# What's the difference between “duality” and “symmetry” in mathematics?

Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H (if G is connected)." In mathematics, duality is a very important phenomenon and one may immediately come up with lots of examples(e.g., dual space in functional analysis). At the same times, there are many different kinds of symmetry in mathematics. This wiki article points out that "A high-level concept related to symmetry is mathematical duality.".

Here is my question:

What's the difference and relationship between "duality" and "symmetry"?

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Duality involves two things, and symmetry can involve many things. – Qiaochu Yuan Jul 8 '11 at 19:03

No, duality and symmetry are not the same thing. Although in many contexts "the dual of" is a symmetric relation, this is not invariably the case (e.g. the dual of the dual of a topological vector space need not be the original).

Moreover symmetry is not just about symmetric relations; it has to do mainly with automorphisms of algebraic, geometric or combinatorial structures. Those structure preserving automorphisms (including trivial the identity mapping) form a group, and we'd refer to it as the symmetry group of the structure.

As you note, there are many kinds of symmetry. Some symmetries have order two but many do not. Indeed the group of symmetries may combine elements that have finite order with those having infinite order, elements that have discrete action with some that are continuous mappings. The symmetries of a right circular cylinder, for example, would include discrete actions like reflection in a midplane as well as continuous actions of rotation about the axis.

If you are looking for a fundamental difference, perhaps it should be noted that duality often involves different categories, i.e. the dual may belong to a different category than the original, while symmetry involves not only the same category but actually a mapping of the same object to itself.

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So, I guess dual in the vector space context is actually a misnomer arising from the finite-dimensional case? – Marek Jul 11 '11 at 21:09
Not a misnomer! The category theory perspective shows us something interesting. For a fixed field K, taking the dual of a vector space gives us a contravariant functor, meaning that it has the effect of reversing directions of arrows. Thus the "taking the dual" functor is not naturally isomorphic to the identity functor (which is covariant). However the double dual is an injective functor that is a natural transformation of the identity functor. – hardmath Jul 11 '11 at 21:55
@hardmath Yes, but with the notion of dual that people like to think of, you are looking for more than just a contravariant endofunctor $F$ with a transformation $\operatorname{Id}\to F^2$. We want that $F^2$ is an isomorphism (it isn't even enough to be an equivalence). It turns out that we can restore duality for vectorspaces by taking on one side vector spaces as purly algebraic objects and on the other topological vector spaces with the profinite topology. What you have without taking topology into account is NOT what I would consider duality. – Aaron Jul 11 '11 at 23:58
@hardmath, I guess it depends on the point of view (and one's definitions) but in my book, duality should correspond to the ${\mathbb Z} / 2{\mathbb Z}$ symmetry so that double dual operation is (a canonical) isomorphism. – Marek Jul 12 '11 at 5:44

An algebraic, geometric or combinatorial object $\Omega$, consisting of a "ground set" $O$, and provided with additional structure elements like a metric, edges, binary operations, etc., may have symmetries. A symmetry is the same thing as an automorphism, i.e., a bijective map $\phi: \ O\to O$ such that any relevant relation among the elements $x\in O$ is preserved. This means that if, e.g., it matters that $x*y=z$ then one should have $\phi(x)*\phi(y)=\phi(z)$. Sometimes a symmetry $\phi$ has the property that $\phi$ is not the identity, but $\phi\circ\phi$ is. Such a $\phi$ is called an involution.

Now the concept of duality is a different matter. A duality is an involution not of a single object, but of a whole theory. Examples are the duality present (a) in plane projective geometry or (b) in the theory of convex polyhedra.

Ad (a): To each theorem in PPG corresponds its dual theorem. The dual of Pascal's theorem about $6$ points on a conic is Brianchon's theorem about $6$ tangents to a conic.

Ad (b): An individual polyhedron, say a platonic solid, may have an interesting set of symmetries. But duality is something far deeper. It says that to any such polyhedron $P$, symmetric or not, corresponds a dual polyhedron $\hat P$, such that incidences among the vertices, edges, and faces of $P$ appear in $\hat P$ reversed. The dual of an dodecahedron is an icosahedron.

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