Tuesday, June 7, 2011

The Fractal Unity of Torah

About fifteen years ago, I was extremely mystically rather than rationalistically inclined. Related to this was my desire to see patterns in everything - especially fractal patterns. These are patterns that repeat at different scales, which appear in various forms in nature (such as fern leaves), and in a mathematical construct known as the Mandelbrot Set - which I eventually used for the cover of The Science of Torah.

I was therefore thrilled when, at that time, I came across an obscure and difficult book by Rabbi Yehoshua Honigwachs called The Unity of Torah. This work set out to answer a question that, to my astonishment, nobody had ever asked before: What is the overall structure of the Torah? Rabbi Honigwach's answer was that the structure of Torah reflects a five-stage approach to its central goal: Taking man from the extremes of egotism, in which society is an anarchy of selfish elements, towards a state of unity between all men and God. This pattern, argued Rabbi Honigwachs, is found in each of the five commandments on the Tablets of Law; it is also found amongst the five books of the Torah; within these books; within each fifth of these books; and so on - although, he noted, there are some places where a different version of this pattern is found, and there are others where it is not found at all.

Needless to say, I was very much taken with this fractal pattern. I figured out a way to diagrammatically illustrate it in ways that made it easier to comprehend; I sent my illustrations to Rabbi Honigwachs, who was pleased at the result. In my book The Science of Torah, I made much use of fractal patterns, and I decided to include an overview of Rabbi Honigwach's approach in an appendix. You can freely download the appendix at this link, and it is also aided with the following schematic:

But my mentor Rabbi Aryeh Carmell was very unhappy with my interest in such patterns, being skeptical of this sort of pattern-seeking in general. I, on the other hand, was obsessed with it! We worked out a compromise, whereby I clearly restricted such discussions to distinct parts of the book, and he made it clear in his approbation that he was not supportive of those parts.

Today, I realize that pattern-seeking is both one of man's greatest strengths and one of his greatest weaknesses. In evolutionary terms, it was a helpful skill to learn; it enabled man to spot a camouflaged predator, and to discover causal relationships in the natural world. But, as a result, people also have a powerful propensity to see patterns even when none exist, and to ascribe causality even where there is none (as in much religion, and alternative medicine). And the thrill of discovering patterns can easily hamper one's objectivity. Of course, there are also those who deny patterns and causality even where they really exist (such as liberals with regard to Islamism, and smokers with regard to the dangers of smoking); but the former problem seems more innate and pervasive.

So what about this fractal pattern in the Torah? I don't have the time, or the objectivity, to study it all again and re-evaluate it. For a variety of reasons, I am certainly inclined to be skeptical. Nevertheless, I remain impressed at how Rabbi Honigwachs freely admitted that his pattern does not work in all places, which indicates a high degree of objectivity, and presented it as a model to be further refined. I was also interested to discover an article in Jewish Bible Quarterly that also discusses it.

Furthermore, with regard to the more limited suggestion of the pattern appearing in the two tablets of the Ten Commandments, it appears much more straightforward. As I noted in The Challenge Of Creation, a similar structure has been observed in the Six Days of Creation by scholars from across the theological spectrum. There is even more reason to expect it in the Ten Commandments, which from the outset are presented in two parts. I present an illustration of this pattern here (I hope it displays properly in different web browsers). Chag sameach!

Beginning with: Selfishness; Isolation; Focus on the Individual
Manifestation between Man and God
Manifestation between Man and Man
Accepting God
Acceptance of God’s existence
Acceptance of others’ existence
Do Not Murder
Other people’s right to exist; Value of life
Do Not Worship Idols
Accepting God’s ownership of the entire universe
Acceptance of their domain
Do Not Commit Adultery
Accepting the other person’s domain
Do Not Take God’s Name in Vain
No unlawful use of God’s Name
No unlawful use of property
(lawful coexistence)
Do Not Steal (Kidnap)
No unlawful use of another’s being, person or property
Observing Shabbos
Testifying to God’s authority
Readiness to cooperate
(particularly through speech)
Do Not Testify Falsely
No destructive talk; hence, cooperate
Honoring Parents
Unity with one’s source
Total unity
Do Not Covet
No resentment of others

Climaxing with: Selflessness; Unity with God and man; Integration into the Community


  1. Here's more fuel to add to your fire:

  2. people also have a powerful propensity to see patterns even when none exist, and to ascribe causality even where there is none

    Agreed, but there is a difference between patterns and causality.

    To erroneously ascribe causality is to falsify reality, period. However to seek out patterns is an important, even crucial part of the creative/cognitive process - whether those patterns are "intended" or merely coincidental.

    It's when people claim that such patterns are "inherent" or "planned" that they run the risk of falsifying reality. At that point it's like any hypothesis - it either survives the test of time/confirmation or not.

    The best way to take patterns in a Torah context... not as pshat but as DRASH - i.e. not as an assertion of reality or a statement of "proof", but as a pedagogical tool, a vehicle to bring meaning and inspiration where none exists.

  3. R'DM
    correlation vs. causation
    confirmation bias

    Joel Rich

  4. Please correct me if I am wrong, but doesn't the RAMBAM stress the importance of "koach hamedameh" which I have seen translated as "the imagination" but which literally means "abiliity to see similarities", which would mean finding pattern in things?

  5. Here is a nice presentation of the Torah's structure, though it doesn't seem to be finished yet, and there doesn't seem to be anything too mystical about it:


  6. AddeRabbi, you beat me to it!
    Y'know, chaver.com also has an article on the structure of the 10 Commandments.

  7. Thanks to those who pointed out that one of the links was broken - I fixed it.

  8. I very much enjoy your occasional forays describing the evolution of your thinking in Torah. [When reviewing my own written notes, I often smile, or cringe, at something I wrote 10 years ago that I would never write today.]

  9. Fascinating! I think you did something similar with the Jewish year in your book on that topic. I'm sure you know that this seems similar to the practice of finding chiastic structures in Tanach as well as studies showing that the patterns in Tanach are similar to those in other Ancient Near Eastern documents. There are those Orthodox Jews who are uncomfortable with these studies, but they happen to prove the unity and historicity of the Tanach.

  10. Mrs. Honigswachs used to tell me how happy her husband was when he once discussed his book with my father because, she said, so few understood his work. I'm sure, therefore, that he truly appreciates you as well! Chag sameach.

  11. Y Ben-David said:
    "koach hamedameh" which I have seen translated as "the imagination" but which literally means "abiliity to see similarities"


    Medameh is pi`el, which would yield ability to either imagine or compare. To me it seems that pattern detection is a higher level of abstraction.

    Moreover, does Rambam actually use that term, or is it a translation from Arabic?

  12. It's a translation from the Arabic.

    Lawrence Kaplan

  13. I was visitor 248400 to this site! What are the chances? And today is erev shavuot, rearrange the numbers and you get 2448, the year of the giving of the Torah! 248 also signifies the positive commandments of the Torah.

    Anyway, its true. People see lots of patterns. It affects people when gambling. Playing roulette and it's been red, black, red, black... red, black, the next spin must be red!

    Rabbi Fohrman gives a great shiur on the Ten Commandments (all his shiurim are great). See:
    And he even says that the Torah isn't a science or history book when speaking about Bereishit.

    In his shiur that uses the same structure as you have here, he notes that the first five commandments aren't bein adam l'Makom but rather between man and his Creator/creators.

    Chag Sameach!

  14. While I am not taken with the fractal scheme proposed, I certainly support the idea of finding connections between different parts of the torah. Instead of some formalism, however, I would emphasize awareness of repeated themes in the torah as one reads both the narratives and legal material.

    Finding patterns and connections is a worthwhile mental effort that is also the basis of scientific advance. In the absence of an objective way of testing such would-be patterns, however, it becomes a largely subjective exercise. It can also become an obsession, as R' Natan noted. One example of such obsessive pattern seeking is the 'torah codes' enterprise. I consider that effort to be arbitrary, useless, and even counterproductive. It certainly should not be used to convince people of the divine authorship of Bereishit. Another example is 'gematriot'. That, too, can become obsessive and unproductive.

    The torah should not be read in a manner that attempts to fit it into some preconceived pattern. Rather, it should be read with thought given to the lessons and ideas that it seeks to impart. Part of that attempt to understand the lessons is to attempt to understand the players - both human and divine. While it is difficult enough to fully understand a fellow human - much less GOD, still the attempt is of value.

    Chag samee'ach

  15. I think it is interesting that the readers who clicked Kefira far outnumber those who clicked Emes on this post - the only such imbalance that I recall seeing since this poling was started. Apparently even most "rationalists" are able to detect the overwhelming presence of patterns in the Torah.

    RNS wrote: "So what about this fractal pattern in the Torah? I don't have the time, or the objectivity, to study it all again and re-evaluate it. For a variety of reasons, I am certainly inclined to be skeptical."

    -- However, the patterns are clearly present for those who are open minded. This is what the "Kabbalistic" system of the ten sefiros and the four worlds, etc., is all about. To be skeptical about this does seem to indicate that one is missing out on many of the most basic principles of the Torah (even if strictly speaking it is not "kefirah").

    RNS, you would do well to hear the voice of critisism this represents, even from those of us who wish you well and stick up for you in other areas, and devote some of your time to reexamining this matter.

  16. A few quotes from Kabbalah which I find allude to a notion of fractals:

    “There is a visionary mirror [Shekhinah] reflecting supernal colors [the upper sefirot], envisioned in that visionary mirror; there is vision within vision, and vision within vision, one above the other, all poised on specific rungs, presiding, called night vision. Through them spread all dreams of the world, these resembling those above” (Zohar 1:196a).

    “And all of the faces look at each other and are included in each other, and from them they spread out in many directions, and tens of thousands, above and below, without measure, number, or account” (Ra’aya Meheimna, Pinhas).

    “Who am I? I am a mustard seed in the middle of the sphere of the moon, which itself is a mustard seed within the next sphere. So it is with that sphere and all it contains in relation to the next sphere. So it is with all the spheres – one inside the other – and all of them are a mustard seed within further expanses. And all of these are a mustard seed within further expanses. Your awe is invigorated, the love in your soul expands” (Moses Cordovero, Or Ne’erav 18b-19a).

  17. Not sure if you saw this new interesting book: ‘Patterns on Parchment - The Structural Unity of The Five Books of Moses’ by Dr. Robert Appleson.
    Appleson builds on the core approach taken by R’ Yehoshua Honigwachs. He also quotes him often.



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