a_plutonium wrote:
> --- quoting Iowa State website which says hyperbolic rectangles are
> impossible ---
>
http://72.14.205.104/search?q=cache:dflBftt61AgJ:orion.math.iastate.edu/msm/DonaldCMSMCCF05.pdf+hyperbolic+parallelogram&hl=en&ct=clnk&cd=13&gl=us
> If three of the angles were 90 degrees then by the argument
> above, the last angle would be less than 90 degrees. Euclidean
> rectangles are quadrilaterals with four right angles. The sum of
> the angles of a rectangle equals 360 degrees. Therefore,
> rectangles do not exist in Hyperbolic Geometry. Therefore,
> squares, quadrilaterals having four right angles and four
> congruent sides, do not exist in Hyperbolic Geometry (Ramsey and
> Richtmyer, 1995). This, however, does not mean that there are no
> regular quadrilaterals in Hyperbolic Geometry. A regular
> quadrilateral is one for which all sides are congruent and that
> all angles are congruent. Regular quadrilaterals and rhombi
> exist in Hyperbolic Geometry because there is a model in which
> such figures exist. "The main property of any model of an axiom
> system is that all theorems of the system are correct in the
> model. This is because logical consequences of correct
> statements are themselves correct" (Greenberg, 1993, p. 52).
> These next proofs focus on the characteristics of rhombi and
> regular quadrilaterals in Hyperbolic Geometry.
>
http://72.14.205.104/search?q=cache:dflBftt61AgJ:orion.math.iastate.edu/msm/DonaldCMSMCCF05.pdf+hyperbolic+parallelogram&hl=en&ct=clnk&cd=13&gl=us
> --- end quoting Iowa State ---
>
> --- quoting a website that constructs a hyperbolic rectangle ---
>
http://www.joma.org/images/upload_library/4/vol1/hypertoolbox/rectang.htm
> Given that the Poincaré half-plane is conformal (i.e. it represents
> angles faithfully), the difficulty with the construction becomes
> clear: angle BCD is certainly not a right angle. This highlights the
> im****tance for students of relying on formal definitions rather than
> on their experiences in Euclidean geometry when studying hyperbolic
> geometry. In hyperbolic geometry, it is possible to have a
> quadrilateral with exactly three right angles!
>
http://www.joma.org/images/upload_library/4/vol1/hypertoolbox/rectang.htm
> --- end quoting ---
>
> Can you see the problem? Iowa State is defining rectangles as having
> to have 4 right angles
> and ignoring the idea that a rectangle does not have to have 4 right
> angles but merely
> needs to have opposite sides parallel and equal arc length. Which
> stands to reason since
> Iowa State then further goes on to say that Hyperbolic Parallelograms
> exist, yet they deny
> the existence of Hyperbolic Rectangles. And the second website goes on
> to construct
> a Hyperbolic Rectangle.
I think the header
``A First Example: Constructing a Rectangle in Hyperbolic Geometry"
is very confusing. Also, the box with
"Rectangle Construction Steps". (This refers to the 2nd site).
The 2nd site does have the sentence:
"An immediate consequence, of course, is that in hyperbolic geometry,
rectangles do not exist." But I think it would be better to
say " quadrilaterals with four right angle don't exist in the
hyperbolic plane.".
Wouldn't it be better to have a ruler with graduations to estimate
lengths?
Anyway, I prefer this Web page at the University of Minnesota:
< http://www.math.umn.edu/~garrett/a02/H2.html
>
for the Poincaré open disk model of the hyperbolic plane.
David Bernier
> This is very confusing to those High School students that Iowa State
> is intending to teach
> Hyperbolic geometry. Confusing to students who go to the Internet and
> see a website that is
> 180 degrees opposite in instruction from Iowa State.
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