Suppose preferences are strictly monotone and strictly convex and
consider any point \((x_{1},x_{2},x_{3})\). Geometrically, consider the
plane (in green in the diagram above) going through the point
\((x_{1},x_{2},x_{3})\) that is tangent to the indifference surface (in
blue) going through the point \((x_{1},x_{2},x_{3})\). The plane
corresponds to some combination of prices and income
\((p_{1},p_{2},p_{3},y)\). If we put the consumer in a situation where
he has disposable money \(y\) and is faced with prices
\((p_{1},p_{2},p_{3})\), he will choose the point
\((x_{1},x_{2},x_{3})\). From that we will be able to learn that the
indifference surface going through the point \((x_{1},x_{2},x_{3})\) is
tangent to the plane corresponding to \((p_{1},p_{2},p_{3},y)\).
Intuitively, since we can do this for all points, we can recover the
indifference surfaces. Using strict monotonicity, this allows us to then
infer the preferences, because it will allow us to rank all the
indifference surfaces.
In fact, the procedure works in any dimensions.