parshat Vayera: Avraham's Special Tent

A famous midrash relates that Avraham had a special tent with entrances to each compass direction so that he would be able to see and accept guests into his home from whatever direction they came. I claim (almost) every midrash is derived from a close analysis of the text. What is the derivation of this midrash?

It would seem that the midrash is derived from the beginning of this week's parsha, Vayera. We read (Bereishit 18:1-2)

א וַיֵּרָא אֵלָיו ה, בְּאֵלֹנֵי מַמְרֵא; וְהוּא יֹשֵׁב פֶּתַח-הָאֹהֶל, כְּחֹם הַיּוֹם.	1 And the LORD appeared unto him by the terebinths of Mamre, as he sat in the tent door in the heat of the day;
ב וַיִּשָּׂא עֵינָיו, וַיַּרְא, וְהִנֵּה שְׁלֹשָׁה אֲנָשִׁים, נִצָּבִים עָלָיו; וַיַּרְא, וַיָּרָץ לִקְרָאתָם מִפֶּתַח הָאֹהֶל, וַיִּשְׁתַּחוּ, אָרְצָה.	2 and he lifted up his eyes and looked, and, lo, three men stood over against him; and when he saw them, he ran to meet them from the tent door, and bowed down to the earth,

So Avraham was sitting in front of his tent, and happens to lift his eyes and see men approach. The narrative continues detailing Avraham's great hospitality, and so the assumption that Avraham would have such a special tent for the purpose of hospitality, and that this is why he was in fact waiting outside his tent, is partially justified.

However, I believe that the specific detail of the 4-door tent (which is then fit into the greater context of hospitality) comes from the words כְּחֹם הַיּוֹם, "in the heat of the day." There are (at least) two possible interpretations of this phrase. One is that כְּחֹם הַיּוֹם is modifying the verb יֹשֵׁב. Thus, his sitting is done in the heat of the day. This is a time or location specifier.

The other is that כְּחֹם הַיּוֹם modifies פֶּתַח-הָאֹהֶל. That is, פֶּתַח-הָאֹהֶל is what specifies the location of Avraham's sitting, and כְּחֹם הַיּוֹם specifies which tent door. It is as if the word אֲשֶׁר were there:
וְהוּא יֹשֵׁב פֶּתַח-הָאֹהֶל אֲשֶׁר כְּחֹם הַיּוֹם - "and he was sitting by the tent door which was in the heat of the day."

That the pasuk would need to specify which tent door implies that there was more than one. Further, the tent door that faces the heat of the day presumably faces East. Thus, there is at least one other door, and it presumably faces in another compass direction. Since there are four compass directions, we might deduce that their were four doors, each facing a different compass direction.

Why construct such a tent? Well, we see Avraham's great hospitality, and one can read into the story that he was on the lookout for guests, and thus we have an explanation for the four doors.

This reconstruction of midrashic analysis done without seeing the midrash inside. Perhaps I would have a different explanation then...

Update: Note that the other popular midrash, that Hashem put out the heat of the sun to keep away guests, and Avraham sat out in the heat of the sun regardless, hoping that guests would come, such that Hashem send angels, is based on reading כְּחֹם הַיּוֹם as modifying the verb rather than the location.

Update: And see this post, about the possible specification of a tent door facing another direction, from the pasuk:

י וַיֹּאמֶר, שׁוֹב אָשׁוּב אֵלֶיךָ כָּעֵת חַיָּה, וְהִנֵּה-בֵן, לְשָׂרָה אִשְׁתֶּךָ; וְשָׂרָה שֹׁמַעַת פֶּתַח הָאֹהֶל, וְהוּא אַחֲרָיו.

10 And He said: 'I will certainly return unto thee when the season cometh round; and, lo, Sarah thy wife shall have a son.' And Sarah heard in the tent door, which was behind him.--