Talk:Square pyramidal number/GA1

GA Review


Reviewer: Olivaw-Daneel (talk · contribs) 01:45, 18 December 2021 (UTC)

Interested in reviewing this. I'll have comments up in a couple days. Olivaw-Daneel (talk) 01:45, 18 December 2021 (UTC)

The article is well-illustrated, but I think it covers the material a bit too tersely – some further elucidation would greatly help. Specific suggestions below.

Lead

 * Currently the first sentence requires the reader to know what a figurate number is – suggest postponing the mention of figurate numbers to a separate sentence that hints at their definition. (Perhaps something like "It can be generalized to a broader category of numbers that are known as figurate numbers...")
 * Added a gloss for figurate numbers. —David Eppstein (talk) 07:20, 22 December 2021 (UTC)

Formula

 * There seem to be two competing definitions here – algebraic and geometric – and I'm not sure which of them is more primary. Since the lead starts out with the geometric version, suggest doing the same here. (It may also be logical to rename this section to Definition).
 * It really was intended as a formula solving the question "how many points are in a pyramid" rather than a definition "these numbers are the values of a cubic polynomial". I rewrote to suggest that ordering more strongly. —David Eppstein (talk) 07:26, 22 December 2021 (UTC)
 * I like the new version. A few comments: the sphere interpretation is not mentioned in ref #1 (suggest adding ref #10 Beiler); numbers of points should be numbers of spheres; height probably refers to the number of layers but could be clearer.
 * Ok, height is now number of layers, and the sphere interpretation cited to Beiler is used more consistently until the Ehrhart paragraph. —David Eppstein (talk) 19:20, 24 December 2021 (UTC)
 * The lead hints at a historical perspective – The study of these numbers goes back to Archimedes and Fibonacci – that one would expect to be covered here. (E.g. When and for what purpose this was studied.)
 * Turns out Archimedes and Fibonacci had different purposes than counting points in pyramids. Added. —David Eppstein (talk) 07:26, 22 December 2021 (UTC)
 * The mention of figurate numbers appears all of a sudden; as in the lead, it would be helpful to gloss them before use.
 * Added another gloss, also using Beiler to source the jump from spheres to points. —David Eppstein (talk) 19:20, 24 December 2021 (UTC)
 * The more general statements about Erhart polynomials are uncited (ref #5 Beck only talks about polytopes with integer vertices)
 * Rewritten to only talk about Ehrhart for integer polyhedra, per the source. —David Eppstein (talk) 19:20, 24 December 2021 (UTC)

Geometric enumeration

 * It would be helpful to add a prefatory remark similar to this one from the lead: As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems
 * ✅ —David Eppstein (talk) 19:43, 24 December 2021 (UTC)
 * Square counting: the refs don't mention square pyramidal numbers (suggest adding ref #1 Sloane)
 * ✅ —David Eppstein (talk) 19:43, 24 December 2021 (UTC)
 * Rectangle counting: this statement seems related to the square counting problem, so it should probably be moved up.
 * ✅ —David Eppstein (talk) 19:43, 24 December 2021 (UTC)

Relations to other figurate numbers

 * In the first sentence, it should be made more clear that the pyramid is made of cannonballs.
 * Reordered to put the cannonballs earlier. —David Eppstein (talk) 19:45, 24 December 2021 (UTC)
 * This fact was proven by G. N. Watson in 1918 – I'm not sure if Watson deserves sole credit. Ref #9 Anglin says that he filled gaps in earlier drafts by Moret-Blanc (1876) and Lucas (1877).
 * Added a mention to earlier incomplete proofs by Moret-Blanc and Lucas. —David Eppstein (talk) 21:02, 24 December 2021 (UTC)
 * The sum of binomials equation is not cited (suggest adding ref #11 Caglayan)
 * Since Caglayan phrases this as a problem rather than a statement, I used two other references, Conway & Guy's Book of Numbers and a paper by Grassl. Grassl explicitly uses the binomial coefficient description of this formula rather than calling them tetrahedral numbers as the other sources do. —David Eppstein (talk) 22:43, 24 December 2021 (UTC)
 * In ref #10 Beiler: pp. 194 → pp. 194–195
 * Ok. —David Eppstein (talk) 21:11, 24 December 2021 (UTC)
 * Last paragraph: a single tetrahedron of slightly more than twice the edge length – why is the length slightly more than twice? (The source says: "Four times the nth square pyramidal number is the (2n)th tetrahedral number".)
 * The edge length of $$P_n$$ is $$n-1$$. The edge length of $$4P_n=T_{2n}$$ is $$2n-1 = 2(n-1)+1$$, slightly more than twice. Edge length is off by one from the index in the sequence of these numbers. —David Eppstein (talk) 22:10, 24 December 2021 (UTC)
 * Ah I see, so that's based on an "edge" as a line connecting the centers of the spheres – do sources define it this way? It seems at odds with the last figure which says $P_{24}$ has side 24. Olivaw-Daneel (talk) 00:01, 25 December 2021 (UTC)

Other properties
Placing it ; please ping when you're done. Thanks. Olivaw-Daneel (talk) 09:47, 24 December 2021 (UTC)
 * Suggest expanding a little to avoid a single-sentence section. Some possibilities: the connection to Archimedes' approximation for $&pi;$ (22/7) mentioned in the ref; some background on the Leibniz formula.
 * I found a connection to approximation theory and added it. While searching for material for this expansion, I also found additional material on the history of these numbers (now split off into a separate history section) and on using them to count certain integer matrices (added to the end of the enumeration section). —David Eppstein (talk) 23:24, 24 December 2021 (UTC)

All comments addressed; I think it's now ready for a second look. —David Eppstein (talk) 23:24, 24 December 2021 (UTC)
 * Just one last comment above about edge length. The article looks great; I really like the history section. Olivaw-Daneel (talk) 00:01, 25 December 2021 (UTC)
 * Ok, changed to "points along each edge" rather than edge length to avoid that ambiguity. —David Eppstein (talk) 00:11, 25 December 2021 (UTC)
 * Great, congrats on the GA. Olivaw-Daneel (talk) 00:23, 25 December 2021 (UTC)