Math CentralQuandaries & Queries


Question from Deb, a teacher:

I have been reading the responses you have to whether or not a cone and cylinder have edges, and whether a cone has a vertex. I have also found differing answers in different books and websites. Your idea that it depends why we want to call it a vertex or edge is interesting but does not help the students next year when they have a different teacher who tells them that whatever decision I make is wrong! It also doesn't help them when they do the provincial testing. Does anyone know if EQAO considers them vertices or edges? I want to know because textbooks and provincial tests often ask the students to count the number of faces, edges, and vertices or to describe a solid by using these, or to identify a solid that they have described. We need to have consistency in what we are teaching!


CONTEXT is important in any discussion. The meanings of the words "edge" and "vertex" depend on the context. Of course, the same thing can be said of words in any language. Probably most words in English have multiple meanings. A man may use "sugar" as a term of endearment for his wife, but when he puts sugar in his coffee the meaning is clear from the context. That is not always the case in English. The word "Indian" causes all kinds of trouble -- does it refer to an Indian from Asia or from America? (The suggested substitute in the US is even worse -- Native American should refer to a person of any race who is born in America.) Let us look at the relevant contexts for the words you are interested in.

A POLYGON is a plane figure bounded by line segments. Its boundary consists of a finite number of points, called VERTICES, and the same number of line segments, called EDGES. Two neighboring vertices are joined by an edge, while neighboring edges meet in a vertex. The boundary of a hexagon, for example, consists of 6 vertices and 6 edges. For beginners it is wise to limit the concept to "convex" polygons -- those for which the angle at each vertex is less than 180 degrees. Sometimes, as when discussing the area, one considers the the polygon to consist of the surface together with its boundary; other times the word polygon just refers to the boundary. It depends on the context.

A POLYHEDRON is a solid bounded by polygons. The surface of each polygon is called a FACE; adjacent faces meet in an EDGE; three or more faces and an equal number of edges meet at a VERTEX. Again, it is convenient to restrict our discussion to convex polyhedra -- where convex here means that you can set any face of the polyhedron on a table (so that only that face touches the table while the rest of it is above the table). The word polyhedron can refer to either the solid figure or its surface, depending on the context. The word vertex has several other meanings in mathematics: the points where a conic meets one of its axes is called a VERTEX. The three-dimensional version of a conic is called a quadric, and the points where a quadric surface meets an axis is called a vertex.

Let's next look at a cone. The Greeks thought of a cone as consisting of a circle sitting in three dimensions, a point, called the VERTEX not in the plane of the circle, and the surface formed by the infinitely long straight lines that join the vertex to each point of the circle. The surface of such a cone is in two parts, called NAPPES, that are joined at the vertex. Again here, the context is important. Often the cone is restricted to the finite surface formed by the line segments joining the vertex to the points of the circle; sometimes the word refers to the solid that is bounded by that surface together with its base circle. Sometimes the concept of a cone is generalized by replacing the circle by any closed curve (think of a string tied in a loop) or by a polygon. Usually when the base of a cone is a polygon, it is called a PYRAMID. Here is a source of confusion: the word vertex is used in two different ways in this same figure. A pyramid is a polyhedron (bounded by its polygon base and its triangular sides), so that all the points where edges come together are called vertices. But a pyramid has one point that is a vertex in its other meaning as the special point in the definition of a cone. The context must be made clear whether in the current discussion you intend the pyramid to be a polyhedron or a cone (as well as whether you intend it to be a surface or a solid). There is no confusion among the different uses as long as the context is made clear.

The word EDGE likewise has several other meanings. In English it refers to the boundary between two surfaces (like the edge of a cliff or the edge of a lake). Thus one can refer to the circle around the base of a solid cone as its edge -- but that is not a mathematical use of the word. It is best to restrict the word edge to refer to a line segment that joins two vertices or bounds a polygon. The word edge is also used in graph theory, where a graph consists of points (called VERTICES), and lines (called EDGES) that join designated pairs of vertices. Here the edges are not necessarily straight lines and they are allowed to cross at places that are not necessarily vertices. Such graphs are abstract objects and require such care with the terminology that the concept is best to left to advanced math classes. I mention these graphs because they provide the best setting for Euler's theorem which is tacitly mentioned in your query. But Euler's theorem itself requires such care in its details that it is best to restrict its use to polyhedra. The theorem says that if the number of faces of a polyhedron plus the number of its vertices always equals two more than the number of its edges. Thus a tetrahedron, which is just a pyramid with a triangular base, consists of 4 faces and 4 vertices, and therefore 4 + 4 - 2 = 6 edges. Please do not try to apply Euler's theorem to a cone since the context is all wrong -- the vertex and edge of a cone have nothing to do with Euler's theorem.


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