# Talk:Square pyramid

I think that the volume of this section is not correct. Please check Wolfram Research, Inc

http://mathworld.wolfram.com/SquarePyramid.html

for correct information.
--Kibria100 13:05, 20 June 2006 (UTC)

The formulae are different because they use different parametrizations. The one given on this page assumes a pyramid with regular faces, edge length a.
The Johnson pyramid should be "half" a regular octahedron in volume. Looks like an extra sqrt(2) ... I'll remove one!
Tom Ruen 23:41, 20 June 2006 (UTC)

I removed this:

Other square pyramids have isosceles triangle sides. For example the Great Pyramid of Giza has isosceles triangles of base 756 feet and slant height 719 feet. That pyramid has the interesting property that the slant height (along the bisector of a face) is very nearly equal to the golden ratio times the height, in which case the area of each triangular face is equal to the square of the pyramid's height.
If the base is 756 ft and the slant-height is 719 ft, then the height is 612 ft; 719:612 is nowhere near the golden ratio. —Tamfang (talk) 21:24, 6 April 2009 (UTC)

You're right, it was wrong.  You can find the correct relationship and some sources at golden ratio#Pyramids. Dicklyon (talk) 04:13, 7 April 2009 (UTC)

It may be worth mentioning in the "Other square pyramids" section, the pyramid (with apex above base centre) which has a height exactly half of the base length; this is the type of pyramid yielded by slicing a cube into six identical pyramids, and hence it is capable of entirely filling a 3D space without leaving any gaps, and without the help of any other polyhedron.
Also, by sticking six such pyramids to the faces of a cube, the result is a rhombic dodecahedron, and this could be mentioned in the "Related polyhedra and honeycombs" section. Certainly that section should be clarified a bit, regarding the tetrakis hexahedron; it should be mentioned that by "short" is meant a pyramid the height of which is less than half of the length of the base.
If citations can be found, these would be worthy additions. — 2A02:C7D:419:2500:207F:B0B3:C1A7:E60B (talk) 09:17, 3 June 2017 (UTC)

I support all these suggestions (above). --JavBol (talk) 20:00, 10 October 2021 (UTC)

@Administrator David Eppstein: Sorry for bothering you with an infobox again... But the infobox of the Square pyramid article should treat only the equilateral square pyramid, shouldn't it? In advance, thank you for your answer! --JavBol (talk) 20:35, 10 October 2021 (UTC)

Why? If this article is about all kinds of square pyramids, what is the justification for ignoring that and only considering a special case in the infobox, as if it were an article with a different focus? —David Eppstein (talk) 04:16, 11 October 2021 (UTC)
Because for the Antiprism article, you told me (on your talk page, i think) to make its infobox focus on convex uniform antiprisms, & you insisted on uniformity more than convexity. Of course, the equilateral square pyramid is not uniform; but it has regular faces. Besides: the head section's 3 figures have regular faces; & the infobox's "Type" row specifies "Johnson J1". --JavBol (talk) 14:57, 11 October 2021 (UTC)
@David Eppstein: But all right square pyramids have the same symmetry group & rotation group, so perhaps this infobox should focus on right square pyramids? Then, "Johnson J1" should be removed from this infobox's "Type" row, shouldn't it? --JavBol (talk) 15:55, 12 October 2021 (UTC)

This review is transcluded from Talk:Square pyramid/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: David Eppstein (talk · contribs) 18:20, 2 November 2023 (UTC)


### First reading
At a first glance, this article looks in pretty good shape, much better than most of our polyhedra articles, with good care to distinguish the equilateral, right, and general cases, and refreshingly free of pyramidology. I think it is missing some topics, though (WP:GACR 3a), I found one serious mistake, and I have some other more minor comments below.
Lead: "the Egyptian pyramid": there is more than one. Why singular when the linked article is plural? (GACR 1a) In the infobox, the claim that this polyhedron is self-dual (clearly true, as for all pyramids) is unsourced (GACR 2c) and is not discussed in the text of the article (GACR 1b,3a). The pyramid net is also unsourced and undiscussed. And the vertex configuration is sourced, but undiscussed and unexplained; the only way to find out what it means is to follow the link. In these cases, I think if there is a source in the body, it doesn't need the footnote to be repeated in the infobox.

I have copyedited the lead. For the infobox, I never knew and missed that images such as the net should be sourced and explained in detail. If I'm not mistaken, there is a source that mentions the net of a square pyramid, possibly in here or [1]. I kinda didn't remember which page mentions it. Can you help me a little bit to find them, because I can't access those sources? Sorry about this one. About the vertex configuration, I think I should remove the footnote (maybe? I'm skeptical about this one) and try to change those notations into arithmetical things, with an example being Triaugmented triangular prism, which is more confusing to me as the notation analogously may write 
  
    
      
        
          
            
              
                4
                (
                
                  3
                  
                    2
                  
                
                ⋅
                4
                )
              
            
            
              
                (
                
                  3
                  
                    4
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}4(3^{2}\cdot 4)\\(3^{4})\end{aligned}}}
  
 into 
  
    
      
        4
        ×
        (
        
          3
          
            2
          
        
        ×
        4
        )
        +
        1
        ×
        (
        
          3
          
            4
          
        
        )
      
    
    {\displaystyle 4\times (3^{2}\times 4)+1\times (3^{4})}
  
???? About the square pyramid is self-dual, yeah, I think I have found the source between the range page from here to here; will add this one later. {X} Dedhert.Jr (talk) 04:02, 3 November 2023 (UTC)
Properties/Right and equilateral square pyramid: "Congruent" is a link to a disambiguation page (GACR 1a). In the second paragraph, the angles are sourced to Uehara, but I don't see them there (GACR 2c). Maybe Uehara was intended only to source the J1 numbering? I think the angles are in the Johnson source. In the third paragraph, Johnson sources the specific symmetry group of the pyramid, but I don't see where it sources the part in the first sentence about this being an example of a more general class of pyramidal symmetries (GACR 2c)

This was started by the user who copyedited the whole article after I took it WP:GOCE. The dihedral angle is supposed to be sourced by Johnson, and Uehara is supposed to be the source of the numbering of Johnson solids. For the symmetrical things, this was initially amalgamated together with the paragraph on the equilateral square pyramid, but then the user broke the paragraph. Dedhert.Jr (talk) 02:22, 3 November 2023 (UTC)
Properties/Surface area and volume: "The surface area A of a square pyramid can be expressed as": this formula is only for right square pyramids.

Yup. This is actually for the right square pyramid. Dedhert.Jr (talk) 02:02, 3 November 2023 (UTC)
Applications, first paragraph: I wonder whether we should separate out the architectural applications from the history of volume calculations, with the calculation history possibly in a different section as it is not really about applications? According to Babylonian mathematics the Babylonians also tried to calculate volumes of pyramids but used an incorrect formula, and according to doi:10.1016/0315-0860(79)90076-4 the ancient Chinese (much later) did so as well (GACR 3a). And while we mention Egyptian pyramids there is no mention of Mesoamerican pyramids.

My thought about the Egyptians finding the volume of a (complete) square pyramid explained in the Application was intended to explain the background of the Egyptian pyramids. From your perspective, I think I might kinda agree with separating the history of how they calculate, and my thoughts about this should be explained in the $ surface area and volume part. Beyond the Egyptians, I never knew that there were many people trying to calculate the volume of a square pyramid as well. About the Mesoamerican pyramid and other problematic things, will try to add it later. {X} Dedhert.Jr (talk) 04:34, 3 November 2023 (UTC)
I have added the Mesoamerican pyramids, along with the sources taken from the article of the same name. Also, I added some more background about the history of calculating the volume by many mathematicians in ancient times in the section that I mentioned earlier. Dedhert.Jr (talk) 11:39, 3 November 2023 (UTC)
Applications, first image: "The shape of an Egyptian pyramid is an equilateral square pyramid": WRONG! It is a right pyramid, and the exact shape is a matter of significant debate (see pyramidology), but it is definitely not equilateral. All of the "explanations" give it a slant height of roughly 
  
    
      
        0.8
        ℓ
      
    
    {\displaystyle 0.8\ell }
  
, but the mathematical ones vary as to whether this comes from 
  
    
      
        4
        
          /
        
        5
      
    
    {\displaystyle 4/5}
  
, 
  
    
      
        (
        1
        +
        
          
            5
          
        
        )
        
          /
        
        4
      
    
    {\displaystyle (1+{\sqrt {5}})/4}
  
, 
  
    
      
        π
        
          /
        
        4
      
    
    {\displaystyle \pi /4}
  
, etc. (The removal of the outer surface of the pyramid makes it difficult to tell this experimentally, but a comparison with what the Egyptians wrote about their mathematics and their other architecture makes the integer ratio explanations more likely than the others.) An equilateral pyramid would have a slant height around 0.866, quite a bit higher and not consistent with the measurements (GACR 6b). For a non-cranky and reliable discussion of this see Herz-Fischler, Roger (2000). The Shape of the Great Pyramid, an entire book reviewing many different theories for the shape of the pyramids.

This is due to the copyedit by the user. Rather than mentioning the shape, I rewrote them, stating that it is a square pyramidal building. Dedhert.Jr (talk) 04:34, 3 November 2023 (UTC)
Applications, stereochemistry: what is the relation between this molecular geometry and the shape described in this article, beyond their similar names? (GACR 3b)

This was intended to explain that the structure of a square pyramid can be found in chemistry. Is there something missing here? Dedhert.Jr (talk) 05:31, 7 November 2023 (UTC)
Applications, augmentation: do you really want to consider the regular octahedron to be an augmentation? If so it would have to be an augmentation of a square dihedron, no? Re "Some of the other Johnson solids ... these are": please fix the comma splice (GACR 1a).

A somewhat confusing question. Are you trying to say that it is false to write that an octahedron may be constructed by augmentation? Also for comma splices problem, I may tighten a little bit. Dedhert.Jr (talk) 02:26, 3 November 2023 (UTC)
If an octahedron is formed by attaching pyramids to every side of a smaller shape, then that smaller shape is the degenerate one that you get by gluing two squares back-to-back. —David Eppstein (talk) 05:41, 3 November 2023 (UTC)
I see, then. This was actually intended to explain examples of polyhedra that may be constructed by attaching a square pyramid base, other than Johnson solids. Explaining the construction of octahedra by gluing two square pyramid bases, as you mentioned earlier, made me think twice or more that I would probably remove this part, and maybe one example is already sufficient to explain the augmentation. Also, maybe we can discuss further about it in either your or my talk page. Dedhert.Jr (talk) 11:46, 3 November 2023 (UTC)
See also: Can Square pyramidal number be discussed in the main article text rather than relegated to a see-also link (GACR 3a)? Maybe we could add Pyramidology as a see-also link instead (or mention it briefly in the context of the Egyptian pyramids).

Sadly. I cannot find a way to describe more about square pyramidal number, because it is just literally a stacked sphere forming a pyramid with a square base. Let me know if you have an idea about this one. Also, for Pyramidology, one of the sections relates the building of Great Pyramid of Giza with the height of golden ratio, I don't think I can add it to the text, because I'm aware that there was a talk page years ago, where two users discussed removing the similar claims. Dedhert.Jr (talk) 02:19, 4 November 2023 (UTC)
Re the golden ratio theory of pyramidology: see the paragraph on the same subject in Kepler triangle#History. (The golden ratio is the purported ratio of slant height to half-base-length, in that theory, while the Kepler triangle gives the half-cross-section shape for the same theory.) —David Eppstein (talk) 06:56, 4 November 2023 (UTC)
Okay, That's another good information about it. Will try to add it. {X} Dedhert.Jr (talk) 16:52, 4 November 2023 (UTC)
Strikethrough the mark, meaning I have added the relation between the Kepler triangle and the mensuration of one of the Egyptian building's design, the Giza pyramid. Dedhert.Jr (talk) 05:21, 7 November 2023 (UTC)
References: Spot-checking found most content properly referenced (GACR 2b) and most references good for what they are referencing (GACR 2c) with a few exceptions noted above. Neither those checks nor Earwig found any significant copying (GACR 2d). All references appear to be reliable publications, consistently formatted in Citation Style 1 (GACR 2a).
External links: Within WP:EL rules, but the relevance of the wheel graph link is unclear. I think the topic is relevant, but should be explained in the article text (GACR 3a), and that with a proper explanation and wikilink, the external link would become unnecessary.

I'm not familiar with the external links here. I thought it would be a good idea to explain generally that all 
  
    
      
        n
      
    
    {\displaystyle n}
  
-sided pyramids can be represented by wheel graph 
  
    
      
        
          W
          
            n
          
        
      
    
    {\displaystyle W_{n}}
  
 for 
  
    
      
        n
        ≥
        3
      
    
    {\displaystyle n\geq 3}
  
 in the article Pyramid (geometry). But since GACR3a states the broad coverage, you are absolutely correct and I made a mistake here. Will try to add it if I have found the source first. {X} Dedhert.Jr (talk) 05:31, 7 November 2023 (UTC)
I added the wheel graph in the section of property. Sadly, it seems that I can't find any sources that mention the skeleton of a square pyramid specifically; so just in case, I add more about the generalization about the skeleton of a right pyramid. Furthermore, some sources say it is the representation of wheel graph 
  
    
      
        
          W
          
            5
          
        
      
    
    {\displaystyle W_{5}}
  
 and some of them say it is 
  
    
      
        
          W
          
            4
          
        
      
    
    {\displaystyle W_{4}}
  
 But for the source I use right now, it mentions the generalization of a regular pyramid as 
  
    
      
        
          W
          
            n
          
        
      
    
    {\displaystyle W_{n}}
  
, judging from the number of sides. Dedhert.Jr (talk) 10:59, 7 November 2023 (UTC)
Neutrality (GACR 4) and stability (GACR 5): no issues.
Media licensing and captioning (GACR 6): only one issue, noted above.
—David Eppstein (talk) 01:46, 3 November 2023 (UTC)

@David Eppstein Thank you for reviewing. Will try my best to accomplish your suggestions one by one. For some reason, I will add {X}, marking that there is an unfinished suggestion, because I sometimes forget it. Strikethrough the mark means I have done it. Dedhert.Jr (talk) 01:59, 3 November 2023 (UTC)
@David Eppstein I think I have accomplish your suggestions here. Please let me know if I missed some of them, and also it seems that I think I have a few questions while I replied, although you have replied back. Beyond that, I also added a visual for the Mesoamerican pyramid to give the difference between two similar buildings, and expand the lead a little. Thank you. Dedhert.Jr (talk) 11:16, 7 November 2023 (UTC)


### Notes
### Quick second reading
@Dedhert.Jr: Most of the changes look like improvements but there are some new grammar issues.
Lead: "with examples being Egyptian pyramids, along with the background of mensuration": the background of mensuration is not an example of architecture.

Removed. Dedhert.Jr (talk) 09:58, 8 November 2023 (UTC)
Surface area and volume: The choice to put the history of the formulas here is ok. But "The Babylonian mathematicians also found its volume, but stated that the incorrect formula is the half product of the height and the sum of quadrilateral bases.[17]": confusing grammar. There was some other incorrect formula and they stated what it was?

Okay. I have copyedited and fixed the grammar here, and I added another source; both citations mention that the Babylonians used the product of the height and half the sum of bases as the volume of a square pyramid, and this formula is incorrect. If you said that "there was some other incorrect formula", I will add it as well, if I have found the sources. Dedhert.Jr (talk) 11:47, 10 November 2023 (UTC)
Note: I have removed the additional source, which says it is non-RS. See here. Dedhert.Jr (talk) 05:37, 11 November 2023 (UTC)
Applications: "However, modern scholars opposed this because it would not be inconsistent with Egyptian mathematics during construction, along with the fact that they did not know the utilization of the golden ratio in their theory of architecture and proportion": more confused grammar. "Not be inconsistent" is a double negative, for one thing, and the grammar of the two supposed reasons for scholars to disagree with this (not really oppose it) is not parallel.
—David Eppstein (talk) 07:39, 8 November 2023 (UTC)

Hope you guys don't mind me butting in here! I've had this article on my watchlist since the PR. Dedhert.Jr, I know you've had some trouble with grammar and stuff on this article, so let me know if I can be of help or if you'd like me to do a pass for copyedits at any point. —TechnoSquirrel69 (sigh) 08:11, 8 November 2023 (UTC)
@TechnoSquirrel69 Thank you for the help, I always need your help to improve the grammar. For @David Eppstein, I'm sorry that I'm currently busy in real life, and I may not respond and improve the article. Nonetheless, I will always keep checking and accomplish the suggestions one by one if I have time for editing. Dedhert.Jr (talk) 09:42, 8 November 2023 (UTC)
Ok, the article appears to have mostly stabilized. I made some minor copyedits to some text that still felt awkward to me, but other than that I think everything from this review was already addressed. I don't see anything important missing (for instance, there are too many modern pyramids for it to be a good idea to add many others). Passing for GA. —David Eppstein (talk) 07:17, 20 November 2023 (UTC)

@Dedhert.Jr: In response to your edit summary, I'd support a full switch to {{sfn}} citations. I was going to go through and do some cleanup with the references anyways, so I might just do both at the same time if you're alright with that. —TechnoSquirrel69 (sigh) 19:47, 20 November 2023 (UTC)

@TechnoSquirrel69 Short citations may be also used for notes, so I agree about to convert them. I will do the cleanup right now. Dedhert.Jr (talk) 23:25, 20 November 2023 (UTC)
Done! I have made some changes here, converting to sfnp and pull down the references. Let me know what do you think. Dedhert.Jr (talk) 00:36, 21 November 2023 (UTC)

@David Eppstein. I'm considering expanding the properties section that includes the height, surface area, and volume of an equilateral square pyramid. Its height can be expressed as 
  
    
      
        
          
            1
            
              /
            
            2
          
        
        ≈
        0.707
      
    
    {\textstyle {\sqrt {1/2}}\approx 0.707}
  
, but the source I have found source I have found explains the case of the unit edge. Does it count as WP:CALC if I multiply that height with random edge length 
  
    
      
        a
      
    
    {\displaystyle a}
  
? An alternative source is from Spanish language, but I'm skeptical whether reliable or not. Dedhert.Jr (talk) 11:51, 31 January 2024 (UTC)

I'd rather go with a good book source and count the scalar factor as the sort of simple calculation allowed by WP:CALC. —David Eppstein (talk) 17:05, 31 January 2024 (UTC)