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kirby urner
Jun 27, 2012, 4:34:04 PM6/27/12
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From: : http://mathforum.org/kb/thread.jspa?threadID=2388791
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Re: Alternative area postulate for geometry
Posted: Jun 27, 2012 1:54 PM
On Wed, Jun 27, 2012 at 10:32 AM, Joe Niederberger
<nieder...@comcast.net> wrote:
<< snip >>
> On a different note though, I think Kirby is 100% correct about the arbitrariness of what ends up being the "unit" of measure for area or volume. "Rectilinear" units are completely conventional, as are rectilinear coordinates for a plane.
>
I think it's well worth investing in cartoons / animations that
develop this alternative understanding, of a x b as representing a
triangle of area a*b.
Because it's sooo different from a couple thousand years or more of
formal innovation around the cube as unit, it makes sense to starkly
segregate this "other" geometry as distinct and different. That's
part of the rationale behind my "Martian Math" branding.
In the cartoons I've been scripting / storyboarding, the Martians show
up with a unit tetrahedron of edges D. D for diameter, of a specific
sphere. Earthlings, on the other hand, have their unit cube, which
usefully, in this story, has edges of all R. R is for Radius = 1/2 of
D.
Pack three pool balls, three ping pong balls or whatever, in a
triangle. That's D x D (the area defined by the centers), or 1. Put
another ball in the valley atop these three, either side. That's an
all-D edged tetrahedron in relation two four closest packed spheres.
This is a canonical beginning. This is our unit of volume (in this
other culture -- we invent the ethnicity as part of the lore, i.e.
science fiction is a part of the package).
> I'll note in passing that the wikipedia entry on area says "Every unit of length has a corresponding unit of area, namely the area of a square with the given side length." One could equally truthfully say "and also a different corresponding unit of area that is the area of a tetrahedron with all edges of length one, and also..."
> The word "namely" might cause people to think the one unit mentioned is the only, which it isn't.
>
> Later, the same wikipedia article states "The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom." I suppose that could be done, but I've never sat through such a development. Also I think saying it " follows directly from the basic properties of area" is obtuse unless the author has pointed out what those "basic properties" are, which the Wpedia author has NOT done.
>
People say "squared" and "cubed" as a matter of reflex around 2nd and
3rd powering. We're not trying to say that's wrong, just that you
might question it, and in so doing, open a door into another world,
where STEM has evolved / is evolving differently.
> I think such a list would include things either like your #1, or Cavalieri. You are correct that #4 is a choice that is made. And the rectangle formula LxW is then "dervivable".
>
> But I disagree that #1 is a "definition" - it is proposition one accepts without proof. Part of that accepting is imlplicit in the fact that area hasn't been defined at all - its left to ituition.
>
> To get more formal and define "area" you have to start thinking about functions from sets of points to real numbers that have the properties you think an "area function" should have.
>
> Joe N
>
I think our notion that 3rd powering must be shown with a cube is
"notional". It's not so much axiomatic as definitional.
Going part way down this path of an Alternative Branch develops
valuable skills in our students.
What is open to question?
Diverging from standard Euclidean thinking by jiggering with the 5th
postulate is not the only way to diverge.
The branch I'm following has some roots in Karl Menger's work, a
dimension theorist. You may have heard of the Menger Sponge:
http://www.flickr.com/photos/17157315@N00/7437399936/in/photostream
There's lots of room for new recruits in this picture. Make a splash
with Flash or whatever. HTML5 and its 3D canvas is a fun tool to play
with. I signed up for some workshops at OSCON this year.
Kirby
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Re: Alternative area postulate for geometry
Posted: Jun 27, 2012 1:54 PM
On Wed, Jun 27, 2012 at 10:32 AM, Joe Niederberger
<nieder...@comcast.net> wrote:
<< snip >>
> On a different note though, I think Kirby is 100% correct about the arbitrariness of what ends up being the "unit" of measure for area or volume. "Rectilinear" units are completely conventional, as are rectilinear coordinates for a plane.
>
I think it's well worth investing in cartoons / animations that
develop this alternative understanding, of a x b as representing a
triangle of area a*b.
Because it's sooo different from a couple thousand years or more of
formal innovation around the cube as unit, it makes sense to starkly
segregate this "other" geometry as distinct and different. That's
part of the rationale behind my "Martian Math" branding.
In the cartoons I've been scripting / storyboarding, the Martians show
up with a unit tetrahedron of edges D. D for diameter, of a specific
sphere. Earthlings, on the other hand, have their unit cube, which
usefully, in this story, has edges of all R. R is for Radius = 1/2 of
D.
Pack three pool balls, three ping pong balls or whatever, in a
triangle. That's D x D (the area defined by the centers), or 1. Put
another ball in the valley atop these three, either side. That's an
all-D edged tetrahedron in relation two four closest packed spheres.
This is a canonical beginning. This is our unit of volume (in this
other culture -- we invent the ethnicity as part of the lore, i.e.
science fiction is a part of the package).
> I'll note in passing that the wikipedia entry on area says "Every unit of length has a corresponding unit of area, namely the area of a square with the given side length." One could equally truthfully say "and also a different corresponding unit of area that is the area of a tetrahedron with all edges of length one, and also..."
> The word "namely" might cause people to think the one unit mentioned is the only, which it isn't.
>
> Later, the same wikipedia article states "The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom." I suppose that could be done, but I've never sat through such a development. Also I think saying it " follows directly from the basic properties of area" is obtuse unless the author has pointed out what those "basic properties" are, which the Wpedia author has NOT done.
>
People say "squared" and "cubed" as a matter of reflex around 2nd and
3rd powering. We're not trying to say that's wrong, just that you
might question it, and in so doing, open a door into another world,
where STEM has evolved / is evolving differently.
> I think such a list would include things either like your #1, or Cavalieri. You are correct that #4 is a choice that is made. And the rectangle formula LxW is then "dervivable".
>
> But I disagree that #1 is a "definition" - it is proposition one accepts without proof. Part of that accepting is imlplicit in the fact that area hasn't been defined at all - its left to ituition.
>
> To get more formal and define "area" you have to start thinking about functions from sets of points to real numbers that have the properties you think an "area function" should have.
>
> Joe N
>
I think our notion that 3rd powering must be shown with a cube is
"notional". It's not so much axiomatic as definitional.
Going part way down this path of an Alternative Branch develops
valuable skills in our students.
What is open to question?
Diverging from standard Euclidean thinking by jiggering with the 5th
postulate is not the only way to diverge.
The branch I'm following has some roots in Karl Menger's work, a
dimension theorist. You may have heard of the Menger Sponge:
http://www.flickr.com/photos/17157315@N00/7437399936/in/photostream
There's lots of room for new recruits in this picture. Make a splash
with Flash or whatever. HTML5 and its 3D canvas is a fun tool to play
with. I signed up for some workshops at OSCON this year.
Kirby
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