Grow 1 by all of itself and you get 2. Grow it in smaller and smaller pieces, each building on the last, and you get a little more every time, but never more than 2.71828…. That number is e. Start on the number line, where e and its powers come from, then push sideways into the plane of complex numbers, where the same recipe walks around a circle.
Station 1
Start at 1 and grow it by all of itself in one step: you land on 2. Now split that growth into n pieces. Each piece multiplies by 1 + 1/n, so it adds 1/n of whatever you have now, including what the earlier pieces added. That growth on growth lands you a little further every time you split it finer. Every landing leaves a mark. Zoom in as the marks bunch up: they crowd toward 2.71828…, but never get past it. That number is e.
Station 2
The recipe takes any push x, not just 1: multiply by 1 + x/n, n times. Push by 2 and look halfway. The first half of the steps push by 1 in n/2 pieces, which is exactly Station 1's recipe, so they land near e. The second half do it again from there. So you land near e × e, and the recipe settles on ex, e raised to the power x. Pushes add while the results multiply.
Station 3
Every push so far went forward, along the number line. Stand a second number line upright through 0 and the line becomes a plane, the plane of complex numbers, where multiplying by i turns a number a quarter turn. Now push sideways. Split a push of iθ into n pieces, and each piece multiplies by 1 + iθ/n. To see what one piece does, call wherever you are z. Multiplying it out, z(1 + iθ/n) = z + (θ/n)iz: you keep z and add a small piece of iz, which is z turned a quarter turn. So the piece is at right angles to the line from 0 to z, and it takes you to the amber point z′ = z(1 + iθ/n). Drag z anywhere and the piece turns with it. Slide Pieces taken to keep going: z′ becomes the new z, and the next piece is at right angles to that, so it leans further round. The dashed circle runs through where you started. A pure turn would keep you on it, but every piece also stretches you a little, so the walk drifts outward. Then shrink the pieces with the Piece size slider: a small sideways piece turns you by about its own size, in radians, and hardly stretches you at all.
Station 4
Station 1 started at 1 and multiplied by 1 + 1/n, n times, so each piece added 1/n of what you had. Change just one thing: make each piece i/n of what you have, and multiply by 1 + i/n instead. Each piece is now 1/n of where you are, turned a quarter turn, so it pushes sideways, and Station 3 showed that it turns you by about 1/n of a radian. All n pieces together turn you about 1 radian. It's Station 2's recipe with x = i, so the sideways walk settles on ei ≈ 0.5403 + 0.8415i, one radian round the circle. There's no π in it. Both walks below use the same n: the forward one is Station 1's, heading for e.
Station 5
Station 2 took Station 1's recipe and pushed by any x. Do the same with Station 4: push sideways by iθ, so each of the n pieces multiplies by 1 + iθ/n. Each piece turns you about θ/n radians, so together they walk a distance θ round the circle of radius 1, and as n grows the walk settles on eiθ. With a few pieces, a small square at each corner marks the right angle. Pushes add while results multiply, just as in Station 2. A push of 2i is two pushes of i in a row, so e2i = ei × ei, two radians round. Halfway round the circle is π radians, so a push of iπ lands on −1.
Station 6
Each sideways piece is at right angles to the line from 0, so it slides along the circle instead of away from it. Pushing sideways by θ walks a distance θ around the circle. So here is eiθ without any pieces: take the number line from 0 to θ, stand it up at 1, and wrap it around the circle. Every number t on it lands on eit. The circle is 2π around, so π/2 reaches a quarter of the way, at i, and π reaches exactly halfway, at −1. Halfway is two quarter turns, so eiπ = i × i = −1, which is Euler's identity, eiπ + 1 = 0.
Station 7
Stations 5 and 6 found the same thing two ways. Put plainly, eiθ is where you end up after walking a distance θ round the circle from 1. Nothing gets added to 1. You only ever multiply, and each symbol has its own job. i sets the direction, θ sets how far, π comes from the circle, and e names where you land. Walk the circle and the readout shows each one at work.
Station 8
Station 2 pushed forward and Station 5 pushed sideways. Push at a slant and you do both. Each piece multiplies by 1 + (a + bi)/n: the forward part a stretches and the sideways part b turns. You land on ea+bi, stretched to ea and turned b radians, so ea+bi = ea × ebi, a stretch times a turn. In between, the walk spirals.