```Date: May 8, 2013 4:20 PM
Author: kirby urner
Subject: Re: Based on the quadrilateral tree

The properties of "gons" in general should be reviewed, as studentsmay forget we're conventionally forcing them to be flat, all vertexesin one plane.  A necklace of four cylindrical beads, or 4-edgedconstruction with ball bearing hinges, is going to be floppy in space,all wobbly, and is not considered a "quadrilateral" except in snapshot moments when the four edges are "in a plane".  A polygon forfleeting instants.What students should be reminded of is the rules of the game (likechess) narrow the permitted / legal moves to an exponentially tinyfragment of what's possible, but this strictness is what makes for therigorous proofs of Euclidean geometry.  Strict definitions excludewhat's irrelevant.  Besides, we have topology for the morenecklace-like thingamabobs.  It's not like math itself is confined byEuclidean definitions.Note that triangles have no choice but to be planar whereas thequadrilateral is the first n-gon able to "hinge" in a way thatintroduces no new vertexes (triangles may be creased, but this addsnew nodes).  A rhombus may be creased along a diagonal to make two"wings", the tips of which may be connected by another edge of equallength.  A tetrahedron is born.  It is not floppy either, being madeof triangles.The wobbly hexahedron frozen in cube moments (unstable) is the hedronof choice for European volume units, such as grams.  A tetrahedron,calibrated to standards (of weight, of size), might sit in someschool's museum as an alternative choice.  It's another possiblemathematics and is accessible to all ages.  I'm something of a tourguide in this area.  I think you understand your own ethnicity betterwhen you have an opportunity to compare it with something more alien.Good mental exercise.KirbyOn Wed, May 8, 2013 at 8:40 AM, Joe Niederberger<niederberger@comcast.net> 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
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