   Next: 5. Quantum Fractals and Up: 4. Notes Previous: 3. Importance of fuzziness

## 4. Geometrical meaning of the parameter It is instructive to have a visual picture of the map of the sphere implemented by the operator To this end let us assume the vector is pointing North, i.e. . Then the result of applying the operator to a point on the sphere is on the same longitude as the original point , but its latitude changes - it moves towards the North Pole along its meridian, the new latitude being given by the formula: (20)

Remark: Here is not exactly the "geographical latitude". It is zero at the "North Pole" ( ), degrees at the equator, and degrees at the "South Pole" ( ).

Each map maps the sphere onto itself in an injective way. For the map is easy to picture. All points of the sphere move towards the North Pole along their meridians, except of the two fixed points: North and South Pole. All of the Northern hemisphere, and a strip below the equator, shrinks, while the other part, near the South Pole, stretches. The amount of stretching can be found by plotting the function - it has a maximum at corresponding to . Thus the parameter gets a simple interpretation: it is the value of coordinate for which shrinking of meridians is replaced by stretching - an equilibrium point. This point is always on the southern hemisphere. For close to zero, where the map is close to the identity map, the equilibrium point is close to the equator. Then, as approaches the value of , corresponding to the sharp projection operator, the equilibrium latitude gets closer and closer to the South Pole. In the limit of all of the sphere shrinks to the North Pole, only the South Pole remains where it was.   Next: 5. Quantum Fractals and Up: 4. Notes Previous: 3. Importance of fuzziness

2002-04-11