EDIT: I really have to admire the kabbalistic coincidence that this node currently has 2 C!'s and a reputation of 22. :)
EDIT(2): Well, it was fun while it lasted.
Throughout his career, Shakespeare shows a deep interest in the idea of doubling1. Nowhere, I think, does this interest manifest itself more thoroughly than in Hamlet. Doubling in the play (and, to a lesser degree, its opposites [1,2]) is so pervasive and subtly deployed that it is literally everywhere.

Below is an extremely loose attempt to classify the various kinds of doubling in the play; I plan to expand and fine-tune this node over time. This list is by no means intended to be exhaustive--or especially rigorous. I've tried not to include the most minor examples unless they happen to be especially clever. Feedback on this node is heavily encouraged.

Events

1. Critics have long pondered over the significance of Polonius' conversation with Reynaldo; I suspect that, among other things, it's supposed to compare with the next scene, where Rozencrantz and Guildenstern are deployed to spy on Hamlet.

2. The play references two funerals, King Hamlet's and Ophelia's2.

3. In the first scene, the characters are visited twice by the Ghost of Hamlet's father while they are talking about what they have twice seen two nights' previously.

4. In his first soliloquy, Hamlet laments the fact that it has been two months since the death of his father. During the production of the Mousetrap, he jests to Ophelia that Gertrude looks merry despite the fact that it has only been "two hours" since his father's death. "Nay, 'tis twice two months," she responds.

5. The Gravediggers twice tell variations on the same joke.

6. In the Mousetrap scene, the characters in Hamlet twice watch the character of a regicide poison a player king in the ear.

7. The play contains two revenge plots: Hamlet's and Laertes'.

Characters

1. Rosencrantz and Guildenstern, Voltemand and Cornelius and the two Ambassadors are all obvious redundancies. Likewise for Polonius/Osric and Marcellus/Bernardo. The play also contains two murdered kings (four if you count the two player kings) and two heroines.

2. Each of the characters in The Mousetrap maps to two different characters in Hamlet:
A. Lucianus, like Hamlet, is both a regicide and a nephew to the king; like Claudius, he is a regicide that operates by pouring poison into ears.

B. The Player King, like Hamlet, is an erratic melancholic; like King Hamlet, he is poisoned via his ear while reclining in his orchard.

C. The Player Queen, like Ophelia, attends to a character that is "so far from cheer and from [a] former state"; like Gertrude, she remarries a regicide.
3. Almost all of the characters in the play are reflections of Hamlet in some fashion or another:
A. Hamlet, Laertes, Fortinbras and Phyrrus are all avenging sons. Hamlet and Laertes both blame Claudius for the death of their fathers. Hamlet and Phyrrus are both seized by inaction at some point in their respective narratives and each avenges his father. Hamlet and Fortinbras both have plans that are thwarted by uncles that are also kings.

B. Hamlet, Rosencrantz, Guildenstern, Osric and Polonius are all courtiers.

C. Hamlet, his father, Bernardo, Marcellus, Francisco, Fortinbras and several other characters are all soldiers.

D. Hamlet and his father share a name (as do Fortinbras and his father).

E. Hamlet, Horatio, Rosencrantz, Guildenstern and Laertes are all students.

F. Hamlet, his father, Gertrude and Claudius are all members of the Royal Family. Each of them is also killed by poison -- poison that Claudius is responsible for.

G. Hamlet and Ophelia are each rebuked by their surviving parent in subsequent scenes (1,2); the surviving parent of each happens to be of the opposite gender. Both also enter scenes reading books (1,2) and there is a contrast between the (possibly) pretend madness of Hamlet and the very real insanity of Ophelia.

H. Hamlet, Horatio, Polonius, Rosencrantz, Guildenstern and Claudius are each "lawful espials" at some point in the play.
4. The Polonius family, one son and one daughter with a single male parent, is a neat inversion of the Hamlet family, one son with one male and one female parent.

5. The entire problem of doubled characters takes on another dimension if one considers that, due to budgetary/personnel constraints, multiple characters were likely played by the same actors on the Elizabethian/Jacobean stage. For instance, pace item 1 in this section, the same actors that played Rosencrantz and Guildenstern probably also played Voltemond/Cornelius and the Ambassadors; likewise for Polonius and Osric as well as Francisco (who only appears in the play's first scene and essentially only exists to tell the audience that he, like Hamlet, is "sick at heart") and Hamlet. Furthermore, if the traditional conjecture is correct and the same actor in King Lear did play both the part of the heroine and the clown3, it seems possible that the same actor in Hamlet could have played both Ophelia and the First Gravedigger4.

Language

1. As George T. Wright has noted5, a highly-redundant "X and Y" construction (generally termed hendiadys -- literally "one by means of two.") is more prevalent in Hamlet than in any other Shakespeare play. For example: "slings and arrows of outrageous fortune", "stand and unfold thyself", "angels and ministers of grace", "the book and volume of my brain", and so on.

2. Many of the major speeches in the play are based around some form of comparison or contrast: Claudius' first speech, Hamlet's initial comparison of Claudius and his father ("hyperion to a satyr"), Polonius' "neither a borrower nor a lender be", his speech before the King and Queen about "why day is day, night night," Hamlet's "to be or not to be", his ironic repetition of his mother's language in the Closet scene6 and his insistance that she "look upon this picture and on this," Claudius' speech at prayer comparing his soul against his crown, etc.

3. For all of the fuss about whether or not Hamlet says "sallied," "sullied," "solid" or something else entirely in his first soliloquy, it's interesting that that speech starts out "O that this too too solid flesh..." and that the speech takes a handful of lines before it returns to what has happened these "two" months past -- a number that is said twice across two lines.

4. Hamlet has a habit of employing verbal redundancies: "nor the windy suspiration of forced breath", "all saws of books, all forms, all pressures past", "words words words", "remorseless, treacherous, lecherous, kindless villain!", etc.

5. Gertrude's response to Hamlet's assault in the Closet scene is that he "hath cleft [her] heart in twain."

6. Laertes welcomes his father in the play's third scene by saying that "a double blessing is a double grace." Similarly, this is Hamlet in scene 5.1:
This fellow might be in ’s time a great buyer of land, with his statutes, his recognizances, his fines, his double vouchers, his recoveries. Is this the fine of his fines, and the recovery of his recoveries, to have his fine pate full of fine dirt? Will his vouchers vouch him no more of his purchases, and double ones too, than the length and breadth of a pair of indentures?

1. One possible reason for this is that theater itself is predicated on a special kind of doubling, between actors and characters.

For further evidence that this interest manifests itself across many of his plays, consider the following lines from Macbeth: "DOUBLE, DOUBLE toil and trouble...", "All of our service / Is in every point twice done, and then done double" (Macbeth1.6.16-7, two lines which also double "done"), "As cannons overcharged with double cracks / So they doubly redoubled strokes upon the foe" (1.2.37-8), "He's here in double trust" and so on.


2. There is also some muted refernce to the fact that Polonius doesn't receive a funeral -- odd in light of the fact that Ophelia, supposedly a suicide, does.
3. See, for instance, http://faculty.goucher.edu/eng211/Shakespeare--cordelia_and_the_fool.htm
4. Also note that the two characters are the only two characters in the play besides Hamlet to sing, that the content of their songs is thematically similar and that, when asked whose grave he is digging, the First Gravedigger responds: "Mine, sir" -- an answer that, if this conjecture is correct, would have delivered several rich layers of meaning to an audience.
5. Wright, George T. 1981. 'Hendiadys and Hamlet.' PMLA 96:168-193.
6. There is also a fairly subtle pun in Hamlet's second line in this scene ("Now, mother, what's the matter?"), since "mater" is Latin for "mother." The manifest prevalence of puns in this play is yet another example of a sort of linguistic doubling.

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