Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: A Point on Understanding
Replies: 66   Last Post: Jan 6, 2013 11:07 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: A Point on Understanding
Posted: Dec 16, 2012 11:22 PM
 att1.html (2.0 K)

On Sat, Dec 15, 2012 at 9:04 AM, Robert Hansen <bob@rsccore.com> wrote:

<< SNIP >>

> Even though the student accepted that 0.333... is 1/3, that acceptance was
> based on nothing more than a presentation (long division). Without being
> able to produce a similar presentation for 0.999... the student is stuck.
>
>

A related demo that may leave a student feeling uneasy has to do with
creating a spherical network of more and more triangles. The algorithm is
such that six triangles surround each vertex except in 12 places where five
triangles serve. The triangles are not quite equiangular such that at none
of the vertexes do we get a full 360 degrees (perfect flatness), but rather
a smidgen less, and this begets convex curvature as viewed from the outside.

In a calculus mindset, looking at this algorithm, we see that the number of
degrees v at each vertex differs from 360 by epsilon, and I may always find
some number of triangles N such that |360 - v| < e, i.e. lim (N->infinity)
|360 - v |, at each vertex, -> 0.

What's wrong with that vision?

Descartes proved that adding the angular deficits of all such vertexes, no
matter their number, yields a constant number, 720 degrees. Ergo Sigma
(360 - v) over all N = 720. This proves the limit at each vertex is never
zero, as every vertex contributes some tiny "tax" or "tithe" to the
invariant constant 720. 720/N > 0. |360 - v| > 0 even as N -> infinity.

Kirby

http://www.scientificindians.com/general-sciences/mathematics/how-geodesic-domes-work

Date Subject Author
12/15/12 Robert Hansen
12/17/12 kirby urner
12/18/12 Robert Hansen
12/17/12 Haim
12/18/12 Robert Hansen
12/26/12 kirby urner
12/26/12 Robert Hansen
12/26/12 kirby urner
12/27/12 Paul A. Tanner III
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/28/12 Paul A. Tanner III
12/28/12 kirby urner
12/28/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Louis Talman
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/27/12 Domenico Rosa
12/27/12 Robert Hansen
12/27/12 Richard Strausz
12/27/12 Domenico Rosa
12/30/12 Joe Niederberger
12/30/12 Paul A. Tanner III
12/30/12 Robert Hansen
12/30/12 Joe Niederberger
12/31/12 Robert Hansen
1/2/13 kirby urner
12/30/12 Joe Niederberger
1/1/13 Paul A. Tanner III
12/31/12 GS Chandy
12/31/12 Robert Hansen
1/1/13 GS Chandy
12/31/12 GS Chandy
12/31/12 Robert Hansen
12/31/12 Joe Niederberger
1/1/13 Robert Hansen
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 Robert Hansen
1/1/13 Haim
1/1/13 Joe Niederberger
1/1/13 Joe Niederberger
1/1/13 Paul A. Tanner III
1/1/13 Louis Talman
1/2/13 Paul A. Tanner III
1/1/13 Joe Niederberger
1/2/13 Paul A. Tanner III
1/2/13 GS Chandy
1/2/13 Joe Niederberger
1/4/13 Joe Niederberger
1/5/13 GS Chandy
1/5/13 GS Chandy
1/6/13 Robert Hansen
1/6/13 GS Chandy