What symbols or objects represent the three witches in Macbeth?

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When the Witches first appear in act 1, scene 1 of Shakespeare's Macbeth, they're not carrying or wearing any objects or talismans. They do, however, refer to their "familiars"—evil spirits which are usually in the form of a small animal like a toad, bird, dog, or cat and serve as a witch's demonic companion or attendant.

FIRST WITCH. I come, Graymalkin.

SECOND WITCH. Paddock calls.

THIRD WITCH. Anon! (1.1.9–11)

"Graymalkin" is an affectionate term for a gray cat, and "paddock" refers to a toad.

The Third Witch doesn't identify her "familiar" until act 4, scene 1, when she calls it a "harpier," or harpy—a flying monster with the head and upper body of a woman and the tail, wings, and talons of a bird of prey.

THIRD WITCH. Harpier cries, “’Tis time, ’tis time.” (4.1.3)

The Third Witch wins the prize for most horrifying familiar.

These “familiars” symbolically help to represent the “weird sisters,” as they call themselves (1.3.33), as witches.

In act 1, scene 3, the First Witch is seen to have, if not a symbolic charm, then a grisly souvenir.

FIRST WITCH. Look what I have.

SECOND WITCH. Show me, show me.

FIRST WITCH. Here I have a pilot's [sailor's] thumb,
Wreck'd as homeward he did come. (1.3.27–30)

Although a cauldron in which to brew their potions might not be something that the Witches ordinarily carry around with them, cauldrons are symbolic of witches in general, and particularly of the Witches in Macbeth.

WITCHES. Double, double, toil and trouble;
Fire burn and cauldron bubble. (4.1.10–11)

There's an interesting stage direction that appears in this scene in the First Folio of Shakespeare's plays, published in 1623. The Three Witches have just finished brewing up their potion and cooling it off with "baboon's blood" (4.1.37).

Enter Hecat, and the other three Witches.

(In later printings of Macbeth, "Hecat" is "Hecate.")

The other three Witches? The original Three Witches have been brewing their disgusting potion for thirty-seven lines, then Hecate shows up with three more witches? Where have these other witches been all this time, and why do they choose to crash the party at this particular moment?

Modern editors change this stage direction to read “Enter Hecate to the other three Witches,” or simply “Enter Hecate,” but it’s interesting to speculate whether this is a printing error or if three other witches actually appeared on stage during the early productions of Macbeth.

There's another implied stage direction in this scene.

MACBETH. ... Why sinks that cauldron? and what noise is this? (4.1.119)

It's easy enough to explain the noise, because “hautboys” (oboes) start to play, but where is the cauldron going?

Scholars believe that there were trap doors in the stage floor of Shakespeare's Globe Theatre which led to an under-stage area called the “cellarage,” which was nicknamed “hell.”

The stage floor was approximately five feet above ground level, so actors and stage hands could move around under the stage. The ghost of Hamlet's father likely ascended from and descended into “hell” through a trap door, and it’s possible that the cauldron—or even the Witches themselves—might have appeared and disappeared in the same seemingly magical manner.

The coming and going of the cauldron can be seen to symbolize the mystical, supernatural powers of the Witches.

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It is interesting to note that there are three weird sisters. Although they are used to represent the supernatural in Macbeth, the number three has religious significance in several religions and even in the Wiccan culture.

The witches give Macbeth three titles  - thane of Glamis, thane of Cawdor, and king.  

Traditionally witches stir pots or cauldrons containing animal and human body parts. The witches do not disappoint. The expected impact adds to the dramatic tension in the play and contributes to the ultimate decision that the audience or reader must make - whether Macbeth is responsible for his own downfall or whether he is controlled by "forces" outside the realm. The very contents of the cauldron will be used as a "charm" to further their aims and

everyone shall share i'th'gains. 

The witches create storms, hail, thunder, lightening, "fog and filthy air" and also sink ships, sail in a sieve, vanish without warning and generally create confusion and disorder. When upset, after a disagreement with a sailor's wife, one of the witches creates a windstorm and brags about  the “pilot’s thumb,” or small bone, she has as a charm. This should strike fear into anyone at the suggestion of her carrying human parts around.

Values in the witches world are reversed from the norm as "fair is foul." Their chanting is hugely symbolic in Macbeth and foreshadows what will happen in Scotland once temptation and ambition get the better of Macbeth and Lady Macbeth.

Darkness represents evil and fear and Macbeth refers to the witches as "secret, black and midnight hags."   

The witches are presumably female but their appearances deceive as they have beards, thus creating ambiguity. Not only do their words and actions cause confusion but even their appearance.  

Hectate, the weird sisters' senior and mentor, uses her power to catch "a vap'rous drop...." and "distill'd by magic sleights" she will cause more confusion in Macbeth's thoughts to ensure he puts "his hopes" above all rational thought.

The apparitions are strong symbols which encourage Macbeth to proceed with his plans. They give him confidence and courage but also add to his paranoia and obsession as he sees a vision of Banquo's sons which definitely does not form part of his own plan for the future. It is relevant to note that it is Macbeth's own insistence that the witches provide answers to his questions which encourages the witches to show him the apparitions and make him irritated and contribute toward his irrational behavior as he feels his issues are not suitably resolved.

The witches, significantly, do not appear towards the end. It is Macbeth's confusion that adds to the realism of the play as otherwise a play with strong supernatural influences may lose some of its efficacy. The witches absence also adds to the conclusion that Macbeth himself is responsible for his own downfall.

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