Tegmark's Level I multiverse needs no exotic physics β only two mundane facts: space is (plausibly) infinite, and any bounded region can hold only a finite number of distinct particle arrangements, because it has finite energy and finite volume (a Bekenstein-bound argument caps its entropy, hence its number of microstates N).
Regions with the same macrostate β same N possible microstates
Birthday-paradox estimate for the first repeat:
k* β sqrt(Ο/2 Β· N) β 1.2533 Β· βN
P(at least one repeat among k regions):
P(k) β 1 β exp( βk(kβ1) / (2N) )
This simulator makes N tiny (8β400) so the repeats happen on-screen in seconds. In the real universe N is roughly the number of quantum microstates in a Hubble volume, of order 10^(10^118) β astronomically larger, but still finite β so the same birthday-paradox logic implies an identical copy of our observable universe exists, just unreachably far away (Tegmark's own estimate: roughly 10^(10^118) metres).
- N slider β how many distinct arrangements a region can hold; smaller N forces repeats sooner (a smaller "alphabet" to draw from).
- Lattice size β how many Hubble volumes are laid out along each axis (gridΒ³ total regions).
- Reseed β draws a fresh random arrangement for every region with a new seed, like sampling a different slice of an infinite universe.
- Show: all pairs / nearest only β toggle between every duplicate pair found, or just the closest one to the centre region.
Real-world relevance: this is the same combinatorial argument used in cryptographic collision attacks (the "birthday attack") and in estimating hash-function collision rates β cosmology, quantum computing and cryptography all lean on the identical sqrt(N) scaling law.