# Talk:Dual polygon

## Contents

## Side bisection vs exterior angle bisection

The dual of this green pentagon, construction by recitification. These edge midpoints form the vertices of the new (dual) black pentagon. |
The dual of this green pentagon, construction by bisecting the external angles of each vertex. A new (dual) pentagon is formed by the intersecting of adjacent line bisections. |

Here's 2 dual constructions, rectification and vertex angle bisection. The first definitely is not symmetric. I don't know about the second one. Tom Ruen (talk) 04:09, 11 December 2008 (UTC)

- I'm guessing "not symmetric" means "dual of dual is not similar to original". In that case the second method is also not symmetric. It almost seems like these two should be meta-dual dualities, with one bisecting the sides while the other bisects the angles, but perhaps that's a dead end -- the angular-dual of a rectangle is a square, but the rectification-dual of a rhombus is not. Joule36e5 (talk) 07:57, 21 May 2012 (UTC)

## Improvement needed

A definition is needed, preferably at the beginning of the article.

I do not understand the comment about cyclic polygons. Can someone make sense of it? --seberle (talk) 19:18, 18 September 2009 (UTC)

## Cyclic/Tangential duality

Two problems here:

- In a cyclic polygon, smaller angles correspond to shorter sides, and bigger angles to bigger sides – further, congruent sides in the original yield congruent angles in the dual (a tangential polygon), and conversely. For example, the dual of an acute isosceles triangle is an obtuse isosceles triangle.

Firstly, as can be seen in taking the rhombus dual of a rectangle, the relationship is between short sides in the original and *larger* angles in the dual. Secondly, not every acute isosceles triangle has an obtuse dual, since the limit case is the self-dual equilateral. I've rewritten the former, and changed the latter to read "highly acute", which seems adequate (I don't think we need to be more specific about the degree). Joule36e5 (talk) 22:37, 15 May 2012 (UTC)

- It depends on whether one takes the interior or exterior angle: I have edited to make the original author's assumption explicit. And yes, you are right about the isosceles triangles, thanks. — Cheers, Steelpillow (Talk) 20:07, 16 May 2012 (UTC)

## Overall organization

The (tangential) dual of a cyclic polygon should be listed under "kinds of duality". Like rectification, it's "not reversible" (there's surely a better term for that), in this case because the dual is not necessarily cyclic, a requirement for generating the dual of the dual. The section about cyclic quadrilaterals should then be moved into this subsection; it's exhibiting a special case of this duality.

The "Properties" section is problematic. "Regular polygons are self-dual" is clearly true, for any type of duality we consider. Beyond that, the properties depend on the duality. Even isogonal-isotoxal duality is shaky: every isogonal polygon is cyclic, and I believe we get the same isotoxal polygon whether we use cyclic-tangential duality or rectification duality, but they produce different answers for which angles are dual to which sides! "In a cyclic polygon, ..." is specific to cyclic/tangential duality, and judging from the diagram, so is the Dorman Luke construction. Joule36e5 (talk) 09:06, 21 May 2012 (UTC)