SAME LAW
Same rule. Different histories.
Held constant(x, y) → (2x + y, x + y) mod 1
Read the study
Three exact histories
Opposite edges join. Dots show the last 13 discrete states; no continuous path between them is asserted.
Distance from A
· · logarithmic distance axis
Knowing the rule is not knowing the history.
Start at step 0, then play. Try a smaller ε and watch when the nearby histories become distinguishable. The rule and its parameters never change.
- Initial distance from A
- B distance now
- C distance now
First distance ≥ 0.1
What stays the same
The map, the square with opposite edges identified, and the grid of spacing 2⁻⁴⁰. A starts at (Q/8, floor(Q/5))/Q, with Q = 2⁴⁰. B adds ε to x; C adds ε to y.
What changes
Only an initial coordinate. The law is part of the initial-value problem; it is not the complete cause. The same complete starting state would reproduce the same history.
What the distance means
Distance is the shortest Euclidean displacement on the wrapped square: for each coordinate, use the smaller of its difference and one minus that difference. The threshold 0.1 is a declared visual separation criterion, not a universal prediction horizon.
Dots mark discrete updates. No path between successive dots is asserted. The strip below plots distance, not the speed of a continuously moving particle.
Exact evolution, finite display
The state is stored as two integers modulo Q. Every update uses exact integer arithmetic. The inverse is (x − y, −x + 2y) mod Q, so distinct starts cannot merge through rounding. Conversion to screen coordinates and the distance calculation use ordinary floating point.
This finite grid has periodic orbits. The exhibit demonstrates finite-time sensitivity for these starts; it does not claim that a finite computer orbit is an aperiodic chaotic trajectory.
Why this map separates nearby starts
The matrix has determinant 1 and eigenvalues (3 ± √5)/2. Before wrapping, a displacement with an expanding component grows under repetition. Once the points separate around the square, their shortest distance can decrease as well as increase.
The construction is the Arnold cat map on a rational grid. Further reading: Hall, Arnold’s Cat Map: An Exposition.