The creator of 3Blue1Brown turns e^{πi} = -1 into a design lecture, and comes away with three transferable principles for designing an explanation.
Designing Math ft. Grant Sanderson (3Blue1Brown) I Config 2026 · Loredana Crisan
27 min total·Actually worth watching closely: ~20 min·3 must-watch clips
- 0:00 – 4:25Listen
Opening: math explanation is design
Figma's Loredana Crisan opens day two and introduces Grant Sanderson, who sets up his thesis with e^{πi} = -1 as the running example: explaining a hard idea and designing are the same job — finding a clear path through an abundance of choice and a large potential for complexity.
The more choice there is, the more design is needed; mathematicians and designers look like different species but are doing the same work.
This stretch is spoken setup with no essential visuals — fine to listen to on a commute or while doing something else.▶ Jump to 0:00Speaker · Grant Sanderson - 4:25 – 7:20Skim
The right mental images for π and i
The symbols in the equation are unpacked one at a time: π is not just the ratio of circumference to diameter but the distance you travel going halfway around the unit circle; i extends our usual notion of numbers from a one-dimensional line into a two-dimensional plane, and the action of multiplying by i is a rotation of 90 degrees.
Complex numbers aren't mystical — they're a practical tool for manipulating two-dimensional space, a new brush in a designer's belt.
The unit circle and 90-degree rotation diagrams are static illustrations — a glance is enough to fix the mental image, no need to watch frame by frame.▶ Jump to 4:25Speaker · Grant Sanderson - 7:20 – 10:55Watch
The spiraling sum: what the equation actually says
e^{πi} is added up term by term: each power of i turns the direction another 90 degrees while the factorial denominator grows so rapidly that the terms shrink, and the whole spiraling sum closes in on -1. That leads to principle one: treat the explanation first as a visual design challenge.
Not every piece of math has to be visualized, but simply asking "What does it look like?" is a lot of low-hanging fruit.
The animation of the spiral converging on -1 (at 464s) carries all the persuasive force here — you have to see it.▶ Jump to 7:20Speaker · Grant Sanderson - 10:55 – 14:26Watch
The dynamics of e: velocity locked to position
Read e^t as a position moving in time whose velocity vector is locked to equal that position: the bigger the position gets the bigger the velocity, and the bigger the velocity the faster the position grows — that runaway feedback loop is the felt sense of exponential growth. Put a constant c in front of the time and the velocity simply becomes c times the position.
Understanding exponentials isn't about memorizing formulas — it rests on one dynamical intuition: the rate of change is locked to equal the thing itself.
The animation of position and velocity vectors evolving in time (at 654s and 753s) is the foundation for the circling argument that follows — worth watching closely.▶ Jump to 10:55Speaker · Grant Sanderson - 14:26 – 18:08Watch
Plugging in i: why the circle is inevitable, and principle two
With i, the velocity is equal in length to the position but perpendicular to it, so you go around the unit circle at 1 unit per second and π seconds lands you exactly on -1 — the equation could never have been otherwise. That leads to principle two: break the thing into components and give each one a clearly defined motivation, the way every character in a novel needs one; and the spiraling view (what is claimed) is set beside the circling view (why it holds) as complements.
A good explanation doesn't just show what's true — it answers why it couldn't have been any other way.
The circling animation from 865s is the core of the whole argument, and the juxtaposition of the two visualizations at 977s only lands with the picture in front of you.▶ Jump to 14:26Speaker · Grant Sanderson - 18:08 – 21:24Watch
The mystery of Escher's Print Gallery
The print that nests deeply inside itself: the man looking at a picture of a boat ends up sitting inside the picture he's looking at — Escher called it the strangest thing he ever made. His genius wasn't the picture nested within a picture, but intuitively realizing there must be a way to pull that inner nested world out and connect it to the outer version.
The transformation Escher found by intuition turns out to be described exactly by the complex exponential — math he himself knew nothing about.
Without the original print (1088s) and the diagram of pulling the nested world out (1226s), the suspense in this stretch doesn't land at all.▶ Jump to 18:08Speaker · Grant Sanderson - 21:24 – 27:22Watch
The log as paintbrush: reconstructing Escher, and principle three
The exponential rolls vertical lines up into concentric circles whose radius scales by e at each step; the log unrolls those circles back into straight lines — an inverse pair of brushes. Take the log of the infinitely self-nested image and you get a tiling you can actually work with; find the right rotation, exponentiate back, and Escher's wild spiral comes out. The transition animation in the talk was made exactly that way: rotate in log space, then present what you get after exponentiating. It closes on principle three: the most memorable application is often the most surprising one.
The first lesson in teaching is making someone want to learn — and a surprise like "imaginary exponentials can serve an artistic goal" is what stays.
The continuous morph of straightening by the log and rolling back by the exponential (1304s, 1486s, 1541s) is the most striking footage in the talk, and the live proof of all three principles.▶ Jump to 21:24Speaker · Grant Sanderson