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:

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

Walter Whiteley