The toy shelf
Four little experiments live here. Each one gives the hand something to do, then points back to the page whose mathematics it carries.
Unit needles fall onto unit-spaced ruled paper. The crossing count gives a running estimate of π, one toss at a time.
The estimate is 2N/C when needle length and line spacing are both one.
Structure shelfAt each peg, a bead chooses left or right. The live histogram gathers those choices into a bell-shaped pile.
Each row doubles paths; the final bin is the sum of twelve independent left-or-right choices.
Boltzmann's entropyA point starts at the center, chooses a corner at random, and jumps halfway there. Repetition draws a Sierpiński triangle from almost nothing.
Each update contracts the distance to the chosen corner by exactly one half.
Lorenz attractorEvery room is occupied, and a new guest arrives. Each resident moves from room n to room n + 1, leaving room one open.
The shift n ↦ n + 1 is a bijection from all occupied rooms to every room except the first.