A Hierarchical Classification of Quadrilaterals

The quadrilaterals in the classification below display a side-angle symmetry, which is discussed more fully in my book Some Adventures in Euclidean Geometry, available as downloadable PDF or printed book.

All quadrilaterals are dynamic and can dragged into the special cases below them, which INHERIT ALL the properties of the linked general cases ABOVE them. However, depending on how the quadrilaterals were constructed in this particular sketch there may be some limitations regarding the different shapes they can be dragged into. Click on Definitions of the Quadrilaterals, and some properties to open a separate, new window with a list of possible definitions and some important properties of these quadrilaterals.


Hierarchical Classification of Quadrilaterals

The classification is too large to fit onto one WebSketchpad webpage and has been divided into two webpages. Click on the LINK button in the sketch to navigate either lower down or higher up in the hierarchy.

a) The above classification is by no means intended as an exhaustive, complete classification of all possible quadrilaterals. One can easily add others such as the self-dual ortho-equidiagonal quadrilateral; in words, a quadilateral with perpendicular and equal diagonals. Similarly, the isosceles trapezoid (trapezium) with perpendicular diagonals is dual to the kite with equal diagonals, and both are special cases of an ortho-equidiagonal quadrilateral. An interesting, easy to prove property of these quadrilaterals are that the midpoints of their sides form a square.
b) Another example is that of the right kite and isosceles circum trapezoid (trapezium), which are both so-called bicentric quadrilaterals; in other words, subsets of those general, self-dual quadrilaterals which are both cyclic and circumscribed (but not shown in the classification scheme above). Bicentric quadrilaterals also have several interesting properties which can be found in books on advanced Euclidean geometry and mathematical journals. See Bicentric Quadrilateral Properties for some properties.

c) Sketchpad 4 and Sketchpad 5 sketches of this Hierarchical Classification Tree (with the exception of the Triangular Kite, Right Kite, Isosceles Circum Trapezoid and Trilateral Trapezoid) has Property buttons in the sketch, and can be downloaded for free from Sketchpad Exchange.

d) A suggested series of learning activities to visually introduce young children to the quadrilaterals and foster the idea of hierarchical class inclusion from the beginning is available at Classifying Quadrilaterals Visually.

1) Though definitions are to some extent arbitrary and we can often choose them as we wish in mathematics, sound pedagogical practice is NOT to just provide 'ready-made' definitions to students, but to involve students in actually defining and classifying the quadrilaterals themselves (e.g. see De Villiers, 1998).

2) Sound pedagogical practice would also involve a classroom discussion and critical comparison of the merits and demerits of different possible definitions for the quadrilaterals, including carefully comparing hierarchical (inclusive) and partition (exclusive) definitions (e.g. see De Villiers, 1994; De Villiers, 2009, pp. 13-28. and Jim King, undated.) A similar analysis and approach is advocated in the 2007 book The Classification of Quadrilaterals: A Study of Definition. by Zalman Usiskin and Jennifer Griffin, and is also highly recommended.

3) Matthew Tydd from Australia has also argued in the AMESA KZN Mathematics Journal for a classification similar to the above, based on diagonal properties and symmetry. Read his paper at: Two Kites (2005).

4) Also read my 2011 paper Simply Symmetric, which briefly discusses and illustrates the value of symmetry in the choice of definitions for quadrilaterals (and some of its advantages over traditional definitions).

5) Martin Josefsson has extensively explored the duality between orthodiagonal and equidiagonal quadrilaterals, as well as their characteristics, in papers published in Forum Geometricorum. Download them respectively at Characterizations of Orthodiagonal Quadrilaterals (2012) and Properties of Equidiagonal Quadrilaterals (2014).

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Michael de Villiers, created 2011. Most recent update: 17 September 2016.