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The Book of Leviticus and the Fractal Geometry of Torah
Richard S. Ellis

The statement of Galileo that "the great book which ever lies before our eyes—I mean the Universe—is written in mathematical language and the characters are triangles, circles, and other geometrical figures" applies as well to the Halakhah. And not for naught did the Gaon of Vilna tell the translator of Euclid’s geometry into Hebrew [R. Barukh of Shklov], that "To the degree that a man is lacking in the wisdom of mathematics he will lack one hundredfold in the wisdom of the Torah."1


 [Mandelbrot Set #1]
 [Mandelbrot Set #2]  [Mandelbrot Set #3]  [Mandelbrot Set #4]
Four images of the Mandelbrot set--each one based on a magnification of an area of the one above it.

The Archangel Gabriel warned the prophet, "God is hidden by 70,000 veils of light and darkness. And were those veils to be lifted, even I would be annihilated." The Book of Leviticus is hidden behind many such veils. Indeed, we wonder about the essence of a God who stipulates (Leviticus 25:29): "A man—if he sells a residential house in a walled town, its redemption period (is) until the end of the year of its sale, a year-of-days shall be its redemption period."2 After reading Genesis and Exodus and scaling all those mountains of significance and profundity, it seems that in Leviticus we are condemned to wander in the desert of the abstruse and the obscure.

This leads to a basic question. Is it possible to synthesize, with the lush landscape of the narratives of Genesis and Exodus, the dry dust of Leviticus’ legal code, delivered ceaselessly from God’s mouth to Moses’s ear at Mount Sinai? Answering this question will yield an unexpected bonus. It will lead us to a mathematical law of nature, elucidated in Benoit B. Mandelbrot’s 1983 book The Fractal Geometry of Nature,3 which offers deep insights not only into the structure of the universe but also into the structure of Torah.

Let me jump directly to a synthesis. The stories of Genesis and Exodus both animate and embody the concepts of the legal code to follow without explicitly calling attention to those concepts. As one commentator remarks, "these stories prepare us to appreciate the necessity and wisdom of the law that comes later."4 Narrative without a legal code is a body bereft of soul. A legal code without narrative is a cold, inaccessible jewel. Of course, this law is crucial because the core of our beliefs as Jews is expressed in the legal code, starting with the Ten Commandments. Thus, the question can be turned on its head. Why isn’t the theology enough? Why do we need narrative? In an extremely insightful devar Torah, Jay Ladin addresses this same issue:

This may be why the Torah contains so much narrative, instead of simply laws and dogmas: because our relationship with God depends on creating a history in which the Divine is apparent. This history depends both on the way we act, individually and collectively, and on the way we imagine and recount our past, on telling stories which connect our lives and God’s, so that the way God "becomes" in our lives is made visible.5

As we will see, these insights will help lift some of the veils of light and darkness hiding Leviticus. While my points could be made in the context of any of the parshiyot in that book, I will focus on the penultimate parashah Be-har (Leviticus 25:1–26:2), which as legal code is particularly rich in allusions to the narratives of Genesis and Exodus. The parashah begins by discussing the sabbatical year for the soil, goes on to stipulate laws of buying and selling property, discusses how one is to act "when your brother sinks down (in poverty)," and ends by discussing some of the laws concerning the treatment of servants. Let us start at the beginning, in verse 1 of chapter 25, keeping in mind a key principle of Torah interpretations: an apparently unnecessary phrase usually signals something important.

Here is Leviticus 25:1, first in Fox’s translation, then in Hebrew: "YHWH spoke to Moshe at Mount Sinai, saying":—"Vayidaber Adonai el Moshe behar Sinai leimor": God then goes on to speak about the sabbatical year for the soil. The parashah’s opening verse sounds familiar. Indeed, God’s pronouncements in Leviticus are almost universally preceded by the same formula: "YHWH spoke to Moshe, saying":—"Vayidaber Adonai el Moshe leimor": In fact, the new element in the opening of the parashah gives rise to its title: namely, the addition of the phrase "at Mount Sinai"—"be-har Sinai." In this context Rashi poses the following question: "What has the matter of the Sabbatical year to do with Mount Sinai that Scripture felt compelled to expressly state where it was commanded? Were not all commandments given on Sinai?"6 A natural expectation is that the sequel will contain a pronouncement that both is of great moment and is somehow intrinsically related to the holy mountain.

This expectation is indeed fulfilled. The text goes on to expand profoundly the concept of Shabbat, which, like a double-arched bridge, spans the eons of time and light-years of space from the present parashah in Leviticus to the giving of the law on Mount Sinai in chapter 20 of Exodus to the creation of the universe in chapter 1 of Genesis. In the next two verses of our parashah (Leviticus 25:3–4) we read about the sabbatical year for the soil:

When you enter the land that I am giving you,
the land is to cease, a Sabbath-ceasing to YHWH.
For six years you are to sow your field,
for six years you are to prune your vineyard,
then you are to gather in its produce,
but in the seventh year
there shall be a Sabbath of Sabbath-ceasing for the land,
a Sabbath to YHWH.

How beautiful and how significant, and not just because this passage gave rise, in academies of higher education, to the reward of the sabbatical year. These verses expand Shabbat, the crown of creation, from a resting point in time to a resting point in space. And not just space, but space in the holy land of Israel, the focus of the significant promises by God to the patriarchs and the goal of the newly freed band of slaves who are to be molded into a nation. In these verses we hear echoes, and in a sense a fulfillment, of chapter 13 of Genesis (13:14–17), when God speaks to Abram after he and his nephew Lot separate:

Pray lift up your eyes and see from the place where you are, to the
     north, to the Negev, to the east, to the Sea:
indeed, all the land that you see, I give it to you
and to your seed, for the ages. . . .
Up, walk about through the land in its length and in its breadth,
for I give it to you.


And echoes of chapter 28 of Genesis (28:13–14), when God speaks to Jacob in his dream at Beit El after fleeing the wrath of his brother Esau:


The land on which you lie
I give it to you and to your seed.
Your seed will be like the dust of the earth;
you will burst forth, to the Sea, to the east, to the north, to the Negev.
All the clans of the soil will find blessing through you and through
     your seed!

These promises to Abram and to Jacob are made in the thick of narrative entanglement. Abram settles in Canaan, Lot in Sodom, God prophesies four hundred years of servitude in Egypt, Hagar gives birth to Ishmael, Sodom and Gomorrah are destroyed, Sarah gives birth to Isaac, Abraham banishes his firstborn Ishmael and almost murders his secondborn Isaac, and on and on. Fleeing Esau, Jacob returns to the ancestral homeland in Haran, marries Leah and Rachel, slaves twenty years for Laban, fights the angel at the Yabbok, confronts Esau, loses his beloved Rachel in childbirth, and on and on. When viewed through the eyes of our slave ancestors, newly freed from Egypt, these narratives reinforce a key idea. The reward of the land of Israel, though promised unambiguously and repeatedly, is centuries ahead in the future, a reward to be bestowed only after centuries of servitude and suffering. When in parashah Be-har we read about the sabbatical year for the soil in Israel, we can perhaps have an inkling of understanding of how our newly freed slave ancestors must have felt when they realized that the reward of the land was near.

"YHWH spoke to Moshe at Mount Sinai, saying":—"Vayidaber Adonai el Moshe behar Sinai leimor": The verses that follow concerning the sabbatical year for the soil hearken back, of course, to the great pronouncement concerning Shabbat at the holy mountain, which itself hearkens back to the hallowing of Shabbat on the seventh day of creation. The Fourth Commandment in Exodus 20 (20:8–11):

the Sabbath day, to hallow it.
For six days, you are to serve, and are to make all your work,
but the seventh day
is Sabbath for YHWH your God:
. . . For in six days
YHWH made
the heavens and the earth,
the sea and all that is in it,
and he rested on the seventh day;
therefore YHWH gave the seventh day his blessing, and he hallowed it.

This link between our parashah and the seventh day of creation is an instance of a deep correspondence resting on the so-called "priestly" world-view, which is exposed most obviously in Leviticus but also in significant portions of the other books. Fox explores this correspondence in great detail and sums up with the following:

In the priestly view, the world is to be an echo of the divine order that is portrayed in the Creation story. . . . The human body becomes symbolic of the cosmos: its life/death boundary is marked, and troublesome flows from it are carefully regulated. The land of Israel becomes symbolic of the cosmos: too much evildoing pollutes it, to the point where it can do naught else but "vomit out" its settlers . . .7


Later we will return to this idea whereby parts of the cosmos—here, the human body and the land of Israel—become symbolic of the cosmos as a whole.

Thus we may explore the vista from the double-arched bridge linking our parashah with the giving of the law in Exodus and with the creation of the universe in Genesis. Certainly this connection between our parashah and Genesis 1 is apt. Then, the concern was the creation of the universe. Now, the focus is on the creation of a nation. The smooth functioning of the universe requires the implementation of mathematical, physical, chemical, biological, and related scientific laws, including gravitation, electromagnetism, quantum mechanics, and natural selection. Analogously, the smooth functioning of a nation requires the implementation of legal, moral, religious, and related ethical laws. The main purpose of the rest of the parashah is to continue the discussion of these ethical laws, which already have been a central focus of the text since the giving of the Ten Commandments.

The theme is picked up in verse 14, where we read the following:

Now when you sell property-for-sale to your fellow
or purchase (it) from the hand of your fellow,
do not maltreat any man his brother!


"Do not maltreat any man his brother."—"Al tonu ish et ahiv." This admonishment is echoed four times later in the parashah when God discusses how one should act when "your brother sinks down (in poverty)" (Leviticus 25:25, 25:35, 25:39, 25:47). Are these commandments concerning brothers enriched or made incarnate by any of the earlier narratives? Of course they are, for the maltreatment of sibling by sibling is one of the main themes of Genesis. Cain kills his younger brother Abel. Ishmael laughs at his younger brother Isaac and is banished. In an ironic foreshadowing of verse 14 of the present chapter, which opens a discussion of ethical behavior during property transfers, Jacob steals first the birthright and then the blessing from his elder brother Esau, whose wrath he subsequently flees, to be sucked into the spirals of jealousy and rivalry between Laban’s two daughters, Leah and Rachel. Jacob’s favorite son Joseph is despised by his brothers, who sell him into captivity, eventually to be reunited with him in Egypt, thus setting in motion the servitude of the Jewish people foretold by God to grandfather Abraham. Keep these narratives in mind, as the Jewish people in their trek through the desert certainly did, and listen again to the words of verse 14:


Now when you sell property-for-sale to your fellow
or purchase (it) from the hand of your fellow,
do not maltreat any man his brother!


Or verses 35 and 36:


Now when your brother sinks down (in poverty) . . . ,
then shall you strengthen him . . .
and he is to live beside you,
Do not take from him biting-interest or profit,
but hold your God in awe,
so that your brother may live beside you!


Feel the resonance and the depth that the narratives of sibling rivalry lend these words. Without these narratives, how could one truly understand the commandments not to maltreat one’s brother or how to act when your brother sinks down in poverty?

God’s aim during the desert trek is to mold the Jewish people into a just and ethical nation. As God emphasizes throughout our parashah and all of Leviticus, the caring that each person must show for his or her fellow human beings is always grounded in God’s own transcendentally ethical behavior. For example, in verse 17 we read the following:

So you are not to maltreat any-man his fellow,
rather, you are to hold your God in awe,
for I YHWH am your God!


Repeatedly, communal morality is grounded in awe for God Herself, and there is a continual refrain back to the Exodus, as in verse 38, which itself is a quote from the First Commandment:


I YHWH am your God
who brought you out of the land of Egypt
to give you the land of Canaan,
to be for you a God!

That man and woman are able even to conceive of God, let alone to hold God in awe, is a consequence of the divine spark that God planted in woman and man on the sixth day. Thus behind the veils hiding Leviticus we discern another double-arched bridge, running parallel to the first, this time arching back from our parashah to the First Commandment in Exodus to the sublime moment in Genesis when man and woman are created (Genesis 1:27):

God created humankind in his image,
in the image of God did he create it,
male and female did he create them.

It is said that the foundation stone of the world lies in the center of the Temple Mount in Jerusalem. Surely this passage, "God created humankind in his image," is the foundation stone of the entire Torah. In closing, I would like to examine the vicinity of this key verse in order to elucidate a mathematical law of nature that offers deep insights not only into how our universe works, but also into how Torah works and, in particular, into how parashah Be-har is related to the whole.

This mathematical law is the fractal law of geometry. As explained by the mathematician Benoit Mandelbrot, a fractal is a "shape having the property that each small portion can be viewed as a reduced scale replica of the whole."8 Mathematical examples of fractals abound. But one can also see fractals in nature as in the shape of a fern, a seacoast, or a branching tree. Whether one looks at these shapes from far away or up close, one sees the same pattern. Consider a tree trunk splitting into boughs, then branches, then limbs and twigs and stems of increasingly smaller size. The branchings of a twig on one of these limbs, branching into every smaller twiglets and stems, has the same shape as the whole. The mathematical study of fractal shapes is called fractal geometry. By means of this field many complicated problems in science can be approached and reformulated more easily. What is important to us is that the fractal law of nature is also a basic philosophical and structural element of Torah.

The essence of a fractal is that the shape of the whole is mirrored in the shape of portions of the whole. Thus in a fractal the whole can be reproduced from such portions. But of course this is the essence of life itself. Each life-form possesses the ability to reproduce itself from a portion of itself. Torah reveals this truth in the record of the third day, Genesis 1, verses 11 and 12, when God creates life:

God said:
Let the earth sprout forth sprouting-growth,
plants that seed forth seeds, fruit trees that yield fruit, after their kind,
     (and) in which is their seed, upon the earth!
It was so.
The earth brought forth sprouting-growth,
plants that seed forth seeds, after their kind,
trees that yield fruit, in which is their seed, after their kind.
God saw that it was good.


The fractal essence of life: each life-form contains the seed to produce the next generation of that life-form. So it is not by chance that after "God created humankind in his image" on the sixth day, God commands man and woman using the same image of fruit. Genesis 1, verses 28 and 29:


God blessed them,
God said to them:
Bear fruit and be many and fill the earth and subdue it!
. . . Here, I give you
all plants that bear seeds that are upon the face of all the earth,
and all the trees in which there is tree fruit that bears seeds,
for you shall they be, for eating.


When the children of Israel will enter the promised land, God the nurturing father will bestow this gift again. Verse 18 of our parashah:


You are to observe my laws,
my regulations you are to keep and observe them,
that you may be settled on the land in security,
that the land may give forth its fruit
and that you may eat to being-satisfied,
and be settled in security upon it.

God, humankind, fruit: woman and man partake of the essences of both God, the divine sustainer, and fruit, the divine sustenance. A unifying image is that of the tree: the tree as a fractal shape, the tree bearing fruit that bears seeds to make another tree, humankind as the apple of God’s eye, as the fruit on God’s tree, the Tree of Life in the midst of the garden, beside it the Tree of the Knowing of Good and Evil (Genesis 3:6–7).

The woman . . . took from its fruit and ate
and gave also to her husband beside her,
and he ate.
The eyes of the two of them were opened
and they knew then
that they were nude.

The Torah is this Tree of Life, etz hayim hi, hidden by 70,000 veils of light and darkness, and you and you and you and I are blossoms on its fractal branches. Let us survey, from big structures to small, first a tree and then Torah. Trunk, bough, branch, limb, twig, twiglet, stem, blossom. Book, parashah, chapter, verse, line, word, letter, breath. The Torah is a fractal because, as the mathematician Benoit Mandelbrot explained, each small portion can be viewed as a reduced scale replica of the whole. The focus in parashah Be-har on Shabbat, on the giving of the land, on ethical behavior, on living in peace and love with our sisters and our brothers, on living in active support of our sisters and our brothers: this is the focus of the entire Torah. Thus the fractal law of mathematics, created in the opening verses of Torah to ensure the smooth functioning of the universe, helps us to explicate this same Torah. Line by line, breath by breath, stem by stem, blossom by blossom, each verse of our parashah, like each verse of Leviticus, mirrors fractal-like the glorious and transcendent vision of the whole.


  1. Joseph B. Soloveitchik, Halakhic Man, trans. Lawrence Kaplan (Philadelphia: Jewish Publication Society of America, 1983), p. 57.
  2. All translations of the Torah are from The Five Books of Moses, trans. Everett Fox (New York: Schocken Books, 1995).
  3. New York: W. H. Freeman, 1983.
  4. Leon R. Kass, "A Genealogy of Justice," Commentary, July 1996, p. 44.
  5. Jay Ladin, "Devar Torah on Parashah Va-era," delivered at the Jewish Community of Amherst on January 11, 1997 (unpublished), p. 10.
  6. Chumash with Targum Onkelos, Haphtaroth and Rashi's Commentary, trans. A. M. Silbermann (Jerusalem Silbermann Family by arrangement with Routledge and Kegan Paul, 5745 [1985]), p. 113.
  7. The Five Books of Moses, trans. Everett Fox (New York: Schocken Books, 1995), pp. 501-502.
  8. Benoit B. Mandelbrot, "Fractal," Infopedia 2.0 CD-Rom (Cambridge, MA: Softkey Multimedia, 1995).


Richard S. Ellis is Professor in the Department of Mathematics and Statistics at the University of Massachusetts Amherst. He recently completed a novel set in Jerusalem, entitled Broken Symmetries.