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Why did the Israelites build a golden calf in the desert? The question is, why a calf?
Why not a golden cat, a golden eagle, a golden lion, golden parachutes or a gold bar?
Of all the images they could build, why a calf?
Note:
One of the theories at http://ancienthebrewpoetry.typepad.com/ancient_hebrew_poetry/2008/04/elyon-bull-el-a.html is that the original word for "sons of Israel" is "bull el". Bull and calf are kind of related.
One answer I learned, heavily steeped in aggadah, was that there was no intent to build a calf. One of the things thrown into the molten gold was something that had the words "aleh shor" written on it, which was previously used to raise Joseph's bones from the Nile (as Joseph was referred to as an ox, or compared to, I forget) so a cow form arose. That is why Aaron in Ex 32:24 says that the calf emerged, not that it was designed as a calf.
For more on this explanation, read http://www.jewishvirtuallibrary.org/jsource/judaica/ejud_0002_0007_0_07514.html and scroll down to "In the Aggadah."
I also learned that the calf was used because it was reminiscent of the Egyptian deity-form so the people would accept it.
Rashi (in addition to citing the above Aggadah), on 32:4 says that Egyptian sorcerers created the shape through sorcery:
As soon as they had cast it into the fire of the crucible, the sorcerers of the mixed multitude who had gone up with them from Egypt came and made it with sorcery. [See commentary on Exod. 12:38.] Others say that Micah was there, who had emerged from the layer of the building where he had been crushed in Egypt. (Sanh. 101b). In his hand was a plate upon which Moses had inscribed “Ascend, O ox; ascend, O ox,” to [miraculously] bring up Joseph’s coffin from the Nile. They cast it [the plate] into the crucible, and the calf emerged. -[from Midrash Tanchuma 19]
For starters: anything about idol worship is going to sound a bit weird to us today.
There are a couple of conjectures out there; the simplest is that cattle were a sign of prosperity (you use them to work, and they give you food), so an idol of that form was very popular. (I'm told we also find it in archeological digs.) Note that 500 years later, King Jeroboam wanted to make his own pseudo-Temple to keep people away from Jerusalem, and thought "hm, how do I make this more popular? I know! I'll put in a golden calf! Hey better yet, why not two?" And a calf, being young, means a promise of future prosperity. Note that if there's an unsolved murder, we take a young calf and publicly kill it -- showing how all that potential promise of someone's life was suddenly ended.
The kabbalistically-inclined Nachmanides observes that in Ezekiel Ch. 1, he sees angels surrounding God, and the one on the north side looks something like a calf; the Hebrews in the desert were particularly concerned of a dangerous hot north wind, so that was believed to help.
According to the Midrash, and some texts of the Kabbalah, ( I don't remember the exact sources), when the Jewish people are described as "seeing Gd's feet", what they saw, was the Merkava (chariot) from Ezekiel.
On one of the sides of the Chariot is a bull. That bull represents Mercy. And since they believed Moshe to have died, they were hoping for the Gd of Israel to be full of Mercy, since they would now be lost.
Another idea that I have heard from year to year, is that the Calf represented a new leadership from the Tribe of Yoseph. (Yoseph is compared to a Bull, so Ephraim and Menashe would be calves) This "battle" of Yoseph(Ephraim or Menashe), vs Yehudah, vs Levi plays out throughout tanach, from Bereshit, until Chronicles and the later Prophets. As to why at that point in time they would prefer the style of leadership that Yoseph granted, it is because Yoseph was known for relieving people of burdens, and helping them through touch economic times. It was a more worldly sort of leadership, based on certain type of equality. (Going through all the points for this would be beyond the scope of this answer)
Another approach mentioned in Let My Nation Serve Me by Rabbi Deutsch (published by Artscroll) relates to astrology. For those unfamiliar with the twelve zodiac signs, they are Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces. In order, they are the sheep, bull, twins, crab, lion, woman, scales, scorpion, archer (specifically his bow), goat, water-drawer (specifically his water bucket), and fish. The cycle begins in Nissan and ends in Adar; this is why many sources discuss Adar Beis as not being subject to the influence of the stars (see the discussion of the war with Amalek in the same book).
I'm sure you see where I'm going with this. The sheep was the god of the Egyptians. When Egypt, represented by Aries, fell, the Jews turned to the next Zodiac, the bull.
Many other explanations are presented there, though off the top of my head I believe they've all been covered by the other answers.
