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gene_sullivan@[EMAIL PROTECTED]
In LogoForum@[EMAIL PROTECTED]
"Wendy Petti" <wpetti@[EMAIL PROTECTED]
> wrote:
>
> Hi, I have a fourth-grade student who has been asking
> me how to draw ovals with Logo (preferably MicroWorlds).
> Is this something that can be accomplished with
> math that a bright nine-year-old could understand?
I believe this is something that can be accomplished
with IMAGINATION ... by a bright nine-year-old.
> If not, is it something that can be accomplished
> with high-school level math that could be explained in
> kid-friendly (and teacher- friendly) terms?
Though I and others have responded to this quite a bit, it
has occurred to me that this question could have and perhaps
should have been addressed at this level of communication in
kid-friendly and teacher-friendly terms. Thus I have
chosen to respond back up at the top of this thread.
Just as a picture is supposed to be worth 1K words, a
depiction -- or visualization -- IS often better than
a description/explanation. Logo and photographs both
allow us to transcend the limitations of a mere stream
of words. Picturing `things' in our mind's eye also
allows us to re-frame a situation or `problem.
Oval ... or ellipse ... or circle viewed at a angle?
I suspect that both kids and teachers can view a hula hoop
from different angles and `understand', first person, that
a circular hoop APPEARS elliptical, just as a polygon
with enough sides drawn on computer screen via logo
APPEARS circular.
A teacher might be aware of different notions or conceptions
of a circle. Some teachers are aware that mathematicians
make much out of NOTHING ... literally. Points -- `being'
infinitesimally small, smaller than an atom of any sort -- `are'
used as building blocks for composing line segments, circles, and
other figments of imagination made out of the NOTHING which these
imaginary -- beyond material science, beyond physics-physical -- math
objects are imagined. Though a `bright nine-year-old' might not
have been exposed to the something-from-nothing metaphysics of
mathematicians yet, and this is not necessary for either kid or
teacher to `get it'. How many kids can't recognize their own shadows
.... or don't know how silhouettes are formed? I'd venture that our
bright nine-year-old could use a hula hoop, circular `ring', or
circular piece of paper to form silhouettes by tilting any of these
circular objects experimentally and observing the oval/elliptical --
(s)he knows-not-what ... nomenclature wise -- shapes that occur.
It seems a rather small leap from this rather Montessori-like exercise
to using logo -- a 3D-capable one preferably -- to draw something
circular on-an-angle (EG non-perpendicular, to us math biased adults)
so as to produce a desired or acceptable shape vis-a-vis whatever his
or her aesthetics or requirements may compel.
---------------------
Here is another way of approaching the situation.
Is there a circle which is not an ellipse ... the way there is a
square which is not a rectangle?
If a circle inscribed within a square IS an ellipse within a rectangle
how might this help us re-think and re-frame?
If would-be, `looks-like' circles in logo are typically drawn as
polygons -- via REPEAT n [FD <side_length> RT (360 / n)] -- then
can we inscribe our circle-cum-ellipse in not just a square, but any
regular polygon?
Hold this thought!
---------------------
Circle bounded by an outer square and inner diamond ... viewed non
perpendicularly ... becomes
ellipse bounded by outer rectangle and inner parallelogram
If an inner `diamond' (EG square, really) is drawn between
the 4 points where a outer square intersect with our circular ellipse
of interest an inner bounding polygon is formed.
Q: if the side count of both our inner and outer bounding polygons are
increased from 4 to 8 to 16 and so on, is our ellipse ever not bounded
by its outer and inner polygons?
If not, then might we graphically display/present an ellipse as a
polygon of n sides as we depict a circle as a polygon of n sides?
Is there ANY regular polygon viewed non-perpendicularly which does not
approximate an ellipse ... the way any regular polygon viewed
perpendicularly approximates, however crudely, a circle?
-------------------------
An intuitive solution to the generation of depiction and approximation
of an ellipse:
1) Draw the first approximation as a diamond whose vertexes
are the midpoints of a bounding rectangular parallelogram.
2) Increase the side count of the internal regular polygon
(EG our diamond originally) bounded by the rectangle
3) repeat step 2 until a close-enough approximation is obtained.
----------------------
While on the topic, if our bright 9-year-old is interested in the
theme of ovals, ellipses, and circles viewed from an angle ... then
why not present the concepts of ...
* elliptical orbits
http://en.wikipedia.org/wiki/Elliptic_orbit
* How *any* regular solid can be used as a template to approximate
a sphere ... by replacing the regular polygon from which it is
constructed with a circular ellipse
Tetrahedron ==> 4 circular ellipses
cube ==> 6 "
octahedron ==> 8 "
Dodecahedron ==> 12 "
icosohedron ==> 20 "
(aside: I sent a picture containing a dodeca-ellipse shape
and several tetra-ellipse shapes, all constructed from hula
hoops, to my friend Pavel a couple of years ago. He quickly
transcribed the dodecaHulaHoop into an Elica program which
is now both viewable online at
<http://www.elica.net/site/museum/Hoola%20Hoops.jpg>
and as a runnable Logo program as one of the 800+ samples
included with Elica, available for free at
http://www.elica.net
)
* comparisons between 2D approximates of ellipses
and 3D approximations of ellipsoids
* comparisons and constrasts between ovoids and
ellispoids (including the special case of spheroids)
http://en.wikipedia.org/wiki/Ovoid_(projective_geometry)
http://en.wikipedia.org/wiki/Ellipsoid
All that has come to mind on ovals, ellipses,
and such of late, Wendy, et al.
Cheers!
Gene
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