Quadrilateral Taxonomy

quadrilaterals

The last post made me want to force ethics onto quadrilaterals.

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15 comments

  1. stevejwright

    Are the chaotic ones strictly quadrilaterals? I can see they can be made with four lines, but they seem to have six sides in total. Quadrilineal but sexilateral?

    Also, I can’t help noticing that Lawful Evil becomes Neutral Good if you tilt your head, and this has me ethically perplexed.

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  2. kiptw

    I notice that the more right angles a figure has, the more likely it is to be lawful. This seems somehow wrong to me, even if they’re not strictly alt-right angles.

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    • Stewart

      The greater the symmetry the closer to lawful good, except that the wedge trapezium and kite should be swapped.

      Missing from the classification are darts (bilaterally symmetric convex quadrilaterals), non-bilaterally symmetric convex quadrilaterals, triamonds (bilaterally symmetric trapezia with three equal sides), bilaterally symmetric trapezia with two equal sides), generalised trapezia (one pair of parallel sides), cyclic quadrilaterals and generalised concave quadrilaterals.

      Criteria are number of symmetries, number of different values of angle, number of different lengths of sides, number of pairs of parallels sides, and number of crossing points in the boundary. One could formalise this by packing these into a pair of numbers and plotting out the positions of the various shapes.

      You might swap the good-evil and lawful-chaotic axes; the “hourglass” is rather regular, i.e. not chaotic.

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      • Lurkertype

        I’m with Stewart on swapping the center and kite ones. The kite’s bilaterally symmetrical, but the one in the middle is lopsided and therefore must be more evil.

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      • Stewart

        I’m aware of the rectilinear cross hexagons, as they occur as self-similar figures, such as the double square (opposites quadrants of a square) and the double golden rectangle (in which the golden ratio is involved in the ratio of areas as well as sides).

        (Self-similarity is another criteria you could add to the mix; all parallelograms, including squares, rectangles and rhombuses are order-4 self similar, and a few are order 2 or order 3; I think you can get arbitrarily close to any wedge trapezium, but the order can be rather high (the lowest order possible is 4) ; and a least one kite is order 4 self similar. Some/all (I don’t recall) cyclic quadrilaterals are self-similar; to the best of my knowledge convex and cross quadrilaterals aren’t, nor are generalised concave quadrilaterals.

        I think that the hourglass, and hence the other cross quadrilaterals are self-affine.)

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