Reading between the lines: How all of math is made from a line.

Imagine a single, slender thread stretched across the vast canvas of existence.

This humble line, the embodiment of simplicity, becomes the seed from which all mathematics grows. Through stretching, translation, and curvature, it transforms into the intricate tapestries of numbers, shapes, and functions that permeate every corner of our universe.

I will show how every area of math can fundamentally be thought of as an extension of a single line.

Geometry

In the heart of geometry, the line forms the fundamental building block.

The easiest intuition of how lines represent all mathematics can be shown by adding additional lines to build polygons, rectangles, and triangles — the shapes that define buildings, bridges, and galaxies.

In elementary school, we took a sheet of construction paper to make paper rings. With a simple arc of the straight edge, the flat paper turned into a flatter, larger version of a fruit loop.

Paper donuts are only the start. We can rotate polynomials unveiling circles, ellipses, and parabolas — the curves shown in planets, orbits, and lenses. Through careful scaling, it stretches and shrinks to form ratios, proportions, and symmetries — the underlying principles of architecture, art, and nature itself.

Algebra

Algebra, the realm of equations and unknowns, also springs from this singular thread.

The line transforms into a coordinate plane, where it becomes the X and Y axes, charting the dance of numbers. Linear equations paint straight paths, while quadratic equations bend the line into parabolas, revealing the secrets of motion and projectile paths. Higher-order equations weave more complex curves, representing phenomena as diverse as population growth and chemical reactions.

A unique feature of lines is their ability to multiply. When one line is multiplied by another, the result is not a line, but a quadratic. Now multiply the quadratic by another line — now a cubic. This sounds remarkably similar to the simple nature of cellular replication.

Calculus

Everyone’s favorite high school class.

The field of calculus finds its start in the line. Summarizing four semesters of material in a sentence, calculus uses lines to approximate how steep or shallow a curve behaves at a single point. Here, the line becomes an infinitesimal segment, stretching and shrinking infinitely, paving the way for concepts like slopes, derivatives, and integrals. These tools unlock the mysteries of change and motion, allowing us to analyze the speed of a falling object, the population's growth rate, or the trajectory's optimization.

Topology & Number Theory

The line’s journey doesn’t end there. In topology, it transforms into loops, knots, and surfaces, exploring the fundamental shapes of space itself. In number theory, its prime factorization displays the hidden structure of integers, while its infinite decimal expansions perfectly describe irrational numbers. Even statistics finds its roots in the line, forming the bell curve, a ubiquitous symbol of randomness and probability. A double pendulum swinging is two lines without a predetermined course.

What's the deal with the line?

This pervasive presence of the line is not a coincidence.

It embodies a fundamental principle of mathematics: universality through transformation. Just as a single note can be manipulated to create an infinite score of music, so too can the line be twisted, bent, and scaled to encompass the vast landscapes of mathematical ideas.

Not to be dramatic, but it reflects the very essence of our universe. There is a reason mathematics is involved with every field of science. From the orbits of planets to the trajectories of electrons, from the growth of a tree to the flow of a river, the world around us pulsates with transformations, translations, and curvatures. By understanding the language of the line, we gain a deeper appreciation for how the world and universe work.

Understanding that mathematics takes something like a Lego piece and builds one step at a time, I hope it’s easy to see that every area of math isn’t as complex as we might think.