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March 1

Why does a dart have only one mouth?

Literature I've seen considers a concave quadrilateral to be an anthropomorphic polygon which, by definition, has two ears and one mouth. Why is it not considered to have two mouths (the vertex with the red spot plus the one without a spot)? Thanks, cmɢʟee⎆τaʟκ 14:02, 1 March 2023 (UTC)[reply]

Because your chin is not your mouth; which is to say that the lower vertex is not concave. --Jayron 32 14:05, 1 March 2023 (UTC)[reply]

And to clarify as well, the common definition of concavity is that a line drawn between two arbitrary points inside the shape passes outside the shape. That can happen around the red dot; so it is a concave (mouth). The lower (undotted) vertex is convex; there is no two points located inside the shape which will pass outside of that vertex. --Jayron 32 14:10, 1 March 2023 (UTC)[reply]

It is a matter of applying the definition of mouth, but the treatment in the article is unclear. The problem is that the given definitions rely on a numbering of the vertices, but no constraint on the numbering is given. Let

n

be the number of vertices. Then we should consider the polygon as a closed polygonal chain with vertices

v_{1},...,v_{n},

in which the vertices are numbered in order along the chain. Furthermore, the notation

v_{(i)}

denotes the vertex

v_{j},1\leq j\leq n,

, such that

j\equiv i~~{\text{mod}}~n.

So

v_{(0)}=v_{n}

and

v_{(n{+}1)}=v_{1}.

Let us number the vertices starting from the bottom one, say clockwise – although because of the symmetry it does not matter. So the blue vertices are

v_{2}

and

v_{4},

and the red one is

v_{3}.

Is the uncoloured bottom vertex

v_{1}=v_{(1)}

a mouth? Applying the definition, this requires the line segment connecting

v_{(0)}=v_{4}

and

v_{(2)}=v_{2}

, the two blue vertices, to lie entirely outside the polygon. It does indeed; it is in fact the same line segment as needed to apply the definition to

v_{3}.

So, accoprding to the definition in the article, the bottom vertex has the same status as the red one. A moment of thought will tell us that in general, in any simple quadrilateral, the vertices

v_{(i)}

and

v_{(i+2)}

enjoy the same status. Corollary: no quadrilateral is anthropomorphic. --Lambiam 16:09, 1 March 2023 (UTC)[reply]

This seems to be something of one of the corollaries of the Strong law of small numbers, which is to say that many definitions break on the smallest numbers. Such a problem only exists for the quadrilateral case. In all other singularly concave polygons, you will never have the problem. It's only because the quadrilateral has a single other vertex that it appears the definition applies, but this is only because "4" is too small a number of vertexes for our definition to work well. Any other larger number of vertexes never have such a problem. --Jayron 32 16:28, 1 March 2023 (UTC)[reply]

This is a clumsy sketch of a pentagon that has two ears and one mouth (please check). --Lambiam 16:58, 1 March 2023 (UTC)[reply]

Yes, but it does not have the problem where a vertex other than the one obvious mouth meets the definition of the mouth. The problem you note for the quadrilateral, where the definition of "mouth" appears to apply to the convex vertex located opposite the obvious "mouth" is only a thing for quadrilaterals, because quadrilaterals are just too small to not screw up the definition. That's what I'm saying. The idea that a convex vertex can be a mouth seems silly, and yet going by the strict definition you cited, it appears to be so for the specific quadrilateral noted above. It is not true for any larger n-gon. --Jayron 32 17:28, 1 March 2023 (UTC)[reply]

What I'm basically saying is that the definition of mouth is inadequate here, because it implies that a convex vertex can be a mouth; which it clearly cannot; the ONLY case where it can be is in a quadrilateral. --Jayron 32 17:32, 1 March 2023 (UTC)[reply]

One can add teeth and the chin is still a mouth (captain, the thing could eat a planet) according to the definition. Wolfram states the same definition for the term "mouth" as we do [1] and cites Godfried Toussaint's "Anthropomorphic Polygons." Amer. Math. Monthly 122, 31-35, 1991. Given that, the quadrilateral should be replaced I would think. Modocc (talk) 22:02, 1 March 2023 (UTC)[reply]

Interestingly and curiously, a subpage of Toussaint's homepage at McGill, created a decade before his death and apparently never challenged, presents the dart as being anthropomorphic.^[2] It's not only quadrilaterals that can have mouths (using Toussaint's definition) that are outies; Fig. 2(b) in Toussaint's paper that is reference [1] in our article also has many innie mouths and one outie mouth. --Lambiam 23:32, 1 March 2023 (UTC)[reply]

I have signed up to JSTOR and verified that his published definitions are precisely defined as we have them. His anthropomorphic polygon shown in Fig. 2(c) is correct (twisted as it is). The quadrilateral though needs to go. Modocc (talk) 00:33, 2 March 2023 (UTC)[reply]

Lambiam, your sketch is, unambiguously, an anthropomorphic polygon. Modocc (talk) 22:53, 1 March 2023 (UTC)[reply]

cmɢʟee, I removed your dart diagram from the article. [3] Modocc (talk) 01:33, 2 March 2023 (UTC)[reply]

Thanks, everyone. Should the article add the condition that the mouth must be locally convex? Nevertheless, I'll update the diagram to Lambiam's to avoid controversy. Cheers, cmɢʟee⎆τaʟκ 03:05, 2 March 2023 (UTC)[reply]

P.S. [4] has a polygon its author(s) declared to be anthropomorphic. cmɢʟee⎆τaʟκ 03:39, 2 March 2023 (UTC)[reply]

This is the same twisted polygon as in Toussaint's Fig. 2(c) referred to above by Modocc; it serves there as a counterexample too, but to a different statement, and is ascribed to Gary H. Meisters, doi:10.1080/00029890.1980.11995014.

We should stick to published sources and not fix published definitions ourselves. --Lambiam 05:30, 2 March 2023 (UTC)[reply]

A little additional question I had regarding the language within the article itself. The article states that in an anthropomorphic polygon, "for exactly three polygon vertices, the line segment connecting the two neighbors of the vertex does not cross the polygon." Should that last part be changed to "cross the edges of the polygon" to avoid ambiguity? I personally read "cross the polygon" as meaning to cross the body of the polygon at first, which caused a bit of confusion, as that property only applies to the mouth. GalacticShoe (talk) 12:28, 2 March 2023 (UTC)[reply]

The formulation of the definition in the article has room for improvement. In the second source referenced in our article Anthropomorphic polygon, an article by Shermer and Toussaint, the definition of mouth reads:

Definition: A principal vertex xi of a simple polygon P is called a mouth if the diagonal [xi-1,xi+1] is an external diagonal, i.e., the interior of [xi-1,xi+1] lies in the exterior of P.^[5]

(Principality follows from the condition following "if", so this prerequisite is superfluous.) Earness can be defined analogously by replacing "exterior of P" by "interior of P". For mouths, one can equivalently require that the diagonal only intersects the polygon in the two vertices it connects, but this approach has no obvious tweak that works for ears. --Lambiam 15:10, 2 March 2023 (UTC)[reply]

That's likely what confused me. Perhaps why the vertex with the red spot in the original image is considered a mouth but not the one without a spot could be explained in more detail in the article? Thanks, cmɢʟee⎆τaʟκ 06:17, 3 March 2023 (UTC)[reply]

It is not you who got confused, but the computational geometers at McGill University who did not fully grasp the implications of their own definitions. The image there labelled "An anthropomorphic polygon with only 2 ears and 1 mouth" has two mouths and does not represent an anthropomorphic polygon. The definition of "mouth" could and probably should be made more restrictive by excluding convex vertices, but fixing infelicitous mathematical definitions is original work we as an encyclopedia should not engage in. --Lambiam 08:02, 3 March 2023 (UTC)[reply]

Right. When I studied at UNC to complete my computer science degree in 1991 from NCSU I read Toussaint's paper and was amused at the time by the fact that his "mouth" could eat a planet. Now, some thirty-years later I have had the opportunity here to point this out. It is a very small world... Modocc (talk) 14:14, 4 March 2023 (UTC)[reply]