01 · Geometry
The infinite cut
Cut the largest square from a rectangle, then from what is left. Most rectangles run out of cuts. One never does, and what it leaves is always itself, smaller.
Enter the instrument02 · Iteration · open since 1937
The Collatz conjecture
Halve it if even; triple it and add one if odd. Every number ever checked falls back to 1. No one has proved that all of them do.
27 · 111 steps · peak 9,232
Trace any number03 · Recursion
The Ackermann frontier
Three rules and two small inputs. A(4, 0) is 13. A(4, 2) has nearly twenty thousand digits, and A(4, 3) could not be written down with every atom in the universe.
265536 − 3
19,729 digits, computed exactly
2003529930406846464979072351560255750447 … 5719156733
number of digits, on a logarithmic bar
04 · Feedback · solved exactly
Improve the generator
If progress makes progress faster, how fast? dx/dt = xp, starting at 1. Below p = 1 growth slows; at p = 1 it is exponential; above it, x reaches infinity at a finite time.
infinite at t = 5.00
Open the model05 · Scale · sourced lengths
The envelope, measured
From a proton to the observable universe is 42 orders of magnitude. Each tick is ten times the last. The smallest thing an unaided eye can resolve, about a tenth of a millimetre, sits near the middle.
Measure it