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Re: Inclusive and exclusive definitions... again!
Posted:
Mar 8, 2010 8:18 PM


*Due to trouble sending this dialogue around the email list, I'm posting it here in the Math Forum board*
From Walter Whiteley:
An interesting chart  and a topic worth continuing conversations. I have a alternatives to how the classification is done  and therefore what is worth naming, and how the names are done. Two perspectives lead to some different classes: (a) if we classify quadrilaterals by symmetries, then some distinctions don't matter so much. On the the other hand, a kite (with a mirror through two vertices) can be nonconvex. By the way, in this classification, parallelogram is the class with halfturn symmetry. (b) If we think about classifying on the sphere  where there is duality between angles and lengths, then some of the alternatives you have get 'paired up'. Interestingly, these pairings carry on into the plane under polarity about a circle  between shapes with four vertices on the circle, and shapes with four edges tangent to the circle. One version of this is linked at the Geometer Sketchpad Users Group site: http://www.dynamicgeometry.com/General_Resources/User_Groups/JMM_2006.html Note that (a) and even (b) actually work well with the names for triangles, and we don't really try to capture the comparable analysis for 5 or more sides. Also, in 3space, with skew quadrilaterals, there is a further set of connections. In the end  3D reasoning is a key goal, so I am happy to do a bit less in the plane if the larger vision opens up (see the link above). These perspectives do come from some types of reasoning one wants to do  and I think naming is best developed to help cue some reasoning / connections etc. So classifying parallelograms by halfturn symmetry, cues us to the fact that most proofs for parallelograms implicitly use this property  or would be easier if we do use this property. For example, when the proof uses a diagonal and that cites ' congruent triangles'  the congruence actually is a halfturn symmetry! Much of the symmetry analysis becomes evident / even essential, when we observe which isometry is used for the 'congruence'. I have some other charts etc. for some of this. On of the criterion: How well does it generalize' is useful, as well as what reasoning / connections does it afford? In terms of the dislike for 'diamond'  many people (including many students) would certainly consider a 'square' oriented with vertices up and down (45 degree angle to the 'standard' orientation' as a diamond. There is a even a commercial in North America which plays on this for a cereal (Shreddies) has a square shape (see en.wikipedia.org/wiki/Shreddies). Walter Whiteley



