Gemara

(Beraisa - R. Akiva): One may never make a tannery in the west of a city, for it (the west) is constant.

(Rav Nachman): The west constantly hosts the Shechinah.

(R. Yehoshua ben Levi): "U'Tzva ha'Shamayim Lecha Mishtachavim" teaches the direction for prayer (the stars travel westward).

Objection (Rav Acha bar Yakov): Perhaps the Shechinah is in the east. They bow to His Presence while departing, like a servant walking (backwards) away from his master!

This is left difficult.

Rav Sheshes holds that the Shechinah is everywhere. He was blind. he told his servant 'point me in any direction to pray, except for eastwards.'

The Shechinah is also in the east. Rav Sheshes did not want to pray eastwards, because Minim (heretics) pray eastwards.

25b (R. Yitzchak): To grow wise, one should face southwards when praying. To grow rich, one should face northwards when praying.

A way to remember this is that (in the Heichal) the Menorah was in the south, and the Shulchan was in the north.

(R. Yehoshua ben Levi): One should always face south. Wisdom will make him rich - "Orech Yamim bi'Minah bi'Semolah Osher v'Chavod."

Question: R. Yehoshua ben Levi said that the Shechinah is in the west (so one should face west)!

Answer: He faces west, and tilts to the south.

R. Chanina (to R. Yoshiyah): You live (in Bavel,) to the north of Eretz Yisrael. You should face southwards.

Berachos 28b (Mishnah): If one cannot come down from his donkey to pray, he turns to face (towards Yerushalayim). If he cannot, he directs his thoughts to the Kodesh ha'Kodashim.

30a (Beraisa): One who is blind or does not know the direction (to the Mikdash) directs his thoughts to his Father in Heaven -- "v'Hispalelu El Hash-m."

In Chutz la'Aretz, one faces Eretz Yisrael -- "v'Hispalelu Elecha Derech Artzam." In Eretz Yisrael, he faces Yerushalayim -- "v'Hispalelu... Derech ha'Ir." In Yerushalayim he faces the Mikdash -- "v'Hispalelu El ha'Bayis ha'Zeh." In the Mikdash he faces the Kodesh ha'Kodashim -- "v'Hispalelu El ha'Makom ha'Zeh." In the Kodesh ha'Kodashim he faces Beis ha'Kapores (the wall in back of the Aron). In back of Beis ha'Kapores he (faces west, towards Beis ha'Kapores, but) considers himself as if he was in front of it.

Conclusion: Wherever one is, he faces towards Beis ha'Kapores. All of Yisrael face one place.

(R. Avin): We learn from "... Tzavarech Banuy l'Salpiyos" -- the Tel (mound) towards which all Piyos (mouths) turn.

Rishonim

The Rif (Berachos 20b) brings the Gemara in Berachos.

Rosh (4:19): Even if there is no danger, one may pray while riding, for the delay would disturb him. He need not turn his face towards Yerushalayim. R. Yosef holds that this is only if there is danger. We hold like our Sugya, unlike the Sugya in Bava Basra in which Tana'im and Amora'im argue about whether the Shechinah is in the east or west. Also there, the conclusion is to pray towards Eretz Yisrael. We face east because we are west of Eretz Yisrael.

Poskim

Shulchan Aruch (OC 94:1): In Chutz la'Aretz, one who rises for Shemoneh Esre faces towards Eretz Yisrael, and intends also for Yerushalayim, the Mikdash and the Kodesh ha'Kodashim. In Eretz Yisrael, he faces Yerushalayim and intends also for the Mikdash and the Kodesh ha'Kodashim. In Yerushalayim, he faces the Mikdash and intends also for the Kodesh ha'Kodashim. In back of ha'Kapores he turns his face towards ha'Kapores.

Source: R. Yonah (20b), cited in Beis Yosef (DH Kosav).

Magen Avraham (1): In back of ha'Kapores means west of the Beis ha'Mikdash.

Shulchan Aruch (2): If one prays in a different direction, he turns his face towards Eretz Yisrael, Yerushalayim or the Mikdash, if he was in Chutz la'Aretz, Eretz Yisrael or Yerushalayim, respectively.

Magen Avraham (2): The Shulchan Aruch discusses a place where the custom is to pray in a different direction, or one who prays on a donkey.

Beis Yosef (DH u'Mah): Mahari Avuhav was astounded why we pray to the north and south. We should face east, like the Tur says! It seems that we should turn our faces east when we pray to the north or south, like it says in Bava Basra that one who prays to the north or south (to get rich or wise) should turn his face towards Eretz Yisrael. Rashi explained that he faces a bit south, i.e. he faces east, and tilts a bit south to get wise or north to get rich. Since we find that tilting the face helps, if one prays to the north or south, it suffices to turn his face towards Eretz Yisrael. Semag says 'we are west of Eretz Yisrael, and face east. Even though it says that one who wants to get rich or wise should pray to the north or south, he must turn his face towards Eretz Yisrael like it concludes there.'

Taz (1): One who prays to the west cannot turn his face to the east. The Gemara discusses tilting the face to the side, but not in back. This is why Mahari Avuhav mentioned north and south, but not west. Even though the Shulchan Aruch discusses Stam another direction, it does not refer only to us, who are west of Eretz Yisrael and must face east. Rather, it is a general rule. Whenever one is, he must turn his face to Eretz Yisrael.

Rema: We face east because we are west of Eretz Yisrael. We do not make the Aron and the direction of prayer where the sun rises, for Minim do so. Rather, we face midday.

Taz (2): The Levush says that since we are northwest of Eretz Yisrael, the Aron should be southeast. If it is in the east, we do not pray towards Eretz Yisrael, and we do like the Minim!

Mishnah Berurah (9): If the Aron cannot be fixed in the east, it should be fixed in the south. Surely it should not be in the west, for then people will Daven with their backs to the Aron! Even if the Aron is in a different direction, one must face east.

Kaf ha'Chayim (9): Some say to face east even if the Aron is in the south and the Tzibur faces south. Others disagree. How foolish are those who put the Aron in the wrong place to 'beautify' the Beis ha'Keneses!

Mishnah Berurah (10): If one prayed to north or south and they alerted him, he may not uproot his legs to face east. Rather, it suffices to turn his face to east, even if he prays with others facing east. If he cannot, or he faces west, he should intend for the Kodesh ha'Kodashim. However, if he is in a Beis ha'Keneses and many see it, the Pri Megadim says that he turns to face like the Tzibur. If the Aron is in the south and people pray to the south, even though this is wrong, he should do so also, but he should turn his face to the east.

Mishnah Berurah (11): We do not face the midday sun, rather, where it is an hour or two after sunrise in Nisan and Tishrei (slightly south of east). All Acharonim say like the Levush, that if we are northwest of Eretz Yisrael, we face southeast. Every country should do according to its location. Lechem Chamudos justifies the custom (to face east), but he agrees that it is better to do like the Levush. If the Aron is in the east, one faces east and tilts his face southeast, unless he is south of the Aron, lest it look like his back is to the Aron. In such a place, the Rav's place should be north of the Aron, so he can face south and grow wise.

Rema: If one wants to fulfill 'to grow rich, one should face north. To grow wise, one should face south', in any case he should point his face east.

Taz (3): One can also face east and point his face north or south (Rashi, brought in Beis Yosef). This is better, for then he does not look like he Davens unlike the Tzibur.

Kaf ha'Chayim (12): The Perishah (4) says that it is better if his legs face north or south and his face faces east, to fulfill "v'Hayu Einai v'Libi Sham."

Mishnah Berurah (13): Where the Aron is in the southeast, there is no need to tilt to the south. One may not tilt north, for this is like turning away from Eretz Yisrael.