Next week, we read Toledot as the weekly portion. Bar Ilan University's faculty of Jewish studies has an excellent series on different interpretations of the portion.

Here is the Talmud Department's Ezra Shvat on the mathematics of it all

Daf Shvui , Number 107

Parashat Toldot, 5756

The "Infinite" Value of the Total of the Angles of the Square

"And Isaac appealed to the Lord on behalf of his wife"(Gen. 25,21), ("on behalf of"- Hebrew: lenochach - can also mean: facing or opposite). Rashi comments: "he stood in one corner and prayed while she stood in the other corner and prayed".

We do not know the midrashic source for this comment by Rashi. However the structure of one corner opposite another exists as a form of prayers to ask for mercy in several Talmudic sources, as follows:

Ta'anit 23b: (Abba Chilkiah, the grandson of Choni Hameagel) said to his wife... Let us go up to the roof and pray ... he stood in one corner and she in another corner...

Tosefta Berachot, Chap 3: (Lieberman Edition, according to Ms. Vienna): Rabbi Yehudah said, when Rabbi Akiva prayed by himself a man would leave him in one corner and find him in another corner because of all the kneeling and bowing that he did (see also Berachot 31a).

Tanchuma, Ki Tissa, 20: "Moses did not leave any corner ... while seeking prayer and mercy. (see also Devarim Rabbah on Deut. 9, 26; Kohelet Rabbah on Eccl. 4,3).

Possibly the archetype for all such sources is connected to the service of the Temple in the burnt offering or peace offering sacrifices about which the Torah says: "and sprinkle the blood round about upon the altar" (Lev. 1,5). This is carried out by means of "two sprinklings which are four" (Zevachim, chap. 5, Mishnah 4), which means sprinkling the blood once on the northeast corner of the altar and once on the southwest corner. This structure of "two that are four" plays an important role in the logic of the ordering of oral laws in the Mishnah (see: beginning of Shabbat, Shavuot, and Negaim). These are laws which can be divided into four possible sides wherein each pair of the sides to the square has something in common.

These sacrifices have a transcendental quality which brings about a connection between he who brings the sacrifice and G-d (unlike the sin-offering which is no more than payment of a debt and a return to a static situation). Perhaps this is the meaning of the semiotic use of the corners in the above mentioned sources. It can be explained that man and the reality that surrounds him are limited by the dimensions of time and place. At any given time it is possible to exist only in one place. The corner enables man to be in two places (two directions) at once, thereby creating the conditions for a spiritual uplifting which makes possible a contact with the infinite which is not limited by those dimensions. For this reason the sprinkling of the blood takes place at one and the same time on two sides of the altar.

This idea is expressed in the semantics of the Bible in reference to the word "yerech". In its singular form it means - opposite one direction (or side). Such as in reference to the walls of the Mishkan (Ex. 40, 22,24) or the camp (Numbers 3,35) or, in the present example, the altar of the burnt offering sacrifice (Lev. 1,11). However, in its dual form (yarketayim, yarkete) it does not mean two directions (or sides), but the juncture between them - at the corner (Jonah 1,5; Isaiah 14,15; Exodus 26,23, see also Rashi on verse 22 there). In this role the word takes on a meaning of something lofty and distant, "the ends of the Earth" (see Radak on 2nd Kings 19,23), a meaning which has infinite value or at very least a value which is not within the range of conscious visibility (see also: Judges 19,18: Jeremiah 6,22; ibid. 25,32; Ezekiel 38,6; Psalms 48,3). Perhaps the conscious field of vision is seen in the Bible as a limitation and the "far side" (= yarketayim both dual and plural form of yerech) has some meditative value which makes possible a kind of rising above or distancing beyond this limitation.

Thus, upon the desire on the part of Isaac and Rebekah to overturn the reality of their barrenness by means of prayer the Talmud compares their prayer (Vaye'etar) to a pitchfork (eter): "Just as the pitchfork turns the grain over from place to place so the prayer of the righteous overturns the attributes of the Holy One blessed be He from the quality of anger to the quality of mercy" (Yevamot 64a). This turnover is brought about when man ascends to communicate on the infinite, divine "wavelength" which is expressed in this world by the limitless "far side" (=yarketayim).

However, it must be pointed out here that this ascent, in the prayers of Isaac and Rebekah, is not brought about through one corner alone, but by the union of the two, one opposite the other, the one of the man and the other of the woman (as in the story of the grandson of Choni, mentioned above) This union is a part of the process of the creation of man in the sense of "Your wife is like a fruitful vine in the recesses of your house (yarketey betecha)" (Psalms 128,3). This is compatible with the transcendental concept of mating in which the union of the two partners needs the participation of a third one, the Holy Presence (Kiddushin 30b; Niddah 31a).

One must read the commentary of Rashi here very carefully. He does not mean to say that Isaac and Rebekah prayed each in his or her corner with their backs towards each other (as the author of "Siftei Chachamim" understood). On the contrary, he draws a picture of a union of the two corners, one opposite the other as Rashi explains in detail in his commentary to Yevamot 64a commenting on the word "Vaye'etar": "... that both of them raised up their voices one opposite the other." (Rashi wishes to strengthen the assumption of Rabbi Yitzchak that both of them were barren and they had to pray for each other). This is also the straightforward meaning of the expression "lenochach ishto - opposite his wife" (see: Numbers 19,4; Lamentations 2,19). The preference of the general term "his wife" instead of the use of her given name, Rebekah, emphasizes the "couple" aspect of the occasion (with perhaps a hint at Psalms 128,3).

It is possible that the semiotic structure, the purpose of which is to effect the union of male and female to create a third entity, and to bring about spiritual ascent and the revelation of the Divine Presence, is not only the union of two corners but primarily the union of two triangles to create a new form - a square. This pattern is discernible in many places in the Bible such as the prophecies of Amos and the proverbs of Agur (Proverbs 30,15-33), and its meaning is the turning of relative truth, that is "the common wisdom" (three), into absolute truth, stable and unchangeable (four). Similarly, G-d does not punish a community unless they continue in the evil ways of their ancestors, immutably to the extent of four generations as it is written: "unto the third and fourth generation of those who hate Me". (Exodus 20,5; see Ramban there; also Genesis 15,16). The Holy Presence is not motivated by "the common wisdom" but by absolute, unchangeable truth.

This is the nature of the prayer of Isaac and Rebekah. Each one is but an angle of a tri-angle, an unstabe reality tending toward death and thus - a lack of eternity. However, in the union of the two they ascend and co-opt the Holy Presence thereby creating the dimension of the square the value of which exceeds the combined values of the two triangles, thus attaining infinity.