Abraham and his nephew, Lot, are blessed with large herds of livestock. To prevent disputes between their respective shepherds, they choose to part ways. Lot moves to the fertile plains of the Jordan Valley adjacent to Sodom and Abraham remains in the Judean Hills near Hebron. After Lot’s departure, G-d appears to Abraham and blesses him, telling him [Bereishit 13:16] “I will make your seed like the dust of the earth, so that if a man will be able to count the dust of the earth, so will your seed be counted”.
Pardon me for sounding like an engineer, but what, exactly, is G-d telling Abraham? Is He telling him that he will have an infinite number of descendants? It is fairly easy to prove that even if the earth were entirely covered with sand, it would consist of less than eight quintillion grains of sand, admittedly a large number but far from being uncountable. To count the grains of sand, all you have to do is start with one grain and keep adding one until you run out of grains of sand. Further, the total global population of Jews has never been greater than 17 million, a peak reached on the eve of the Holocaust in 1939. Even if we implement the explanation of Rabbi Samson Rafael Hirsch, who posits that G-d was referring to the number of Jews who ever lived, the total is still significantly lower than eight quintillion. It is most definitely not infinite. Finally, what is so exceptional about having large amounts of progeny? Indeed, the Torah tells us [Devarim 7:7] “Not because you are more numerous than any people did G-d delight in you and choose you, for you are the least of all the peoples.” As far as G-d is concerned, quality trumps quantity. What, then, is G-d promising Abraham?
One of the ways of interpreting a problematic section of the Torah is by identifying “key words”, words that seem to be overly repetitive in the section under analysis. In the episode that describes the parting of Abraham and Lot, the key word is definitely the word “eretz” – “earth” or “land”. This word appears no less than nine times over the fourteen verses that comprise the episode. More interestingly, the word “eretz” never appears in connection with Lot. For example, the Torah tells us [Bereishit 13:12] “Abram dwelt in the land (eretz) of Canaan, and Lot dwelt in the cities of the plain (arei ha’kikar), and he pitched his tents until Sodom.” What is so special about “the earth” and why does Lot have no part in it?
In the verse immediately following G-d’s promise to give Abraham eight quintillion children, G-d tells Abraham [Bereishit 13:17], “Arise, walk in the land [of Canaan], to its length and to its breadth, for I will give it to you”. The Ramban, a famous medieval commentator, draws our attention to the Talmud in Tractates Bava Kama [9a] and Bava Batra [100a] that teach that a person who wants to purchase a field gains possession of that field by walking around its borders. G-d was telling Abraham that the Land of Canaan was being gifted to him and that all he had to do to acquire it was to take a stroll around its borders. What is so strange about this verse is its opening word, “Arise” (“Kum”). Was Abraham sitting on a couch that he needed to get up? Why doesn’t G-d tell him to put on his hiking boots?
Full disclosure: I warn the reader that we will be addressing these questions via mathematical concepts discovered in the twentieth century. While I do not claim that the Torah requires twentieth-century mathematics in order to be understood, I do claim that when viewed through the prism of modern mathematics, the Torah reveals some beautiful shapes.
In 1967, the Jewish French mathematician Benoit Mandelbrot published a paper called “How Long is the coast of Britain? Statistical Self-Similarity and Fractional Dimension”. In the paper, Mandelbrot expounded upon a paradox first discovered by Lewis Fry Richardson, who observed that the measured length of various geographic borders was a function of the measurement scale. Richardson discovered that if the coastline of Britain is measured with a measuring stick 100 kilometres long, then the length of the coastline is about 2800 kilometres. If the length of the measuring stick is shortened to 50 kilometres, then the length of the coastline grows to about 3400 kilometres, an increase of more than twenty percent! As the measuring stick is made even smaller, the length of the coastline grows even larger. Amazingly, the growth is unbounded, meaning that if the measuring stick becomes infinitesimally small, the length of the measured coastline becomes infinitely long. The reason for this strange behaviour is that the coastline of Britain, or any coastline for that matter, is not straight. Coastlines possesses features such as small inlets and abutments, jutting in and out of the water. These features have their own little features that become visible only when measured with a very small measuring stick. This geometric structure can be easily verified using Google Earth. Pick a coastline, any coastline, and zoom in slowly. Prepare to be blown away. After a few zooms, you will see things you did not know existed.
But wait a minute, say the unwashed masses unfamiliar with fractal geometry. How can a line that goes from Point A to Point B have infinite length? The answer is that a coastline is more than just a line. A coastline is part one-dimensional-line and part two-dimensional-plane. Just as a disc can be “unwound” into a line of infinite length, so, too, can a coastline.
I suggest that when the Torah uses the word “eretz” – “the earth” – in our episode, it is referring to the infinitely intricate shape of the earth and its borders that Abraham will inherit. When G-d tells Abraham that his progeny will be as uncountable as the “dust of the earth”, He does not mean that Abraham will have an infinite number of descendants. He is telling Abraham that just as the shape of a coastline is more than just a line, his descendants will be more than just a nation. They will bear the responsibility for bringing G-d’s infinite presence into this finite world. They will touch the Divine. In the same way that a coastline transcends length, somehow encompassing infinite length in a finite form, so, too, will Abraham’s descendants transcend number, somehow encompassing an Infinite Being in a finite form.
This explanation begets a paradox. When G-d tells Abraham to “walk in the land, to its length and to its breadth”, He is commanding Abraham to perform a task that he seemingly can never perform. Abraham should require an infinite amount of time to perfectly trace the land’s infinite borders. But this, we know, is only an illusion. We’ve all walked on the beach, covering an infinite distance in only a few steps. This is another message that G-d is giving Abraham: while your mission might seem beyond your capability, know that it is eminently performable. All you must do is to “arise”, beyond what you believe you are capable of doing, beyond the person you believe you are capable of being. G-d gives Abraham a similar message later on in the parasha. After Abraham complains to G-d that he has no children and no one to inherit him, the Torah tells us [Bereishit 15:5] “He took him outside, and He said, ‘Please look heavenward and count the stars, if you are able to count them.’ And He said to him, ‘So will be your seed.’”. Another famous medieval commentator, Rashi, comments on G-d’s taking Abraham “outside”, writing “He said to him, ‘Go out of your astrology,’ for you have seen in the signs of the zodiac that you are not destined to have a son. Indeed, Abram will have no son, but Abraham will have a son.” Do not limit yourself, Abraham. Your borders are infinite if only you could see them from above.
Our borders separate and they define. Borders separated between Abraham and Lot. Those same borders defined Abraham’s mission: to imbue our finite earth with infinite holiness.
Ari Sacher, Moreshet, 5779
Please pray for a Refu’a Shelema for Yechiel ben Shprintza and Tzvi ben Shoshana.
 This episode takes place while Abraham is still called Abram.
 In Jewish law, the transfer of money is insufficient to transfer ownership. Typically, some kind of physical action is required. Examples include physically raising an object, pulling it, or placing it into one’s own physical domain.
 See this link for an image: http://fractalfoundation.org/OFC/OFC-10-4.html
 The west coast of Britain shows this effect particularly clearly.
 To quantify this phenomenon, Mandelbrot defined the concept of self-similarity, later called “fractal dimension”, but this is way beyond the scope of this shiur.
 This is just a modern version of Zeno’s paradox.