Editing Talk:Johnson solid (section)

==Third solid in the first diagram==

IIn the first diagram, showing three solids, the caption says that only the first is a Johnson solid, but the third solid (the orange one) seems to be just a cube, so shouldn't it be a Johnson solid too?  A convex polyhedron is just a subset of <math>\mathbb{R}^3</math> satisfying certain properties — the line segments drawn on it are not part of its description, so they should not change it from being anything but a cube. [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 03:27, 13 March 2026 (UTC)

:Convex meant < 180 degrees for dihedral angle. Those lines represent the coplanar faces, so the cube's checkerboard faces is not a Johnson solid. And a convex polyhedron is a finitely many bounded planes. [[User:Dedhert.Jr|Dedhert.Jr]] ([[User talk:Dedhert.Jr|talk]]) 04:57, 13 March 2026 (UTC)

::I understand what you are saying, but it seems that you are using a definition of convex polyhedron different from the  [[Polyhedron#Convex_polyhedra|the definition on Wikipedia]].  According to Wikipedia, a convex polyhedron is just a bounded intersection of finitely many half-spaces, or the convex hull of finitely many points, restricted in either case to intersections or hulls that have nonzero volume.  You instead seem to have in mind a definition in which a convex polyhedron is not just such a subset of space, but such a subset ''equipped with'' a collection of lower-dimensional subsets called faces.  So for you, the 2 x 2 x 2 cube with six 2 x 2 faces is different from the same cube with twenty-four 1 x 1 faces, four on each side, even though as subsets of space, they are the same, hence the same in Wikipedia's definition.  [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 06:20, 13 March 2026 (UTC)

:::By the way, the definition of convex polyhedron given in the present article does not agree with the definition in the reference it cites, because it is missing the condition that the polyhedron not be contained in a plane. [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 06:47, 13 March 2026 (UTC)
:If you're really saying that you believe that a cube with subdivided coplanar faces is a Johnson solid, then I suppose you should submit that to some math journal! hehehe. If you're saying we should include all this stuff about 'lower-dimensional subsets' etc. in the definition, then I have to disagree. Here's a compromise, let's add the word '''strictly''': " also known as a Johnson–Zalgaller solid, is a '''strictly''' convex polyhedron whose..."
:That keeps the definition layman-accessible, while being more precise. [[User:Introscopia|Introscopia]] ([[User talk:Introscopia|talk]]) 14:28, 13 March 2026 (UTC)

::Redefining convex polyhedron to include lower-dimensional subsets to allow subdivisions of faces as part of the definition is the ''opposite'' of what I'd like!  I am just pointing out that according to Wikipedia's definition, a convex polyhedron does not come with subdivisions of its faces.  If we are going to use a different definition in this article, we should say so. 

::It may also be better to avoid the jargon "strictly convex", especially since it contradicts [[Convex set#strictly convex|this definition of strictly convex]].

::Perhaps we could make it easier for a layman to understand by saying something like "A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane.  Its boundary is a finite union of polygons, no two in the same plane; those polygons are called the faces.  A Johnson solid is a convex polyhedron for which the faces are ''regular'' polygons." [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 16:17, 13 March 2026 (UTC)
:::I must disagree that that our use of "strictly convex" contradicts that definition. The extra vertices that were created in that cube in subdividing its faces are not [[Extreme points]]. [[User:Introscopia|Introscopia]] ([[User talk:Introscopia|talk]]) 19:22, 13 March 2026 (UTC)
::::[[Convex set#strictly convex|The definition]] says "A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C."  (I guess they meant "for all x and y in C, every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C." - I'll go ahead and fix the quantifiers at that other article.)  All I meant is that according to this definition, the solid cube is not strictly convex, because if x and y are different points on the same face, it is not true that all the points in the segment joining x and y other than x and y are in the ''interior'' of C.  Only figures with curved boundaries (such as ellipses) have a chance of satisfying this definition of strictly convex.
:::: On the other hand, it does seem as if there is a ''different'' definition of strictly convex that is commonly used when convex polyhedra are thought of as a collection of polygons some of which a priori may be coplanar.  I guess that is the one you are using, which is OK. [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 23:27, 13 March 2026 (UTC)
:::: Another way of stating the issue, if you want to use the characterization at [[Convex set#strictly convex]] in terms of extreme points, which says "A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point":  A convex polyhedron is a closed convex subset, but it never satisfies this extra condition to be ''strictly'' convex, because any point in the middle of a face is a boundary point that is not an extreme point.  Tetrahedra, icosahedra, etc. are never strictly convex according to this definition. [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 23:44, 13 March 2026 (UTC)
:::::I spelled out what "strictly convex" means, and moved the technical discussion of what "strictly convex" means for polyhedra into a footnote. I did also remove the orange cube, to avoid confusion. [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 22:40, 14 March 2026 (UTC)
::::::I must continue to disagree. The Parenthetical you added "(No two coplanar)" in the opening sentence is, sorry to be harsh, really really awful for flow and readability. "Strictly Convex" means no coplanar faces. You already admitted this yourself. There is literally no other conceivable thing the adverb 'strictly' could be doing next to the word 'convex' other than specifying that coplanar faces are excluded.
::::::As for the orange cube, I never liked it anyways. I do believe however the "gyroelongated triangular bipyramid" (aka the trigonal trapezohedron) would be a nice and relevant exemple to replace it with, because it is one of a few very salient gaps in the construction tree of the pyramid family of Johnsons. [[User:Introscopia|Introscopia]] ([[User talk:Introscopia|talk]]) 02:00, 15 March 2026 (UTC)
:::::::Or we can say it "augmented octahedron" [https://lsusmath.rickmabry.org/rmabry/dodec/delta/photos/7V5.html#:~:text=This%20page%20is%20linked%20from%20here;%20go,%22augmented%20octahedron%22%20(I'll%20also%20call%20it%20%227V5%22)%2C.] [[User:Dedhert.Jr|Dedhert.Jr]] ([[User talk:Dedhert.Jr|talk]]) 05:47, 15 March 2026 (UTC)
::::::::@Introscopia: I see what you mean about the parenthetical breaking up the flow; that's a fair criticism.  I removed it (it wasn't really necessary), and added explanation to the footnote instead.  @Dedhert.Jr: I added a citation to an article by [[A. G. Khovanskii]] for the fact that convex polyhedra are never strictly convex in the sense used in modern convex geometry.  
::::::::As for adding a third polyhedron image, I think the two of you know the examples better than I do, so I leave it to you.  It looks as if the augmented octahedron fails the definition because some of its faces are rhombi instead of regular polygons.  By the way, it looks as if some people use "augmented octahedron" to refer to a [https://polytope.miraheze.org/wiki/Augmented_octahedron different polyhedron].  Maybe there are just many kinds of augmented octahedron?  I don't know the terminology well.  Thank you both, [[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 15:27, 15 March 2026 (UTC)