Oil and water refuse to mix. Yet mayonnaise is mostly oil, and it sits on a spoon like a solid. The trick is one egg yolk, a lot of whisking, and a surprising amount of surface.
Pour oil into water and it floats in a layer. Whisk it hard and you get a cloudy mess, but only for a few seconds. The oil is now in droplets, yet every time two droplets touch they merge, because merging shrinks the total area of contact between oil and water, and the mixture prefers less of it. Soon a single layer of oil sits on top again.
Egg yolk contains proteins and phospholipids such as lecithin. Each has one end that likes water and one that likes oil, so they line up on the skin of every oil droplet with the oily end inside and the watery end out. A droplet wearing that coat does not merge when it bumps into another. The mixture stays as a crowd of separate droplets.
A flat slice through a bowl, not to scale. The white tick marks 74%, the most that equal spheres can fill in 3D. Circles in a flat slice can pack to 90.7%, so the slice reads a little differently.
Cutting oil into droplets creates a startling amount of skin. The surface of a droplet divided by its volume is 6 over its width, so a tablespoon of oil (about 15 mL) in droplets 5 micrometres wide has about 18 m² of surface, roughly the floor of a small room. Droplets in mayonnaise average between 2 and 10 micrometres across, which puts that same tablespoon at 9 to 45 m².
All of that skin needs yolk. One yolk holds a fixed amount of coat, so it can only cover so much surface. Try it above: whip a lot of oil into very small droplets and the yolk runs out. Droplets with a thin coat count as bare, and bare droplets that touch merge, so the drops settle at a larger size. This is why more yolk gives smaller droplets, and why a spoonful of oil dumped in whole just floats: a drop that big is not held by any coat until the whisk breaks it up.
Every stroke of the whisk splits droplets in two, and each split makes new bare surface that the yolk has to reach. Add oil faster than the yolk can coat it and the bare droplets merge, grow, and rise. That is a split mayonnaise: the same ingredients, but the droplets have merged back into a layer of oil.
Traditional mayonnaise is about 65 to 80% oil and 5 to 8% egg yolk. In the United States, the FDA requires at least 65% vegetable oil by weight before a product can be called mayonnaise. Identical spheres cannot fill more than about 74% of the space, a result known as the Kepler conjecture, and mayonnaise sits right around that limit. The droplets are packed so tightly that they can only squeeze past each other with difficulty, and the thin film of yolk and water between them is all that keeps them apart. A mixture of two liquids ends up behaving like a soft solid.
Less oil means looser packing and a runnier sauce. It is one reason low-fat versions need thickeners.
The usual fix is to start again. Put a fresh yolk, or a spoon of water, in a clean bowl and whisk the split mixture in a little at a time. The new yolk supplies new coat, and your whisk splits the merged oil back into droplets. In the bowl above you can do the same with a second yolk and the auto-whisk.
Pour two spoonfuls and whisk until the drops are small. Then pour six spoonfuls without whisking and watch the big drops rise and merge. Next whisk them back and add another yolk to see the droplets shrink. Finally keep pouring until the bowl is about three-quarters oil and watch the droplets crowd together until the mixture gets stiff.