January 2008 (UTC)

Simplifying Complex Functions

I was wondering if MuPad had the ability to handle complex functions more intuitively. In particular, is there a way to set up a complex function and have MuPad extract the numerical values for the Real and Imaginary parts? With all of the tests that I've done I have ended up evaluating the real and imaginary parts and then plugging them into MuPad, which is quite complicated, especially when graphing the Absolute Value. Vjasper (talk) 06:57, 20 January 2008 (UTC)

In particular, how did you handle the Gamma function? I can't find anything that isolates the real and imaginary parts of the function. Vjasper (talk) 07:04, 20 January 2008 (UTC)

Well the Im, Re and abs (note the case) give the imaginary part, the real part and the absolute value, respectively. Does that answer your question ? -- Xedi (talk) 12:18, 20 January 2008 (UTC)

I understand what each of the graphs represent, I'm just curious what you actually plugged into MuPad in order to get them. That is to say, if you take the 3D example code, what actual function did you use for, say, the Real part of the Gamma Function? The reason I'm asking this is because, as far as I can tell, there isn't a good way to isolate the Real output of the Gamma function for complex input. In the case of things like, for instance, cos(z), you can expand it to cos(x + iy), then use the additive angle laws for Cosine in order to produce cos(z) = cos(x)*cosh(y) - i*sin(x)*sinh(y) (the hyperbolic functions coming, of course, from their relationship to the complex trigonometric functions). In this case you could easily use the left hand and right hand sides of the equation (those with and without an imaginary coefficient) to produce a graph of your Real and Imaginary parts, respectively.

My question is basically, how did you manage to do this with functions that do not have an explicit (or even implicit) formula with which you could substitute complex values in, and then try to solve and isolate the real and imaginary parts? With something like the Gamma function, there is no real way to express it (other than the integral, which isn't very helpful when dealing with complex functions), so how were you able to find some kind of function to give MuPad to graph? Vjasper (talk) 19:29, 20 January 2008 (UTC)

As I said, just use the Re, Im and abs. For example, for the gamma function, instead of the 8*sin(x-cos(y))+(x^2+x*y) I got at my example code, just plug in Re(gamma(x+I*y)) to get the real part, or Im(gamma(x+I*y)) for the imaginary part, or abs(gamma(x*I*y)) for the absolute value. Hope that helps. -- Xedi (talk) 21:56, 20 January 2008 (UTC)

Oh, wow. I didn't realize quite what you meant, but that's ridiculously simple. Thank you. Vjasper (talk) 04:24, 21 January 2008 (UTC)

Your rendering of LambertWAll.png is just indescribably beautiful

Your rendering of LambertWAll.png is just so beautiful. I don't know what else to say.

Does MuPAD select the hues for you? The color triad is exceedingly elegant.

-Aaron Hefel hefel.wordpress.com —Preceding unsigned comment added by 128.255.95.87 (talk) 17:18, 31 January 2008 (UTC)

No, I just chose the colors myself.

In MuPad, I chose : Red->CornflowerBlue for the real part, CadmiumYellow->SapGreen for imaginary part, and CadmiumYellow->Red for the absolute value. I just fiddled around with the colours until I obtained something nice.

Anyway, thanks. -- Xedi (talk) 17:26, 31 January 2008 (UTC)

The residues page graph

I'm just wondering how that function was evaluated, is it the real part or imaginary part? and what is does the hue display? Complex arguement perhaps? Just wondering =} 193.217.64.142 (talk) 22:58, 19 February 2008 (UTC)

You're right that I didn't say precisely so in the article, I corrected that. So the image is the absolute value of the function described in the text, and the hue just describes the height of the absolute value. There's absolutely no information in that plot about the imaginary part. Thanks for your interest and your correction. -- Xedi (talk) 23:10, 19 February 2008 (UTC)

Ok thank you, i was just wondering what the beautiful plot was. T.Stokke (talk) 14:55, 20 February 2008 (UTC)

Graph for counterexample

You seem to generate nice graphs! Can you please produce one for the counterexample? Schmock (talk) 20:14, 21 February 2008 (UTC)

Yes, sure, just added it to the article !

Be sure to ask if you have any other ideas for mathematical diagrams, I'm happy to help.

Thanks -- Xedi (talk) 04:02, 22 February 2008 (UTC)

Great, thank you! -- Schmock (talk) 10:24, 22 February 2008 (UTC)

Nice picture.

I like your pictures. They really help me understand my math books. --Sbluen (talk) 20:08, 1 March 2008 (UTC)

Thanks, it's nice to know that. Which ones in particular helped, and how ?

If you know of any topics for which it would help to have some more visualisation, I'd be happy to know. -- Xedi (talk) 20:18, 1 March 2008 (UTC)

Right now, I'm studying hyperbolic trigonometry, and your [[Image:HyperbolicAnimation.gif] helped me understand it. --Sbluen (talk) 20:44, 1 March 2008 (UTC)

My question on maths reference desk

You seem to be having trouble with all the questions (to whit 2, 4, 5 and 6) that involve the tap being on and the plug out at the same time. Under these circumstances, four gallons of water flow into the tank every minute and 6 gallons flow out. So what is the overall change in the amount of water every minute? Algebraist 13:08, 28 April 2008 (UTC)

• 2 gallons? —Preceding unsigned comment added by 217.171.129.79 (talk) 13:15, 28 April 2008 (UTC)

• If it is 2 gallons, is Q2. 10 minutes? Q4 6 minutes? Q5 14 gallons? And the last one 0 gallons? --217.171.129.79 (talk) 13:19, 28 April 2008 (UTC)

Yes, total flow = what flows in - what flows out, so here you get that the total flow is 4-6 = -2 gallons per minute (the minus sign shows that water is flowing out). So you got Q2 right.

For Q4, there are 20 gallons initially, and for the first four minutes, 2 gallons are flowing out per minute. After these first four minutes, how many gallons are left ? Then water flows out at 6 gallons per minute.

• So four minutes of water flowing in fills it by 2 gallons, so there are 8 gallons in the tank. So the answer is 8 ÷ 6? So 1½? —Preceding unsigned comment added by 217.171.129.79 (talk) 13:39, 28 April 2008 (UTC)

For Q5, start at 20 gallons, it flows out at a rate of 6 gallons per minute for 2 minutes, then flows out at 2 gallons per minute for 3 minutes. Can you work out how much is left ?

• 2 gallons?

For Q6, after 4 minutes there are 16 gallons as you said. After that, water is flowing out at a rate of 2 gallons per minute for six minutes. Can you see how many gallons will be left ?

• 4 gallons?

Hope that helps -- Xedi (talk) 13:35, 28 April 2008 (UTC)

Did i get them right this time?

Answered on the Mathematics reference desk. -- Xedi (talk) 14:56, 28 April 2008 (UTC)

Wikimedia Commons

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--OsamaK 06:59, 4 June 2008 (UTC)

Wright omega function

In this edit summary, you referred to "incorrect capitalization" but you left the initial "O" as capital. What is that? I would think since the lower-case Greek letter is used, one would also write that as lower case. Michael Hardy (talk) 17:08, 14 July 2008 (UTC)

Sorry, I was not aware of any convention about omega/Omega, I always thought it was only a useful way of distinguishing lowercase and uppercase when writing in LaTeX. Feel free to move the article again to change that capitalization if you feel it is better. I would've thought capitalization is better though, as "Wright Omega" is a proper noun. --Xedi^Talk 12:07, 15 July 2008 (UTC)

Evolute animation awesome, how about Parallel curves

I was reading about Parallel curve and the diagram didn't make much sense, but your linked Evolute figure is awesome. 'Twould be lovely if Image:Evolute_and_parallel.gif were expanded to animate the different ways of establishing parallel curves — the only one that makes sense to me is "Tracing the center of a circle rolled along the curve." Anyway, thanks again, that pic is worth 2000 words. -- Skierpage (talk) 22:49, 29 July 2008 (UTC)

Hi, thanks for the kind words ! I've made an animation about parallel curves, you can have a look at it on the parallel curves article. Hope it helps. --Xedi^Talk 13:00, 30 July 2008 (UTC)

Fuel economy

Hello Sam, Thanks for the answer regarding fuel economy. But doesn't this also depend on the mix of gas/oil consumed? Isn't there some unstated assumption here? Best regards, Jonpol (talk) 16:55, 18 August 2008 (UTC)

Great work!

Hello Sam, please have a look at http://www.luschny.de/math/factorial/FastFactorialFunctions.htm were I exhibit your Gamma plots. Thanks a lot for your fine work!

Cheers Peter —Preceding unsigned comment added by 92.226.228.69 (talk) 23:18, 19 December 2008 (UTC)

Benford's law picture

I don't understand File:BenfordDensities.png. What is the set of which the average density is computed? What's the metric that you're using? If you asked me what the "average density" is for, say, the three-digit natural numbers starting with 1, I would say it's shaped like a top-hat, with some constant value ("one number per unit interval") between 100 and 199, and then zero everywhere else. Clearly that's not what you drew.

Moreover, when I look at that picture it seems to very much imply that the "average density" (whatever that is) of, say 198-199 is larger than 100-101, and therefore it seems to imply that Benford's law would say the first three digits 198 is more likely than the first three digits 101. Of course Benford's law says the opposite. So how precisely does the height and shape of the curves give information about Benford's law?

Don't get me wrong, it's obviously a beautiful picture. I'm open-minded to the possibility that when properly explained this image will shed light on Benford's law and be a nice addition to the article. So I'm eager to hear a more full explanation. :-) --Steve (talk) 04:28, 26 December 2008 (UTC)

Hey. The 1 curve, for example, at a point n on the x-axis, returns the value A1(n)/n, where A1(n) is the number of numbers between 1 and n whose first digit is 1. So each curve basically says how many numbers starting with a there has been, and divides that by that total amount of numbers encountered. The average densities are not drawn (because they would clutter the diagram a bit too much), but they are just the constant lines with y value the average value that the curve takes. If you look carefully at the limsup and liminf of the first curve, you should see that limsup = 5/9 and liminf = 1/9, and the average value is just that predicted by Benford's law, ie log10(2). For the 9 curve, on the other hand, limsup is 1/9 and liminf=1/81, and the average value that the curve takes is log10(10/9).

The idea is just that we would want to compute the so called natural density of the set of numbers with first digit 1, which is the limit as n tends to infinity of A1(n)/n, but the diagram shows that this does not converge as liminf != limsup. But the next-best thing to the actual limit is the average value of A1(n)/n, which corresponds to Benford's law prediction.

I hope that provides a good enough explanation, I know I didn't explain much in the article, so if you feel anything was unclear please correct that in the article. Thanks ! -- Xedi^Talk 07:31, 26 December 2008 (UTC)

OK, I moved the figure and changed the explanation somewhat. Let me know if you approve, it is your figure after all. :-) --Steve (talk) 19:02, 27 December 2008 (UTC)

Featured picture

Hi there. I thought I should draw your attention to Wikipedia:Featured picture candidates/HypotrochoidOutThreeFifths.gif; I nominated your image HypotrochoidOutThreeFifths.gif to be a featured picture and it seems to be doing quite well. I just noticed all your other images, though, and many seem to be of a similar standard of excellence. Great stuff! -- BlastOButter42 ^See Hear _Speak 21:37, 20 May 2009 (UTC)

Congratulations!

An image created by you has been promoted to featured picture status Your image, File:HypotrochoidOutThreeFifths.gif, was nominated on Wikipedia:Featured picture candidates, gained a consensus of support, and has been promoted. If you would like to nominate an image, please do so at Wikipedia:Featured picture candidates. Thank you for your contribution! wadester16 | Talk→ 18:59, 26 May 2009 (UTC)

File:HypotrochoidCurve.png listed for deletion

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The Mathematical Illustrator's Barnstar

For your many beautiful and educational mathematical illustrations and animations, I hereby award you the Mathematical Illustrator's Barnstar. -- JovanCormac (talk) 09:06, 29 July 2009 (UTC)

NowCommons: File:EpitrochoidOn1.gif

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Math Overflow

Math Overflow is a new article on WP. Feel free to edit. Mhym (talk) 01:17, 17 February 2010 (UTC)

MuPad

Do you know if it is still possible to obtain MuPad free as a student, or at least on some reduced price? Thank you, T.Stokke (talk) 15:12, 24 February 2010 (UTC)

Sadly I don't think so; it's no longer for sale. As you know, MuPAD was discontinued and its symbolic toolbox was incorporated into MATLAB. Which is a great shame in my opinion, seeing as MuPAD had amazing possibilities for mathematical visualisation. The possibilities in Mathematica are probably greater, but I find the language to be more annoying, and most things just don't look as nice!

You can always try to send an email to the MuPAD team; I think I might do that myself actually! Xedi^Talk 01:47, 25 February 2010 (UTC)

You were right, I sent them a mail and they said it is no longer possible to obtain it for free and I have to buy entire MATLAB together with the symbolic math toolbox to get it.

What is personally a tad annoying is my university has MATLAB but the linux MuPad/Matlab here refuses to draw correctly in X-11 (or perhaps the problem is deeper considering it doesn't draw correctly even when I export graphics), and the windows Matlab doesn't have symbolic toolbox at all.

Also $100 is a bit steep on a student budget, however I really am considering it.

Just a little rant, sorry to be spamming up your talk page and as always, beautiful plots. T.Stokke (talk) 14:01, 27 February 2010 (UTC)

3d Plot of Complex Function

May I ask you if you could plot the absolute value of the Gamma function on z = x+I*y for x = -4.1..4.1 and y = -2..2, where the colour in each grid is the local complex argument? I believe this will contain all information about the function in a single plot. May I also ask if you succeed that you post the MuPad code here? Thank you, 129.240.72.39 (talk) 10:47, 1 March 2010 (UTC)

I now realize how much like homework this sounded, the idea was to make it look like the hand drawn graph on the gamma function wiki page. I have been fiddling around with some colour functions but can't seem to get a satisfactory result. I can manage to make a linear interpolation between two colours in rgb, but what I would like is a high resolution interpolation in hsv, just like ColorType Rainbow is in MuPad. Any input you can have would be much appreciated, if the question is unclear just say so.

Ps. in the case of the Gamma function where there are poles etc. do you use an Adaptivemesh or just a high resolution SubMesh?

Thank you for your attention 85.167.25.63 (talk) 18:43, 1 April 2010 (UTC)

Sorry for not answering before, I haven't touched MuPAD in quite a while and no longer have it installed, sadly.

I have some code on [page], maybe it can answer your questions. To be honest, a lot of the code came from the documentation which is really well done. I don't remember off the top of my head how to change the color scheme, but the code I linked to seems to give the answer: you can just copy the code I gave to draw a 3D graph and then use a colour function, I remember spending a lot of time figuring out nice custom colour functions. I'm not sure if you mean the grid itself or the surface should have the colour, but both cases should be of similar difficulty.

Thanks. --Sam^Talk 19:06, 1 April 2010 (UTC)

I think you are a student in Warwick

Correct me if I am wrong, but you are a math student in Warwick University, right? If so, please contact me as I do have some questions concerning the math department! --Dyaa (talk) 18:00, 9 March 2010 (UTC)

You're entirely correct! What do you want to know? --Sam^Talk 19:15, 9 March 2010 (UTC)

Animated GIFs

WP recently changed the way it handles thumbnails so many of the animated GIFs you've created are showing up in articles as static GIFs. I started a thread at Wikipedia talk:WikiProject Mathematics but I thought you might want a heads-up.--RDBury (talk) 09:35, 2 June 2010 (UTC)

POTD notification

POTD

Hi Sam,

Just to let you know that the Featured Picture File:HypotrochoidOutThreeFifths.gif is due to make an appearance as Picture of the Day on July 25, 2010. If you get a chance, you can check and improve the caption at Template:POTD/2010-07-25. howcheng {chat} 17:59, 23 July 2010 (UTC)

Neat! Thanks. --Sam^Talk 23:21, 23 July 2010 (UTC)

ADE corrections

Hello: Under ADE classifications I pointed out two correctuins needed:

Reference 13 should be Martin, Pablo; Singerman, David ... and (3 bullets up) should read Abdellah

Is this OK with you?

Thanks, John McKay 23:06, 14 August 2010 (UTC) —Preceding unsigned comment added by 173.178.19.124 (talk)

Sure, I just changed the references to have the proper names.

Next time you want to suggest an improvement, try using the talk page instead (in this case at Talk:ADE classification).

Thanks. -Sam^Talk 09:47, 15 August 2010 (UTC)

Van-der-pol

Hello, Mr. Derbyshire.

I was curious about the picture of the Van-der-pol's oscillator you did. It is a direction field plot, but I don't know how you can do this. I wanted to make the same picture with this online plotter: http://www.ies.co.jp/math/java/calc/DiffEqu/DiffEqu.html but I don't know which equation I need to type in.

If you could please help me out. Thank you. —Preceding unsigned comment added by 91.67.130.30 (talk) 17:06, 3 November 2010 (UTC)

Hi. If you take a look at the article on the Van der Pol oscillator, you'll find that it is a second order differential equation. The webpage you linked to only allows first order differential equations. I found an applet that allows you to look at phase diagrams for second order differential equations here [1]. It's not great, as you don't see the individual vectors, but it's not too bad either. You can probably manage to find better ones. --Sam^Talk 20:20, 3 November 2010 (UTC)

Programs (What does it cost to make such plots?)

Hi Sam.

Your surface plots like have the best quality I have ever seen.
Unfortunately such plots usually look like this or like this .
However, I've also seen poor MuPAD plots, e.g. here. So what is it, what you made different? I'd like to create graphics like these on my own. Can you tell me, how much approximately I have to pay for a program, that can make plots of this quality? Or do you think it's possible to get an old (pre-MATLAB) MuPAD copy from someone? Greetings, Lipedia (talk) 23:22, 23 January 2011 (UTC)

Hi, sorry for the delay in answering.

The MuPAD plots are just standard MuPAD, I think they look nice because it is a recent version. Sadly MuPAD was discontinued. Now I tend to use a mix of Mathematica and POVRay to make nice pictures, but I'll agree that the one Mathematica makes aren't usually as nice. I haven't really spent much time tweaking the Mathematica settings to get it to look nice though.

I've got some code at this page, which you may want to take a look at.

Thanks for your interest. --Sam^Talk 08:30, 9 March 2011 (UTC)

Great work

Hello. I just want to commend you on your creation of the image of the complex roots of the degree-24 polynomials with coefficients 1 and −1. Truly breathtaking; it's my desktop background at the moment. Cheers, —Anonymous Dissident^Talk 12:25, 9 March 2011 (UTC)

Thanks! It's always quite surprising which patterns can appear from seemingly simple problems. --Sam^Talk 00:03, 10 March 2011 (UTC)

Notification: changes to "Mark my edits as minor by default" preference

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Thank you for your understanding and happy editing :) Editing on behalf of User:Jarry1250, LivingBot (talk) 18:31, 14 March 2011 (UTC)

Dimension formula for GLn irreps

Thank you for making this edit. However you should not have marked it a minor edit; there is no way it matches the criteria. I'll try to modify the example to avoid the confusion due to the accident n = number of boxes, that is irrelevant but suggests it is of importance. It would be nice to have a reference; not that I doubt the truth, but many texts that mention the hook length formula for Sn or that treat GLn representations do not mention this formula (and it does not seem obvious from Weyl's dimension formula either). Marc van Leeuwen (talk) 08:06, 22 May 2011 (UTC)

Thanks for your interest. I tend to tag most of my edits as minor, a bad habit I still have to kick. I added a reference. --Sam^Talk 09:23, 22 May 2011 (UTC)

I see now that I made a few mistakes while writing that, as you said. Thanks for the heads-up. --Sam^Talk 10:17, 22 May 2011 (UTC)

HypotrochoidOutThreeFifths.gif

I like your image File:HypotrochoidOutThreeFifths.gif and want to feature it at Portal:Mathematics as a "Picture of the month". Unfortunately, large animated GIFs like this aren't animated when they're scaled down for display (see Portal:Mathematics/Featured picture/2011 12 Portal:Mathematics/Featured picture/2012 01 and its talk page). Could you possibly create another version that starts with the first frame(s) showing the completed curve before it is removed from the graph and then incrementally drawn again (IOW, move the final few "static" frames of the completed curve to the beginning of the sequence)? That way, the non-animated version of the GIF will show the full curve, but the looping animated version will still look the same as before. (I'd do it myself, but I don't have the right software.) - dcljr (talk) 00:36, 9 November 2011 (UTC)

I've moved the image to Portal:Mathematics/Featured picture/2012 01 to allow more time to change the image. Looks like you've been away from WP for a while, so I'm going to request help at Commons... - dcljr (talk) 18:24, 29 November 2011 (UTC)

The Cosmic Software

Hi! I would like to ask you if you have already read my book. If you have, do you have any good or bad opinion about it? Sincerely, Nándor Böröcz — Preceding unsigned comment added by 84.1.181.16 (talk) 18:10, 30 November 2011 (UTC)

Thank You Sam

You have a real talent. When I am browsing Wikipedia and I see an arrestingly beautiful graph, I usually guess correctly that it's one of yours. Your graphs are artistic expressions of the highest caliber. Thank you for sharing them with the world. — Preceding unsigned comment added by 134.10.12.43 (talk) 08:14, 14 December 2011 (UTC)

Request for new image at Exponentiation

The current image under section Exponentiation#Zero to the zero power shows a 3D plot of z=|x|^y, x,y∈ℝ. My impression is that the only purpose of for plotting the absolute value is to give a mirror image for a different perspective. I think that this is potentially confusing and adds very little, though the image is in many respects very good. You seem to have been the original uploader of the image. Would you be able/prepared to replace this with a similar image showing only x≥0?

As a secondary request, three of the four green curves through (x,y)=(0,0) have no formula that is evident from inspection of the curves. It might be worthwhile replacing these with curves of constant y/x, where their form is illustrated perhaps by a (feint) plane hanging from each of these curves down to the plane z=0 (or some similar device; it could also be done via the caption). This will more clearly illustrate the limit as approached from any fixed direction other than along x=0.

I hope I'm not being too presumptious in posing this request. You seem to produce some very since grpahics. — Quondum^☏✎ 12:46, 14 February 2012 (UTC)

Sure, I'll do it! Thanks for the request; feel free to ask about other images you think could be interesting. -Sam^Talk 13:21, 14 February 2012 (UTC)

Sorry for taking this long, I've completely revamped my 3D plot rendering scheme. I think I have it up and running now, so I'll be able to contribute quite a few images!

Here's what I have so far, is this close to what you had in mind? I can add some curves of constant height on top, but there doesn't seem to be much point given that there already are level curves on the surface. Opinions? -Sam^Talk 03:14, 16 February 2012 (UTC)

Hey, you're not at my beck and call :) – your willingness is much appreciated. The reduction to x ≥ 0 is perfect, and I like your choice of countour spacing. I think the original red and green curves added significantly to the ease of understanding, including their visible indication of the limiting point, notwithstanding the contours, and it would be nice to add them again (with the green curves being at say y = x and y = −x, and all the three red curves as they were). We'll have to see how to make it visible that the green lines are defined by a vertical plane.

You seem to have shaded the surface according to the value of z. There is a curious feature in the neighbourhood of the point (0,0,1) where the tangent plane goes through a wild gyration (the horizontal tangent line as one moves down the z-axis suddenly spins through π at this point creating a little plateau) that is core to the limiting behaviour at that point and that would be nice to highlight visually. I have no idea whether this is possible with your setup, but shading the surface according to how it would reflect light from a chosen source angle (i.e. as a function of the angle/gradient of the surface, tuned to exploit how we interpret shading as a visual cue) would probably do this, with lighting angle chosen carefully. This feature is not obvious with the current rendering. Or perhaps one will only see it with a massively expanded version? I'm not sure.

It may be interesting to change the perspective to be from the right of where it is now – viewed from where x is negative. This would expose the point (0,0,0) to view, and would show a little of the surface from behind as one moves up the z-axis, emphasizing the tangent plane along this axis.

And all this is purely my opinion, though I hope others would agree with me. — Quondum^☏✎ 08:41, 16 February 2012 (UTC)

OK, I've added those curves. I tried changing the lighting, and adding some reflections, but I don't think it looked very nice and in my mind it distracts from the mathematical accuracy of the image. Here are two shots I have: one, two. Any further comments? I can add some vertical planes cutting out the green lines, but that clutters the image a bit too much... -Sam^Talk 22:46, 16 February 2012 (UTC)

I prefer the perspective a little from "behind" where the bottom red curves can be followed, namely image two. I think it will be sufficient to label the green family of curves in the caption (as you have now done in the article). The one curve ({{math|y=0) could of course be red or green; it is a member of both families. I think my only further comment would be to determine the scale; it seems my thought of magnifying it achieves little by way of showing local detail, but including the features further afield (scaled as in your earlier image gives a better overall sense of the function and three-dimensional shape (and probably "wow" factor). Maybe you want to ask others, or simply put in what you feel is best? I think you have more than adequately achieved what I'd hoped for. — Quondum^☏✎ 04:39, 17 February 2012 (UTC)