What are affine spaces in physics?

An affine space is where relationships become linear - straight lines instead of curves. In physics, finding the right affine space means discovering the coordinate system where complex, nonlinear behavior simplifies into something you can write as y = a + bx.|
Think of it as finding the "natural viewpoint" for a phenomenon. Earth's orbit looks like a complex curve from one angle, but from the right perspective it's a simple ellipse. Finding the affine space is finding that right perspective where the math becomes clean.
Why log-log is THE affine space for power laws:
Power laws describe scale-invariant systems - systems that look the same at different scales. Bitcoin's price, earthquake magnitudes, city sizes, income distributions. The hallmark: when you zoom in or out, the pattern repeats.
In normal coordinates, P = A·t^5.7 looks curved and complicated. But take logarithms of both sides: log(P) = log(A) + 5.7·log(t). Boom - it's a straight line. The log-log plot transforms the multiplicative growth into additive growth.
This isn't arbitrary. Scale invariance MEANS "multiplication by a constant shouldn't change the pattern." Logs convert multiplication to addition. So log-log coordinates are the natural affine space - the perspective where scale-free systems reveal their true, linear structure.
When someone tests P^(1/k) vs time and finds the best fit at k≈6, they're doing it backwards. They're forcing the data to be linear in the wrong space. We already know the right space: log-log. Bitcoin's 15-year power law (R²=0.96) proves it.
The physics insight: Nature doesn't care about our coordinates. But scale-invariant processes have a natural coordinate system where they become simple. For power laws, that's log-log. Finding the right affine space isn't curve-fitting - it's discovering the symmetry.