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Topic: Based on the quadrilateral tree
Replies: 14   Last Post: May 8, 2013 7:00 PM

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kirby urner

Posts: 1,528
Registered: 11/29/05
Re: Based on the quadrilateral tree
Posted: May 8, 2013 4:20 PM
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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.

Kirby


On 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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