What is a fractal?

A fractal is a shape based on the principle of self-similarity. Self-similarity is when the parts of a whole resemble the whole. For example, a pine needle is long and skinny, just like the pine tree it is part of. The human finger is made up of three sections (tip, middle, base), like the arm  (hand, forearm, upper arm) and the body (head, body, legs). Even human-made structures can show self-similarity: for example, a rectangular building that is made up of rectangular bricks.

Fractal shapes may seem paradoxical because they are both finite and infinite at the same time. Let's look at three examples to explore this concept.

Fractal Triangles


In this triangle, each time an upside-down triangle is added to a rightside-up triangle, three new rightside-up triangles are created, increasing the perimeter of the entire triangle (imagine tracing all the lines of the triangle with a pencil). This can go on indefinitely, all within the boundaries of the original triangle, resulting in an infinite perimeter, yet a finite area.

Golden Rule Rectangle


The same is true for the so-called Golden Rectangles that continue to appear at smaller and smaller scales in this fractal shape.

Fractal Nature

And fractals—such as those found in trees, the body, and buildings, to name a few—are everywhere. Including in language.

How is language a fractal?

Having established that a fractal shape is one which has finite area yet infinite perimeter, let's take a look at how language is a fractal.

To begin, let us establish that a language is an entity that has finite “area”. Simply put: there are only so many words in a language, and if you were to “measure” the area of a language—let’s say by counting the total syllables of the whole of that language’s lexicon—you would come to a fixed number.

However the “perimeter” of the language has the same propensity for infinite expression as do the roughness of the leaves of trees, or the ever-iterating curve of the Golden Spiral. For the potential of combining and recombining the words of the language is virtually limitless.

Consider the following example, using the verb “I like”.

This verb, like all verbs, is a “branch” for, or a “container” of vocabulary. From the verb grow smaller, and more numerous, “branches” and “leaves”. How many “branches” could potentially grow from the verb “I like” ? I like... coffee. I like... tea. I like... baseball. I like... etc. And from those smaller branches grow even smaller branches such as... ”with milk”, “with honey”, “very much”, and so forth.

Since we have established that the lexicon, or “area” of a language is fixed, yet the “perimeter” of the language is potentially infinite (you could combine and recombine words and phrases indefinitely, and never have to repeat yourself), we can see that the characteristics of fractal shapes are the same characteristics displayed in language.

How does this help?

First it is helpful to understand why a self-similar, scaling pattern should be an efficient design for a natural organism.

Consider the role of the leaves of trees, which is to “catch” sunlight for the tree. It would make sense that the leaves spread out as much as possible, to access as much sunlight as possible. And the recurring pattern of many smaller branches on one bigger branch is the most elegant solution. (Compare, for example, to one single shaft coated in leaves.)

And it's less work for the builder to make rectangular bricks which fit perfectly on one another, the resulting shape also being a rectangle. (Compare to building with bricks that are spheres.)

In learning language, we must access the naturally occurring, organic efficiency of language's fractal shape.

In Plain Terms

Learn verbs! And study them in order of “frequency of usability”. Some common ones are: “be”, “have” and “go do”. Verbs are like the seeds from which virtually infinite sentences grow out of.

Allow sentences to grow from the verbs you know—without a verb there is no sentence! Language is the trunk of the tree, and verbs are the next biggest branches. Upon those branches grows vocabulary—smaller and smaller branches and leaves. (To complete the metaphor: the roots—underground and unseen—are thoughts; and the rain, which stimulates thought, is external, sensory stimuli.)


Language, like a tree, can only grow in certain conditions. It must be cultivated as if it was a natural organism. Verbs must be understood as “containers” of vocabulary, because without them there would be no place for these metaphorical leaves . By learning the most high-frequency verbs, in the right order, we gain more “leaves” and so efficient access to the most “sunlight” possible. Obtaining more “sunlight” is our goal, as that is the communicative potential in a language.


Hero Illustration by Malachi Ray Rempen - Itchy Feet