The properties of "gons" in general should be reviewed, as students may forget we're conventionally forcing them to be flat, all vertexes in one plane. A necklace of four cylindrical beads, or 4-edged construction with ball bearing hinges, is going to be floppy in space, all wobbly, and is not considered a "quadrilateral" except in snap shot moments when the four edges are "in a plane". A polygon for fleeting instants.
What students should be reminded of is the rules of the game (like chess) narrow the permitted / legal moves to an exponentially tiny fragment of what's possible, but this strictness is what makes for the rigorous proofs of Euclidean geometry. Strict definitions exclude what's irrelevant. Besides, we have topology for the more necklace-like thingamabobs. It's not like math itself is confined by Euclidean definitions.
Note that triangles have no choice but to be planar whereas the quadrilateral is the first n-gon able to "hinge" in a way that introduces no new vertexes (triangles may be creased, but this adds new nodes). A rhombus may be creased along a diagonal to make two "wings", the tips of which may be connected by another edge of equal length. A tetrahedron is born. It is not floppy either, being made of triangles.
The wobbly hexahedron frozen in cube moments (unstable) is the hedron of choice for European volume units, such as grams. A tetrahedron, calibrated to standards (of weight, of size), might sit in some school's museum as an alternative choice. It's another possible mathematics and is accessible to all ages. I'm something of a tour guide in this area. I think you understand your own ethnicity better when you have an opportunity to compare it with something more alien. Good mental exercise.
On Wed, May 8, 2013 at 8:40 AM, Joe Niederberger <email@example.com> wrote: > What bugs me about this post is the the taking of the *tree* structure to be fundamentally important, rather than investigating the properties of 4-gons and seeing what structure they naturally lead to, all labels aside. > > The notion of a (graph-theory) tree, though, being both mathematical and ubiquitous even through non-math circles as a organizing principle, lends a pseudo-mathematical "rigor" and officious weight to the whole misbegotten proceeding. > > Get the (correct, non-tree) structure of properties across, put the labels on afterwards, note the historic confusions for what they are. > > Cheers, > Joe N