# Talk:Alhazen's problem

I don't think there is such a thing as "Islamic Mathematics";
mathematics is a science that's it. I never heard the term "Christian
Mathematics” for instance. So, why this page should use the term
"Islamic"? People could say is an Arab mathematician, or German etc,
whatever is the case. But what religion has to do with this.217.55.172.180 (talk) 18:27, 4 July 2013 (UTC)

I have rephrased it, while keeping the link to Mathematics in medieval Islam. I think that the idea is that although there is only one universal field of mathematics, it was developed at different times in different parts of the world, and so it is convenient to break up the history of mathematics and mathematicians into different articles for different cultures. Richard75 (talk) 19:54, 11 February 2015 (UTC)

This would be much easier to understand if there was a diagram. Richard75 (talk) 19:49, 11 February 2015 (UTC)

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Sorry but I don't speak fluent english'
I think there is a misunderstanding with reference:

Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine.
Katz presents the Alhazen's method to compute volumes, first attempt after Archimedes and before Cavalieri in calculus. There is nothing to do with Alhazen's problem. See A.I. Sabra, Ibn al-Haytham' lemas for solving Alhazen's Problem for the Alhazen's problem proof. HB (talk) 14:24, 10 January 2019 (UTC)

This review is transcluded from Talk:Alhazen's problem/GA1. The edit link for this section can be used to add comments to the review.
Nominator: David Eppstein (talk · contribs) 20:17, 10 June 2025 (UTC)
Reviewer: MathKeduor7 (talk · contribs) 17:23, 20 June 2025 (UTC)
Co-reviewer: Gramix13 (talk · contribs) 05:06, 28 June 2025 (UTC)

It is the first time I review for GA status. I've read and understood WP:GACR6. Now I'll immediately begin reading the entire article very carefully. I'll keep you informed. MathKeduor7 (talk) 17:23, 20 June 2025 (UTC)
P.S. I've read and understood the other criteria at Wikipedia:Good article criteria. This will take some time. MathKeduor7 (talk) 17:28, 20 June 2025 (UTC)
I've just started reading the article's current revision. MathKeduor7 (talk) 17:37, 20 June 2025 (UTC)
 Comment: I've read the very beginning: "Alhazen's problem is a mathematical problem in optics concerning reflection in a spherical mirror. It asks for the point in the mirror where one given point reflects to another."

I think it is a clear presentation, and I've understood it easily by looking at "File:Alhazen-pb.svg" (the image of a reflection in a circular mirror in the plane) illustrating the problem. I haven't checked the references yet, because I plan to first check if the article is well-written and understandable. So, for this little part I've read:  Done MathKeduor7 (talk) 17:50, 20 June 2025 (UTC)
 Comment: I've read the following: "The special case of a concave spherical mirror is also known as Alhazen's billiard problem, as it can be formulated equivalently as constructing a reflected path from one billiard ball to another on a circular billiard table. Other equivalent formulations ask for the shortest path from one point to the other that touches the circle, or for an ellipse that is tangent to the circle and has the given points as its foci."

I think this is also straightforward clear, the layman will probably only need to click to know what is a concave spherical mirror, and imagining a circular billiard table and billiard balls is easy. The equivalent formulations are interesting and not obvious. So, for this little part I've read:  Done MathKeduor7 (talk) 18:01, 20 June 2025 (UTC)
 Comment: Quote: "Although special cases of this problem were studied by Ptolemy, it is named for the 11th-century Arab mathematician Alhazen (Ibn al-Haytham), who formulated it more generally and presented a solution in his Book of Optics. It has no straightedge and compass construction; instead, al-Haytham and others including Christiaan Huygens found solutions involving the intersection of conic sections. According to Roberto Marcolongo, Leonardo da Vinci invented a mechanical device to solve the problem. Later mathematicians, starting with Jack M. Elkin in 1965, solved the problem algebraically as the solution to a quartic equation, and used this equation to prove the impossibility of solving the problem with straightedge and compass."

Historical remarks, reference to notable solutions of some mathematicians (and Leonardo). Not much to comment here, it's obviously clear. Personally, I've got very curious about Leonardo's mechanical device! But let's keep reading in the order. I think it would be a good idea to mention from when is Ptolemy, so that the reader doesn't need to click to know how much time it took for the 1965 algebraic solution. I mean, like not everyone knows Alhazen is from the 11th-century (as it says), and not everyone knows approximate Ptolemy's epoch. @David Eppstein: What do you think of telling (as context) something like "Ptolemy (XXX – YYY AD)"? (or something like that) Just my two cents of course! It's fine as it is:  Done  MathKeduor7 (talk) 18:26, 20 June 2025 (UTC)
I'll continue later. I have some commitments now. MathKeduor7 (talk) 18:27, 20 June 2025 (UTC)
 Comment: I would only change "Researchers have extended this problem and the methods used to solve it to mirrors of other shapes and to non-Euclidean geometry." to "In the ??th century, researchers have [...]." To better inform and make emphasis on the long timeline and evolution (not to mention it's still attracting interest!) of this old problem. Just my two cents of course! It's fine as it is:  Done MathKeduor7 (talk) 19:24, 20 June 2025 (UTC)

Ok, dates for Ptolemy and later researchers added. —David Eppstein (talk) 21:41, 20 June 2025 (UTC)
My analysis so far:

The lead is really good and meets all requirements of MOS:LEAD.
 Comment: I'll now download all the references used in this article. It will take some time. MathKeduor7 (talk) 19:28, 20 June 2025 (UTC)
From now on, I will comment on criteria 1 and 2 of WP:GACR6 for each part of the text. Let's begin! MathKeduor7 (talk) 19:41, 20 June 2025 (UTC)
 Comment: Quote: "The problem comprises drawing lines from two points, meeting at a third point on the circumference (boundary) of a circle and making equal angles with the normal at that point (specular reflection). It belongs to geometrical optics (in which light is modeled using rays rather than waves or particles), and catoptrics, the use of mirrors to control light: it can be used to find the path of a ray of light that starts at one point of space, is reflected from a spherical mirror, and passes through a second point. Although this is a three-dimensional problem, it can immediately be reduced to the two-dimensional problem of reflection in a circular mirror in the plane, because its solution lies entirely within the plane formed by the two points and the center of the sphere." P.S. Reference given was downloaded from http://www.jstor.org/stable/2589403

Yes, it's well-written and summarizes the cited reference content with his own words (as far as I can tell, everything is in accordance with the given reference information). So:  Done MathKeduor7 (talk) 19:49, 20 June 2025 (UTC)
 Comment: The second paragraph of the "Formulation" section cites six different references. It will take some time to read them all, so: that's it for today! MathKeduor7 (talk) 19:54, 20 June 2025 (UTC)
P.S. I've just noticed some references are to books, not to downloadable journal papers, so I'll use Google Books preview feature! Cya, MathKeduor7 (talk) 19:58, 20 June 2025 (UTC)
 Comment: There are used thirty-nine references in total in this article. Later I will make a list of the ones I have access to and the ones  I don't have. Then we will have to think about what to do to for me to check the ones I do not have access yet. I'll probably make this list tomorrow. MathKeduor7 (talk) 01:14, 21 June 2025 (UTC)
P.S. I never tried https://wikipedialibrary.wmflabs.org/ . Maybe it can help. MathKeduor7 (talk) 01:16, 21 June 2025 (UTC)
Thankfully, many are freely accessible, and so I managed to get access to all of the first six ones very easily (the only exception: it was a hard one... the book of 100 great problems). This is enough for the next part of the review! Cya, MathKeduor7 (talk) 01:30, 21 June 2025 (UTC)

You may be able to see much of the relevant part of 100 Great Problems through Google Books: [1]. I have a pdf but I didn't record where I found it. —David Eppstein (talk) 01:49, 21 June 2025 (UTC)
Thank you! MathKeduor7 (talk) 01:51, 21 June 2025 (UTC)
 Comment: I ask for one week to read and understand thoughtly the first six references (so that I can perhaps give good suggestions to the main author of the article). So: GA review is frozen for likely a week (not more than this, and I may be able to get it sooner!). In addition: I have some commitments next week, and Professor David Eppstein informed me on his user talk page that he will be traveling. That's it for now. Btw, this problem is interesting, and I want to understand it better, I am having fun reading about it. :) MathKeduor7 (talk) 08:34, 21 June 2025 (UTC)
 Comment: As a preparation for the next step of the review, I'll list the first six references the article is using. That's for convenience. MathKeduor7 (talk) 06:54, 23 June 2025 (UTC)
[1] Neumann, Peter M. (1998), "Reflections on Reflection in a Spherical Mirror", The American Mathematical Monthly, 105 (6): 523–528, doi:10.1080/00029890.1998.12004920, JSTOR 2589403, MR 1626185
[2] Dörrie, Heinrich (1965), "Alhazen's Billiard Problem", 100 Great Problems of Elementary Mathematics, translated by Antin, David, Dover, pp. 197–200, ISBN 978-0-486-61348-2
[3] Chen, Tieling; Ilukor, Paul; Koo, Reginald (March 2024), "The one-cushion escape from snooker in a circular table", Recreational Mathematics Magazine, 11 (18): 99–109, doi:10.2478/rmm-2024-0006
[4] Peterson, Ivars (March 3, 1997), "Billiards in the Round", The Mathematical Tourist, retrieved 2025-06-05
[5] Poirier, Nathan; McDaniel, Michael (2012), "Alhazen's hyperbolic billiard problem", Involve, 5 (3): 273–282, doi:10.2140/involve.2012.5.273, MR 3044613
[6] Drexler, Michael; Gander, Martin J. (1998), "Circular billiard", SIAM Review, 40 (2): 315–323, Bibcode:1998SIAMR..40..315D, doi:10.1137/S0036144596310872
That's it for now. MathKeduor7 (talk) 06:57, 23 June 2025 (UTC)
 Comment: Quoting the 2nd paragraph of "Formulation" section: "The same problem can be formulated with the two given points inside the circle instead of outside.[1] In this case the solution describes the path of a billiards ball reflected within a circular billiards table,[2][3] as Lewis Carroll once suggested for billiards play.[4] Because the two chords of the circle through the given points and the reflection point form equal angles with the circle, they form the two equal sides of an isosceles triangle inscribed within the circle, with the two given points on these two sides. Another equivalent form of Alhazen's problem asks to construct this triangle.[2][5] For points near each other within the solution, in general position, there will be two solutions, but points that are farther apart have four solutions.[6]"

I intend to analyze this paragraph tomorrow. MathKeduor7 (talk) 07:57, 23 June 2025 (UTC)
Drexler/Gander is also JSTOR 2653338 in case that's easier for you to access. —David Eppstein (talk) 08:53, 23 June 2025 (UTC)
Thank you, Professor David Eppstein! I had already got all six ones! ^^ Btw, when I think about it... Sometimes I will need help here, maybe from my great friend GregariousMadness, who has university library access and a lot of expertise in geometry. ^^ @GregariousMadness: Hi, friend! Do you have some time to help me reviewing this article? MathKeduor7 (talk) 10:13, 23 June 2025 (UTC)
@GregariousMadness: Would you be interested (and have time to) review the section "Alhazen's_problem#Algebraic"? Many nice references and a lot of advanced math! It sounds fun, haha. ^^ MathKeduor7 (talk) 10:23, 23 June 2025 (UTC)
 Comment: "The same problem can be formulated with the two given points inside the circle instead of outside." can be understood by a toddler and is backed by the Ref [1] in its very first page. Clearly written and very well referenced, so:  Done MathKeduor7 (talk) 10:35, 23 June 2025 (UTC)

The entire 2nd paragraph of "Formulation" section is pretty clear and well-written (even more with the pictures), so all I need is to check if everything is fully backed by the given references. And I'll do that tomorrow! MathKeduor7 (talk) 11:54, 23 June 2025 (UTC)
 Comment: "In this case the solution describes the path of a billiards ball reflected within a circular billiards table,[2][3] as Lewis Carroll once suggested for billiards play.[4]" follows from the stated references (note: Refs [3] and [4] don't mention the word "Alhazen" specifically, but they are about related mathematical billiard tables with point-sized balls, but again, the article just says short infos about them with Lewis Carroll etc, and you can check it at https://mathtourist.blogspot.com/2020/05/billiards-in-round.html for example, that it's relevant to the article IMHO), so, one more positive check towards GA:  Done MathKeduor7 (talk) 23:50, 23 June 2025 (UTC)
Note: I need some days of rest. Checking references is exhausting, but I'll go till the end! MathKeduor7 (talk) 00:34, 24 June 2025 (UTC)
 Comment: Quote: "Because the two chords of the circle through the given points and the reflection point form equal angles with the circle, they form the two equal sides of an isosceles triangle inscribed within the circle, with the two given points on these two sides. Another equivalent form of Alhazen's problem asks to construct this triangle.[2][5] For points near each other within the solution, in general position, there will be two solutions, but points that are farther apart have four solutions.[6]"

It follows from the references (fully backed by them), but I don't know if the writing shouldn't be improved. I thought of it as very hard to parse, hard to read... P.S. I'll try to explain why! MathKeduor7 (talk) 06:17, 25 June 2025 (UTC)
For now: this part is IMHO  Not done (but we will improve it I think. and it will eventually become  Done!) MathKeduor7 (talk) 06:17, 25 June 2025 (UTC)
@David Eppstein: Professor, please give me two or three days to explain why I think this is hard to read. MathKeduor7 (talk) 06:18, 25 June 2025 (UTC)
Nevermind. It's clear enough, I was just making a confusion. So:  Done MathKeduor7 (talk) 10:05, 26 June 2025 (UTC)
P.S. @David Eppstein: Thank you for the clarifications! MathKeduor7 (talk) 11:08, 26 June 2025 (UTC)
Note: I've just discovered that for GA review I don't need to check every ref, just a sample (WP:GANI says: "in-depth review is provided in all other cases. This must include a spot-check of a sample of the sources in the article to verify that each source supports the text in the article that it covers, and that no copyrighted material has been added to the article from the source.") MathKeduor7 (talk) 19:02, 25 June 2025 (UTC)

I think it's pretty clear from the sample I analyzed and Professor David Eppstein's historic of successful GA nominations on Wikipedia that he has (with extremely high level of certainty) done a great job here, but: I must check a representative sample anyways... I must. MathKeduor7 (talk) 19:02, 25 June 2025 (UTC)
 Comment: Before checking more references/sources, I'll make an analysis of the rest of the article: the question "Is it well-written and understandable?" will be my guide. This will be the next step of the review. MathKeduor7 (talk) 07:59, 26 June 2025 (UTC)

 Comment: Quote: "Another way of describing the problem, for points inside or outside the circle, is that it seeks an ellipse having the two given points as its foci, tangent to the given circle. The point of tangency is the solution point to Alhazen's problem. A ray from one focus of the ellipse to this point of tangency will be reflected by the ellipse to the other focus, and because the given circle has the same angle at the point of tangency, it will also reflect the same ray in the same way. The smallest such ellipse has as its point of tangency the point of the circle whose sum of distances to the two given points is minimum.[6][7] The equivalence between finding the path of a reflected light ray and minimizing the total length of the path is Hero's principle, later reformulated in Fermat's principle as minimizing the total time traveled by the ray.[8] More generally, as James Gregory observed, for any analogous three-dimensional reflection problem, the point of reflection is also a point of tangency of an ellipsoid having the source and destination of the reflected ray as its foci.[9]"

Yes, it is well-written and understandable. Very easy-to-follow too. MathKeduor7 (talk) 08:26, 26 June 2025 (UTC)
 Comment: Quote: "Ptolemy included the problem of reflection in a circular mirror in his Optics (written in the second century CE), but was only able to solve certain special cases;[10][11] al-Haytham formulated and solved the problem more generally.[10] Al-Haytham was inspired by Ptolemy's work, and modeled his own book on Ptolemy's, but differed from it in important ways; for instance, Ptolemy used a model of visual perception in which visual rays travel outward from the eye to the objects it sees, while al-Haytham reversed this to the still-used model in which light rays travel inward from objects to the eye.[12][13]"

Yes, it is very well-written and understandable. Historical notes are easier than math information. MathKeduor7 (talk) 08:28, 26 June 2025 (UTC)


### Solutions section
This section is more technical than the previous ones. I admit that most of the math is beyond me. I could just check the references or maybe ask someone (who understands the math used here) to help me on this. MathKeduor7 (talk) 08:31, 26 June 2025 (UTC)

Maybe I could find a co-reviewer at Wikipedia talk:WikiProject Mathematics. I'll try this later. MathKeduor7 (talk) 08:34, 26 June 2025 (UTC)
If I don't find a co-reviewer, I'll just check if the references back the claims. MathKeduor7 (talk) 08:37, 26 June 2025 (UTC)
It is intentional that the earlier parts of the article are less technical than the later parts. See WP:TECHNICAL It is especially important to make the lead section understandable using plain language, and it is often helpful to begin with more common and accessible subtopics, then proceed to those requiring advanced knowledge or addressing niche specialties. I don't think it's possible to make this topic completely nontechnical, but there still may be parts that are unnecessarily confusing and that should be improved. —David Eppstein (talk) 10:57, 26 June 2025 (UTC)
Okay! I'll try my best to understand the technical parts, and make maybe silly comments and questions, but that may be helpful even so! MathKeduor7 (talk) 11:12, 26 June 2025 (UTC)
Note: There are some references not in English (I'll not check them): in Catalan, Latin, Italian, German and French. I won't check because Google translate is not that reliable IMO, and, also, because I don't need to. MathKeduor7 (talk) 08:51, 26 June 2025 (UTC)
Note: I'll use a small (five references) simple random sample, check them, and if everything is alright this article will be promoted to GA. MathKeduor7 (talk) 10:13, 26 June 2025 (UTC)

1) From when is Pappus' Collection? (placed "when tag") MathKeduor7 (talk) 14:36, 26 June 2025 (UTC)
 Fixed https://en.wikipedia.org/w/index.php?title=Alhazen%27s_problem&diff=1297595461&oldid=1297493788 MathKeduor7 (talk) 06:25, 27 June 2025 (UTC)
2) File:Alhazen hyperbola.svg needs repair, I told "The curves are too thin and the yellow ones are hard to see." 
MathKeduor7 (talk) 14:36, 26 June 2025 (UTC)
3) Ditto File:Alhazenproblem-tangentellipsetoacircle.png has a similar problem and it's a raster image, so good luck making again from scratch! MathKeduor7 (talk) 14:40, 26 June 2025 (UTC)
4) Ditto Alhazen-pb.svg (too thin curves are bad for accessibility, this one is .svg, so: easy to fix for those who know how to) MathKeduor7 (talk) 14:51, 26 June 2025 (UTC)
This is more controversial, but I still think what I said is right. MathKeduor7 (talk) 14:53, 26 June 2025 (UTC)
I made the Alhazen hyperbola lines thicker. I don't want to change the color palette because those colors were specifically chosen to be accessible to color-blind people, and I don't want to change the other images because they're not by me and that's not really how images on Commons work: unlike Wikipedia content, Commons content is made by individuals and it's not generally considered appropriate for others to change them. To some extent image legibility can be adjusted by changing the size at which the image is displayed and that's why the hyperbola one is bigger. Also, although advice on image legibility is helpful, this is somewhat beyond what GA criterion 6 is about. —David Eppstein (talk) 05:24, 27 June 2025 (UTC)
Understood! Thank you. MathKeduor7 (talk) 06:25, 27 June 2025 (UTC)

I'll choose randomly among the English-language ones and the ones I did not check yet, five references to be checked. I'll keep you informed. MathKeduor7 (talk) 06:30, 27 June 2025 (UTC)
The drawn selected references were: [35], [24], [36], [14], [13]. P.S. I used https://www.gigacalculator.com/randomizers/random-picker.php to make the selection. MathKeduor7 (talk) 06:50, 27 June 2025 (UTC)
I need some time to get access to those sources and read them: review is frozen for some days. MathKeduor7 (talk) 06:52, 27 June 2025 (UTC)
For convenience:
[13] Smith, A. Mark (1990), "Alhazen's Debt to Ptolemy's Optics", in Levere, Trevor H.; Shea, William R. (eds.), Nature, Experiment, and the Sciences: Essays on Galileo and the History of Science in Honour of Stillman Drake, Boston Studies in the Philosophy of Science, vol. 120, Springer Netherlands, pp. 147–164, doi:10.1007/978-94-009-1878-8_6, ISBN 9789400918788
[14] Knorr, Wilbur Richard (1993), The Ancient Tradition of Geometric Problems, Dover Books on Mathematics, Courier Corporation, p. 345, ISBN 9780486675329
[24] Highfield, Roger (April 1, 1997), "Don solves the last puzzle left by ancient Greeks", Electronic Telegraph, 676, archived from the original on 2004-11-23, retrieved 2008-09-24
[35] Gander, Walter; Gruntz, Dominik (November 1992), "The billiard problem", International Journal of Mathematical Education in Science and Technology, 23 (6): 825–830, doi:10.1080/0020739920230602
[36] Glaeser, Georg (1999), "Reflections on Spheres and Cylinders of Revolution" (PDF), Journal for Geometry and Graphics, 3 (2): 121–139, MR 1748025
Those are the selected refs. MathKeduor7 (talk) 12:39, 27 June 2025 (UTC)

I don't think I can help you with pdfs of these. My copy of Knorr's book is physical (but you only need one page that you should be able to get through Google books), the last three you should be able to get yourself, and I don't seem to have subscription access to Smith (often I do for Springer but not this time). —David Eppstein (talk) 13:21, 27 June 2025 (UTC)
Very good news: I got access to all those five randomly selected references. MathKeduor7 (talk) 13:18, 27 June 2025 (UTC)
Update: I'll take today to read those sources, make the checking with what the article says using them as sources and then we will enter the final stage of this GA review (probably tomorrow!). MathKeduor7 (talk) 17:59, 28 June 2025 (UTC)

For some context, I have been following this GA review loosely since it began, and I notice a request for a co-reviewer on WT:WPM to review the section on algebraic solutions to this problem. I do want to let the reviewer know that, in the future, it would best to follow the instructions in WP:GAN/I#2O to request a second opinion. With that said, I can try my best to look into the details of this section (specifically for correctness), but I will communicate if I need to step away from reviewing this. Gramix13 (talk) 05:06, 28 June 2025 (UTC)

I am reviewing the first paragraph in the section, and I am not quite sure if the solution presented by John D. Smith actually counts as a new solution, let alone an algebraic one. He writes down a solution in a section titled "Analytical solutions of Alhazen's Problem", and he states before the start of the section that "We now obtain both of [Huygens and Sluse's] solutions by complex numbers." There is an equation 1 that appears algebraic, but its unclear that this is the true solution in the source considering it doesn't stop there in finding the solution. I would appreciate an explanation on how his solution differs from the solutions found by those two and how it is algebraic, just in case I have misread this source from my skimming. It might be helpful to add an explanatory footnote clarifying how Smith's solution differs from that of Huygens and Sluse. Gramix13 (talk) 05:47, 28 June 2025 (UTC)
I also do not see how this source verifies the claim of Galois theory proving the existence of unconstructible numbers ("Galois" does not appear in a direct search), if that could also be clarified I would appreciate it. Gramix13 (talk) 06:25, 28 June 2025 (UTC)
The source does verify Waldvogel's solution on the unit circle, with the coefficient in the fourth power being written (yet still equivelent) as 
  
    
      
        
          
            
              a
              b
            
            ¯
          
        
      
    
    {\displaystyle {\overline {ab}}}
  
 instead of 
  
    
      
        
          
            a
            ¯
          
        
        
          
            b
            ¯
          
        
      
    
    {\displaystyle {\overline {a}}{\overline {b}}}
  
 (I prefer the former since it looks more readable), although I noticed that this source does not cite Waldvogel for this algebraic solution. Perhaps this solution has originated elsewhere? Gramix13 (talk) 06:09, 28 June 2025 (UTC)
It looks like I missed a citation for this source that was in the paragraph discussion this particular solution, so that is on me. I think I prefer how this source writes the solution, as 
  
    
      
        
          z
          
            1
          
        
        ,
        
          z
          
            2
          
        
      
    
    {\displaystyle z_{1},z_{2}}
  
 is more clear that these are complex numbers of the two points we are studying, and 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is easier to see as the variable in the quartic rather than 
  
    
      
        ζ
      
    
    {\displaystyle \zeta }
  
 which I feel distracts from the equation, but feel free to disagree with me on this. Gramix13 (talk) 06:14, 28 June 2025 (UTC)
I also think it would more straight forward to say that the solution in the equation occurs when "the angles of incidence and reflection are equal, a light ray from 
  
    
      
        
          z
          
            1
          
        
      
    
    {\displaystyle z_{1}}
  
 to 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is reflected at 
  
    
      
        u
      
    
    {\displaystyle u}
  
 and goes through the point 
  
    
      
        
          z
          
            2
          
        
      
    
    {\displaystyle z_{2}}
  
" (to quote the same source), or some rewording of this, as this is how the source initially presents the problem, and not by minimizing distance. While I do see a mention of a metric later in the article, I think this explanation at the very least complements the description of the solution on the unit circle. Gramix13 (talk) 06:20, 28 June 2025 (UTC)
"A deceptively easy problem" does not appear to necessarily use Galois theory to show the existence of an unconstructable solution. What the author actually does is he constructs an irrational cubic solution, and such a number is cited to not be constructible. To me, that doesn't quite sound like Galois theory is necessary, it really just involves algebraic extensions of the rational numbers whose degree is a power of 2. I would recommend looking into explaining this solution to start with first in that paragraph since I think the justification on that solution being unconstructible would be easier for the reader to understand than citing Galois theory, which even undergraduates might not be familiar with. Gramix13 (talk) 06:38, 28 June 2025 (UTC)
On second thought, trying to explain Galois theory might not be worth the digression, so I would try to balance explaining the background and application of Galois theory to this problem with needing to avoid unnecessary tangents. I will leave this judgement up to the nominator on how best to approach this. Gramix13 (talk) 07:10, 28 June 2025 (UTC)
I have finished reviewing the freely available sources I could access in English. I am unable to verify the contents of Carréga & Haddad (2016) as the source is entirely in French (which I do not speak). I am unsure what is meant by "reflection point" and I think it could benefit with some clarification. I think it would also greatly benefit the reader to take time to explain the Galois theory used in that source, I feel that the average reader wouldn't know what even the symmetric group on four elements is. I mean I myself know what that means, but I have taken courses in abstract algebra to understand that, so for people without that abstract algebra background they might feel lost trying to understand what it means and feel it is inaccessible, especially from a problem that on its surface appears purely as one of optics/geometry and not of algebra. Gramix13 (talk) 06:49, 28 June 2025 (UTC)
One thing I feel is missing is an actual calculation of a solution from the derived algebraic solutions, after all I think that is part of the charm in having a way to algebraically find a solution to the problem. I do think this would be in service of illuminating the solution itself and not be for the sake of calculation. Otherwise, I don't think I currently see anything in this section that I feel could be rewritten, expanded, or shortened. I've leave some time for the nominator and reviewer to look over my comments I have made, as well as giving me some time to relook this section over the coming days if I find that I have missed something that would be important for this review. Gramix13 (talk) 07:01, 28 June 2025 (UTC)
Some responses:
I changed the overline to cover both a and b as suggested.
A sentence in the article states "James Gregory and many others attempted to find an algebraic solution to the problem". The Smith source provides an algebraic solution in eqn (1), labeled by it as an "analytic solution", and states (p.196) "James Gregory attempted an analytical solution of Alhazen's Problem but without the same success". So to me this seems to clearly source at least the Gregory part of that sentence. The later sentence saying that a solution was provided by Smith refers to eqn (1); it is clearly an algebraic solution, it is provided by smith without attribution, and it is actually exactly the same solution as for Waldvogel.
As for the choice of variables in the equation, I was copying the choice made by Waldvogel.
Re "I would appreciate an explanation on how his solution differs from the solutions found by those two and how it is algebraic" and re "how Smith's solution differs from that of Huygens and Sluse": because it is an equation in complex numbers rather than a geometric solution involving a conic curve, obviously. The perspective that these might really be the same thing is anachronistic in a discussion of the history of this problem.
I removed Smith from the list of sources for the use of Galois theory because, as you say, Smith does not mention unconstructability at all.
Re "I noticed that this source does not cite Waldvogel for this algebraic solution. Perhaps this solution has originated elsewhere? ": it appears more likely to me that they both developed the same solution independently, but without a clear source comparing their priority we cannot make a statement about this ourselves.
Re: "I also think it would more straight forward to say that the solution in the equation occurs when "the angles of incidence and reflection are equal ... not by minimizing distance": Unfortunately this does not work. Consider two points outside the circle at equal distances from it, forming some small angle from the center. The algebraic formulation has two solutions at the two points where the angle bisector crosses the circle. One of the solutions is valid optically (the closer one); the other one involves rays that cross to the inside of the circle, reflect on its far side, and cross back again to the outside. Both meet your equal-angle description, but only the closer one can be followed by a light ray (with an opaque rather than half-silvered mirror). I think there is also another kind of wrong algebraic solution where the two points lie on two lines that have equal angles of incidence but where one of the points is on the opposite ray of its line. So the intention of using distance here was to disambiguate the solutions. But I made it vaguer, saying instead that the valid solutions can be found among the roots.
Re the use of Galois theory by Elkin: the title of our article on the specific construction used by Elkin to reduce a quartic to a cubic is Resolvent (Galois theory). Are you suggesting that "Galois theory" is not an appropriate disambiguator for that article? Anyway, I added more detail both on Elkin's polynomial and on how he uses it to prove unconstructability.
Re "I am unsure what is meant by "reflection point" and I think it could benefit with some clarification": a point on the sphere at which the light ray reflects from a to b.
—David Eppstein (talk) 08:09, 28 June 2025 (UTC)
I will reply to only two of these point since I agree with most of what you are saying:
I found that example you gave explains well how the angle of incidence doesn't necessarily give the unique valid solution. On reflecting on this, I think having the mention of the distance is better since that's how the source specifies the "minimizing root" of the quartic equation follow Theorem 1.3 in the text.
I was unaware that resolvents are a concept of Galois theory, and I probably glossed over that when I read the source initially, I appreciate the clarification on that. I think Galois theory would be a fair descriptor for this non-constructible solution then.
Gramix13 (talk) 17:46, 28 June 2025 (UTC)

Congratulations, Professor David Eppstein! This article is GA quality level. MathKeduor7 (talk) 22:11, 28 June 2025 (UTC)

Thank you David Eppstein for improving this article! May I suggest a slight improvement in the illustration above: yellow is being used to color too many different lines. I'd suggest using green for the bisector line (which could be a halfline) and sides of the angle subtended at the circle center (which would benefit from a green point as well). Then, only the asymptotes would remain in yellow. I tried to rerun the Python code but ran into a package installation hurdle. Many thanks. fgnievinski (talk) 18:53, 24 July 2025 (UTC)

It's possible but my main concern here is image clutter: I want the circle, hyperbola, and reflection path to stand out as prominent in the image. The intention behind coloring everything else yellow was to make the auxiliary construction lines visible but more in the background, so that they do not distract from the more important parts. —David Eppstein (talk) 20:41, 24 July 2025 (UTC)
Fair enough. How about making the yellow asymptote lines dotted or dashed? It really takes a toll on the reader to pull out which line is which. Take for example the bisector line and the asymptote parallel to it, they are so close. Also, the line perpendicular to the bisector line passing through the center could go as it has no special meaning; the center point would suffice. fgnievinski (talk) 03:27, 25 July 2025 (UTC)