```Date: Mar 8, 2010 8:18 PM
Author: Allan Turton
Subject: Re: Inclusive and exclusive definitions... again!

*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 non-convex.   By the way, in this classification, parallelogram is the class with half-turn 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 3-space, with skew quadrilaterals, there is a further set of connections.   In the end - 3-D 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 half-turn 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 half-turn 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
```