According to the Bible, God created the world through speech. God spoke and the world came into being. But just exactly how did God do that? How does speech, which is immaterial, create the world, which is material? And what is "speech" to a God who is utterly transcendent? It cannot be the same as human speech for it takes place in a nonmaterial realm and, furthermore, it is creative. The Sefer Yesira, which dates from the early centuries of this era and is actually a group of texts which has been edited into one whole, sought to provide some answers to these questions. According to Sefer Yesira, God's "speech" was not talking in the sense of someone speaking, but rather a manipulation of the letters of the Hebrew alphabet. These letters, Sefer Yesira teaches, are not merely linguistic symbols. They are real, having existence outside the human mind. They are made of a special spiritual substance and, hence, could be formed, weighed, shaped, etc. by God. Creation, then, was the process of shaping the letters so as to form reality. There are several accounts of this process in Sefer Yesira. One such account found in chapter 2, tells us:
Mishna 4: He set the fundamental twenty-two letters in a wheel, as a wall, in two-hundred-and-thirty-one gates. The wheel moves forward and backward. And the sign of the matter is: "There is nothing in goodness above pleasure and nothing in misfortune below a [leprousl lesion."
Mishna 5: How did He combine, weigh, and permute them? Aleph with all of them and all of them with Aleph. Bet with all of them and all of them with Bet. They turn round and round, and there are two-hundred-and-thirty-one gates. It results, then, that all creation and all speech go forth from one term.
The problem of this essay then, is: according to this text, just exactly how did God arrange the letters of the Hebrew alphabet, and how did God manipulate them so as to create the world?The Solution:
The Sequence and Arrangement of the Letters
Beginning with Mishna 5, the basic sequence of the letters can be reconstructed. According to it, each letter is to be combined with all the other letters but not with itself ("Aleph with all of them"). That would yield 462 pairs of letters (22 letters x 21 other letters). By deleting the mirror-images (e.g., AB, BA), one arrives at 231 basic pairs. Using the English alphabet, this method would yield the following sequence: AB,AC,AD . . . BC,BD,BE ... CD,CE,CF ... YZ.
Since, however, the text says that these pairs are to be made two ways -- forward ("Aleph with all of them") and reverse ("all of them with Aleph"), there are actually two sets of the 231 basic pairs. The basic sequence, then, consists of two sets of the 231 basic pairs. Using the English alphabet, the following sequences are generated: forward AB,AC,AD . . . BC,BD,BE ... CD,CE,CF ... YZ; reverse: ZA,YA,XA ... ZB,YB,XB . . . ZC,YC, XC. . . ZX. Figure 1 charts both sequences according to the Hebrew alphabet.
Given this basic sequence, how does one put it into its two forms, the wall and the wheel, as specified in Mishna 4? The word "wall," though apparently not well-attested in the early manuscripts, suggests a chart or a table and it is possible to display the basic sequence in the form of a chart. This has been done in Figure 1 where the lower line of each row represents the 231 basic pairs moving forward ("Aleph with all of them") while the upper line of each row represents the 231 basic pairs moving in reverse ("all of them with Aleph"). The word "wheel," in some texts rendered in the plural, suggests a wheel with spokes, or a star, and it is possible to display the basic sequence in the form of two such stars. This has been done in Figure 2, Figure 3, and Figure 4 (birdseye view) and Figure 5 (eye-level view) which present the 231 basic pairs moving forward, and reverse, reading clockwise.
The solution I propose would seem quite straightforward were it not for the history of the interpretation of this passage. First, Eleazar of Worms (l2th-l3th century) had a different method for assembling the 231 pairs: Each letter is combined with the one immediately following it; then, with the letter which is two letters from it; then with the letter which is three letters from it; until only the first and last letters can combine. Using the English alphabet, this generates the sequence: AB, BC, CD, DE ... AC, BD, CE ... AD, BE, CF ... AZ. Second Saadia, followed by pseudo-Abraham ibn Daud and various epitomizers, understood the Hebrew galgal ("wheel") as "sphere," i.e., as the celestial, diurnal sphere. According to these thinkers, the 231 basic pairs were set into the surface of the diurnal sphere which is in continuous rotation.[8 ] This mixture of Ptolemaic astronomy with neopythagorean magic vastly confused the issue though the tradition of a wheel, circle, or star can be found in Shabbatai Donolo, Judah Barceloni, and Eleazar of Worms.[9 ] It must be added though that, while charts of the basic sequence abound, I have found only one instance of a representation of it in the form of a circle or star. It seems to me, however, that the solution I propose of letters paired "forward" and "backward" and arranged in a "wall" (chart) and a "wheel" (circular diagram) is much simpler and much closer to the text.
The "Gates," The "Sign," and Their Function
Having deciphered the basic sequence (231 pairs of letters, forward and reverse) and the basic arrangement of the sequence (as a wall and as a wheel), the reader remains with the question: What exactly were the "gates" and how did they work? Saadia seems to have understood each pair of letters in the basic sequence to be a "gate." Eleazar of Worms seems to have understood the "gates" to be a series of letters, paired (or grouped) according to one of his magical alphabets. Thus, in his instructions on how to create and destroy a homunculus (Heb., golem), he seems to have required that these alphabetic series be recited in order: "`If for goodness' --[If] one comes to create some creature with them, one should recite them according to their order. And, if one wants to return it to the dust, one should [recite them in] the opposite order. . . . This is the meaning of `If in goodness....'" Neither interpreter, however, sought to define the "gates" by superimposing one set of the basic sequence upon the other, i.e., the forward ("Aleph with all of them") upon the reverse ("all of them with Aleph"). Such a definition may have been the intent of Judah Barceloni, who gives instructions on how to cut leather to make such a device. None of the interpreters claim to have a long tradition behind them and their efforts may, therefore, represent nothing more than individual medieval attempts to reconstruct the possible meaning of this early text.
The solution I propose to the problem of the "gates" is to set the 231 basic pairs of the reverse sequence above the 231 pairs of the forward sequence such that the second letter of the upper line of pairs coincides with the first letter of the lower line. This has been done in Figure 1. Each such grouping of two pairs, then, forms a "gate," there being 231 "gates" in the chart as is required by both Mishnayot. This suggestion has some basis in Barceloni's Commentary and in that of Isaac the Blind. It also conforms to the solution I propose to the problem of the "sign."
"And the sign of the matter is: `There is nothing in goodness above pleasure and nothing in misfortune below a [leprous] lesion.'"
The commentators, uniformly, construe this "sign" in a moralistic vein. Thus, Saadia, Donolo, Eleazar and the others interpret: "If you put your mind to this Book for `good,' [i.e.,] to amplify the exaltedness of God, `there is nothing greater than pleasure.' But, [if you put your mind to this Book] for `misfortune,' `there is nothing below a [leprous] lesion.'" The commentators also, uniformly, notice that the letters of the word for "pleasure" -- ONG -- yield the letters of the word for "[leprous] lesion" --NGO -- when the letter `ayin is put at the end, and not at the beginning, of the word.[16 ] This venerable tradition of commentary, however, is unconvincing. First, it introduces a moralistic note into an otherwise clearly magical and speculative text. Second, it fails to show how the quotation fulfilled the function for which the author-editor of Sefer Yesira designed it -- to act as a "sign" within the permutational scheme of the whole system.
I propose that the quotation, moralistic though it may sound, actually functions as a mechanical-magical "key," or "sign," by which the "correct" grouping(s) of "gates" can be identified. This can be shown as follows: The quotation contains four key words: TVBH (pronounced tovah, meaning "goodness"), ONG (pronounced `oneg, meaning "pleasure"), ROH (pronounced ra`ah, meaning "misfortune" or "evil"), and NGO (pronounced nega`, meaning "a [leprous] lesion"). Disregarding the meaning, note that three of these key words have three letters. The remaining key word ought, I think, to reflect the same structure. I propose, therefore, emending the text to read either TVB (pronounced tuv, meaning "goodness," this being the masculine form of the noun) or TBH (pronounced tovah, same meaning, but with "defective" spelling).
Set theory in mathematics provides that any four-unit set can be converted into a three-unit set when it reaches the form AB-BC, i.e., when the medial term of the two units is common. The key to the permutational system, therefore, lies in one or more of the following "gates": TV-VB (or, TB-BH), ON-NG, RO-OH, NG-GO. It is necessary, then, to move the elements in the chart until one or more of these key "gates" appears. To do this, we kept the letters on the lower lines of Figure 1 (the starting chart) fixed and moved the letters on the upper lines to the left. Thus, the "denominator" remained fixed while the "numerator" moved. Each such movement of the upper lines one unit to the left we called a "rotation," the pairs of letters going from position 1 to position 21 when they reached the edge of the chart. Further, there are 11 upper lines on the chart and we moved all of the upper lines simultaneously. After each move of one unit (i.e., after each "rotation"), the computer printed out a new chart. There are, thus, 21 charts (one for each rotation), each of which contains 231 gates, yielding a total of 4,851 possible gates. Actually, since we did not know which chart would yield the key words, the computer performed an additional (and, it turned out, an unnecessary) operation. After printing out the first set of 21 charts, the computer moved each lower line ("denominator") up one rank on the chart. This we called an "advance." After each advance, all 21 possible rotations of the upper lines were performed, yielding 21 charts for each advance. Since there are 11 advance positions and 21 rotation positions for each of the advance positions, the computer had to generate a total of 231 charts. Each chart has 231 gates, yielding a total of 53,361 gates. Had we, then, set the forward sequence above the reverse sequence (i.e., reversed the "denominator"and "numerator" lines) and repeated the whole procedure, we would have had to examine 462 charts with 106,722 gates.
To find the "correct" gates, the ones which fit the mechanical keys or "sign," it was necessary only to keep the lower lines on Figure 1 fixed and move the top lines 17 spaces to the left. This can be seen in Figure 6, where the "gate" NG-GO appears twice, at coordinates 3:12 and 3:10 (where it is upside-down). The other "correct" chart is generated from Figure 1 by moving the upper line 12 spaces to the left. This can be seen in Figure 7, where the "gate" TB-BH appears twice, at coordinates 2:3 and 2:7 (where it is upside-down). The quotation from the Mishna, then, functions to arrest the movement of the pairs of letters against one another at a certain point (or points). The "sign," thus, is not a moralistic preachment but a device for identifying the correct "wall." For when the proper key word has been reached, it acts like a tumbler in a lock, falling from the upper line to the lower line and locking the movement of the sets. Actually, since the key words appear rightside-up and upside-down, there is a "double- lock" effect in both "correct" charts. To be sure, there is also a "correct" setting (or settings) for the "wheel" or "star." This can be seen, for the NG-GO "gate" only, in Figure 8 and Figure 9 (birdseye view) and in Figure 10 (eye-level view).
The final question to be confronted is, how was this chart (or charts) used? We not know. The text of Sefer Yesira does not tell us. None of the other early rabbinic gnostic-mystical or later scientific texts tell us, though some mention the use of the alphabet in creation. I do not know of any Hellenistic or early Islamic parallels that might explain such usages. The attempt to adjucate the merits of the two proposed emendations also sheds no light on the subject and it is even possible that both charts were meant to be used.[22 ] Personally, I favor the solution of using only the NG-GO chart because it requires no emendation of the texts. In any case, we still have no idea at all of how the "correct" chart (or, charts) was used. It does seem clear, though, that the author-editor of Sefer Yesira intended this pericope to be an account of the speech of God, possibly reflecting the magical practices of his day.
Even in the exciting world of magic and cosmogonic speculation, it is passing strange that the Golem of twentieth-century science should have come to the aid of the researcher to explain something of the mighty deeds of the Creator.
[*] This material first appeared in my Understanding Jewish Mysticism (New York: Ktav Publishing: 1978) 22-29. It was subsequently published as an article in History, Religion, and Spiritual Democracy: Essays in Honor of Joseph L. Blau, ed. M. Wohlgelernter (New York: Columbia University Press, 1980) 114-29. I reproduce it here with expansions and clarifications. My thanks to the Information Technology Division of Emory University for their help in putting this material into web-ready form.
 For the literature on Sefer Yesira, see G. Scholem, Major Trends in Jewsish Mysticism (N.Y.: Schocken, 1941; hereinafter Trends), 75-78, and idem., "Kabbala," Encyclopedia Judaica (Jerusalem: Keter, 1971) 10:507. See also G. Vajda in Introduction à la pensée juive du moyen âge (Paris: Vrin, 1947) and his studies in Revue des études juives, vols. 7, 10, 12, 13, and 16. (The reference in 10:87, n. 32, to Haberman's article in Sinai should be to vol. 20 thereof and not to vol. 10). There is a translation of the Sefer Yesira in German and several into English (see Trends, 427). See my own translation and explicatory commentary in Understanding Jewish Mysticism (N.Y.: Ktav, 1978) 12-46 as well as the translation and commentary of A. Kaplan, Sefer Yetzirah: The Book of Creation (York Beach, ME: Samuel Weiser, 1990).
The second-century origin of Sefer Yesira proposed by Scholem has been hotly contested by J. Dan (see e.g., "Three Phases of the History of the Sefer Yezira," Frankfurter Judaistische Beiträge, 21  7-29). He argues that the silence about Sefer Yesira in pre-tenth-century Jewish texts argues against an early date. In addition, and perhaps more probatively, Dan argues that Sefer Yesira most resembles a rationalistic scientific text of the kind that was produced in the eighth and ninth centuries (16-17). Finally, he argues that Sefer Yesira was adopted as a "mystical" text only later in the twelfth century (18-22), a tradition that continued into the modern period including Scholem. There is much merit to his argument which I could not have seen when I wrote about Sefer Yesira in 1978-80.
 The manuscript tradition of Sefer Yesira is very irregular. See I. Grunewald, "A Preliminary Critical Edition of Sefer Yesira," Israel Oriental Studies, 1 (1971) 132-77 for an attempt at a critical text. However, N. Séd, "Le Sefer Yesira," Revue des études juives, 132 (1973) 513-28 contains a strong critique of Grunewald's theory and a coherent summary of the various manuscript traditions. The "best" manuscript appears to be Vatican, Hebrew, 299ff. 66a-7lb. To allow easy access to the Hebrew original, I have chosen to translate the text ostensibly used by the Vilna Gaon which appears in Sefer Yesira (Heb.), ed. unknown (Jerusalem: Lewin-Epstein, 1964-65), part 3, 25-26 (hereafter SY). The Mantua edition (reprinted. ibid., at the end) has slightly different readings.
 Some manuscripts, including the Vatican Heb. 299, and Grunewald, Critical Edition, 148, omit the reference to the wall. Others include it (L. Goldschmidt, Das Buch der Schöpfung [Frankfurt: Kaufmann, 1894] 55). Some read 221, and not 231, gates (see below). The wording of the quotation also varies: Some read "if in goodness there is nothing" and some read "above" twice (see below).
 The nature of this sequence was already clearly seen by Saadia Gaon (10th century) in his Commentary to the Sefer Yesira (edited and translated into French by M. Lambert, Commentaire sur Sefer Yesirah (Paris: E. Bouillon, 1891); edited and translated into Hebrew by Y. Qafih, Sefer Yesira, Kitab al-Mabadi, im Perush R. Saadia (Jerusalem: Deror, 1972). In the Commentary, Saadia notes that his manuscripts read "221 gates," but he argues against that reading on mathematical grounds. He then explains the basic sequence, calling each pair a two-lettered "word" (see Qafih, 118-19, n. 42, where all this is clearly set forth). It seems also to have been envisioned by Isaac the Blind (see G, Scholem, Hakabbalah be-Provence [Jerusalem: Akadman, 1953] p. 10 of the Appendix).
aleph A lamed L
bet B mem M
gimel G nun N
dalet D samekh S
hey H ayin 0
vav V peh P
zayin Z tsade C / [Sigma]
het X qof Q
tet T resh R
yod I shin $ / [psi]
kaf K taf & / [phi]
 Eleazar's calculation for the 221 pairs follows a similar principle: Each letter is listed; every other letter is listed; every third letter is listed; every fourth ... every eleventh, every twelfth ... For the purpose of this sequence, the alphabet is viewed as a continuous, repeating ribbon. Using the English alphabet, the sequence is: ABCD ... ACEG ... ADGJ ... AM ... ANOB. See his Commentary to Sefer Yesira (Przemysl: M. Spiro, 1888) 5a-b and 17a-20a for the sequences and 4b-5a and 15b-17a for the number of gates. See now Kaplan, 112-20, for these charts. See also Kaplan's geometrical correspondences (110-11) though, in my opinion, such speculations complicate unnecessarily the study of this already complicated text.
 For Saadia, see ed. Derenbourg, 50-55 (Arabic), 73-78 (French); ed. Qafih, 84-88. For pseudo-Rabad, see SY part I, 79-80 in great detail. An example of a modern epitomizer would be I. Kalisch Sefer Yezirah (New York: Frank and Co., 1877), 49.
 For Shabbtai Donolo, see SY part 2, 134-35. For Judah Barceloni, see Commentar zum Sepher Jezira von R. Jehuda b. Barsilai, ed. S. J. Halberstam (Berlin: M'kize Nirdamim, 1885), 209-10 in great detail. He seems to have envisioned actually a sphere within a sphere. For Eleazar of Worms, see Commentary (ed. Przemysl) 4b, 15b.
 He thus interprets "amplifying the exaltedness of God" as the performing of a magical, creative act using the same magical alphabets God used (ed. Przemysl, 4b-5b.) See also Scholem, Ursprunge, 26. See also Commentary (ed. Przemysl) 15b-22a. See the excellent essay by G.Scholem, "The Idea of the Golem," On the Kabbalah and its Symbolism (New York: Schocken, 1965) 158-204 with special attention to 165-73, 184-95 as well as the full-fledged study of M. Idel, Golem (New York: SUNY Press, 1990). For Saadia, see above.
 Barceloni, Commentar zum Sepher, 210. An interesting parallel may be the text in "Hilkhot ha-Kisse," Merkabah Sheleymah, ed. S. Mussajoff (Jerusalem: 1922; reprt, Jerusalem: Makor, 1972), 2lb: "All the letters were called before God in pairs, except Lamed (sic)."
 Scholem asserts that Sefer Yesira is a magical as well as a speculative text and connects it with the magical traditions in talmudic literature (cf., e.g., Ch. 1, Mishna 4, and the last Mishna of the text together with my commentary in Understanding Jewish Mysticism) though recall Dan's objections to that (cited above). Sefer Yesira most certainly was viewed as a magical text by Eleazar of Worms and his medieval followers (Scholem, Trends, ch. 2, "The Idea of the Golem," etc.). Scholem, however, appears to have misinterpreted these pericopae as linguistic lessons (Ursprung und Anfänge der Kabbala [Berlin: de Gruyter, 1962], 29 and "The Name of God," Diogenes, 79  75).
 Heb.: 'eyn be-tovah le-ma`alah me-`oneg, ve-'eyn be-ra`ah le-mattah mi-nega`. Alternate reading: 'im be-tovah, le-ma`alah me-`oneg; v-'im be-ra`ah, le-mattah mi-nega`. Conflated reading: 'im be-tovah, 'eyn le-ma`alah me-`oneg; ve-'im be-ra`ah, 'eyn le-mattah mi-nega`. Grunewald, Critical Edition, 148, shows the many variations. The English here follows the first reading. The alternative reading yields: "If in goodness, it is above pleasure and, if in misfortune, [it is] below a [leprous] lesion." The conflated reading yields: "If in goodness, there is nothing above pleasure and, if in misfortune, there is nothing below a [leprous] lesion." As we shall see, it makes no difference which reading is used.
 It must be clearly stressed that the author-editor of the Sefer Yesira had only to draw a maximum of 18 charts containing 4,158 gates to reach his goal. This was not an overly burdensome task. Note, too, that the number 18, when transmuted into Hebrew letters by gematria (10 = yod, 8 = het) yields the word hai, which has the meaning "alive." The letters of this word are, to this day, worn as a piece of jewelery (subconsciously, an amulet) by many Jews.
 I am deeply indebted to Mr. Russ Burns of the computer graphics program at Brown University for his patience and skill in generating these programs, especially on such short notice. My thanks, too, to Professor Sol Bodner, Visiting Professor of Engineering, who photographed the graphic displays. The "wheels" displayed in Figures 2-5 and 9-10 could be moved manually and, when doing so for the eye-level views (Figures 5 and 10), the patterns slipped in and out of perspectival infinity, which was quite stunning.
 The emendation to TV-VB has one advantage and one disadvantage. The advantage is that all the key words except NG-GO then appear only in the upper line (Figure 6) as follows: TV (6:8), VB (2:21); ON (10:17), NG (3:12), and RO (11:7), OH (5:5). One can then read the sign as follows: "All the key words before the word `below' are `above' (i.e., in the upper lines) while the word that occurs after `below' (i.e., NGO) is the sign (because it appears in the lower line)." The disadvantage to this emandation is that it destroys the stylistic unity of the quotation. The emendation to TB-BH also has one advantage and one disadvantage. The advantage is that it preserves the stylistic unity of the quotation: the two-syllable structure of the key words, the internal rhyme, and the alternating masculine-feminine form of the nouns. The disadvantage is that the words "above" and "below" lose their meaning, since NG-GO is not contained in the upper line only. In neither case is there support from the manuscripts. In both cases, it may be that we are dealing with a deliberately deceptive style which is a common element in magical texts.
 There appear to be two other magical alphabetic sequences in the Sefer Yesira: the sequence of "mothers," "doubles," and "simple letters," each of which "rules" over several aspects of existence (chapters 3, 4, 5) and the sequence of alphabetic permutations called "stones" and "houses" (at the end of chapter 4). They are discussed in Understanding Jewish Mysticism. See also N. Séd, "Le Sefer ha-Razim et la méthode de `Combinaison des Lettres,'" Revue des études juives, 130 (1971) 295-304. Scholem, too, has pointed out that the text of Sefer Yesira contains disconnected elements.
 This name for the computer was coined by Scholem at the dedication of the computer in Rehovot, Israel. That speech is included at the end of his The Messianic Idea in Judaism (Schocken, N.Y.: 1971).]
It may seem strange and even unwarranted to have utilized twentieth-century mathematical analytic tools on a second-century magical, or eighth-century speculative, text. I must, therefore, stress again that, for the author-editor of Sefer Yesira, the formulation of the chart and the circles was conceptually and practically simple and rather straightforward. It is only we who had to utilize extended tools of analysis to arrive at his conclusions. I wish to acknowledge with thanks the criticism and cautions of my teacher, Professor G. Vajda, and my colleague, N. Séd, in formulating these issues.