One lineage — or 3, or 50, every one of them started from the identical figure — copied for up to 400 generations, with no population, no death and nothing competing. A shape is a list of independent parameters: 18 for the polygon, 26 for the fish, 11 for the flower, counting the colours and the stroke width as well as the geometry. Each generation every parameter takes its own Gaussian step of standard deviation σ and is then clipped to the range that keeps the figure drawable, the Mutation size slider setting σ as a fraction of that range — about 0.017 at setting 1, 0.041 at 3, 0.065 at 5. The Δ on each card and the pairwise Δ under the stage are one and the same measure, a root-mean-square over the K parameters:

Δ = √[(1/K)·Σ(δ_k/range_k)²]Δ = 0.10 means the traits differ by a tenth of their range on average — normalising by the range is what makes the number comparable between a polygon, a fish and a flower, which have neither the same parameters nor the same units

Every parameter is therefore an unbiased random walk, and everything the room shows falls out of that one fact.

x(t) − x(0) ~ Normal(0, σ²t)no step is aimed at anything, so the cloud of 50 lineages spreads without travelling: the variance grows with time, the average does not move

E[Δ²] = σ²t, so Δ ≈ σ√tthe "Δ from origin" on each card — divergence grows with the square root of elapsed time, so twice as different takes four times as long

for two lineages, E[Δ²] = 2σ²t, so Δ_pair ≈ √2·Δ_origintwo independent walks instead of one: the divergence line under the stage should read about 1.41 times the figure on the cards, and over 500 simulated runs it comes out at 1.40–1.43

Both laws hold until the walks start meeting the bounds, and then they stop, because the parameters are clipped rather than reflected: Δ bends away from σ√t and flattens towards a ceiling a little under 0.5. At the middle Mutation size the fit is good to a few per cent for the first 25 generations and about a quarter low by generation 100; at the smallest setting it holds out to roughly 100. That ceiling is the same wall the Branching Room meets from the other side — two lineages that have both saturated are as different as this model can make them, and more time will not separate them further.

Ancestor A splits into B and C at generation 0; where the other three splits fall — B→D,E · E→F,G · F→H,I — is yours to set. The five surviving lineages (C, D, G, H, I) then mutate independently until generation N, every one at the same rate. The step is smaller than in the Copying Room, by design: σ is about 0.007 at Mutation size 1, 0.012 at 3 and 0.017 at 5, against 0.017, 0.041 and 0.065 there — two to four times gentler. A run here is five times as long by default, and at the Copying Room's step the shapes would reach the bounds of their traits, and stop recording anything about the tree, well before the last split.

Why the reconstruction goes wrong. Each trait is an independent random walk — one Gaussian step per generation, of a standard deviation σ that the Mutation size slider sets — so two lineages that parted t generations ago have each taken t steps away from their common ancestor.

E[(x₁ − x₂)²] = 2σ²tfor one trait measured in two lineages, x₁ and x₂, that split t generations ago: the distance between them grows as √t, so it is proportional to elapsed time only under the square root

The expectation is proportional to time since the split; any one pair of lineages is a single draw from a distribution whose spread is as large as its mean. That gap between expectation and draw is the whole of the discrepancy between the true tree and the reconstructed one. Traits are also bounded, so a lineage that has wandered as far as the shape allows stops getting more different, and a very deep split ends up looking no older than a moderately deep one.

Haploid Wright–Fisher: each generation is N individuals drawn with replacement from the one before it, so a mutation at frequency p is passed on a binomial(N, p) number of times. Every trajectory drawn here is a real run of that model — none is scripted — and the batch behind the tally is re-run each time you press Simulate, which is why the numbers move a little from press to press.

What the theory predicts. The mutation starts at p₀ = 1/N, and for this model the diffusion approximation gives three results the room can be checked against.

u(p₀) = p₀ = 1/Nprobability of fixation — a neutral allele takes over exactly as often as it is already common, so roughly one population in N

E[T_fix | it fixes] ≈ 2N generationshow long the rare winner takes to get from one copy to every copy

E[T_loss | it is lost] ≈ 2·ln N generationshow long the usual loser lasts

At N = 100 those last two are about 200 generations against about 9 — which is why the losing population is finished while the winner has barely started, and why the batch behind the tally has to run on the order of N populations before it finds a winner to show you.

The first two generations are sampled one individual at a time on the wheel, in real time, so you can see where a frequency actually comes from. After that each generation is drawn in one go, without the wheel, and the run moves on at about five generations a second: fast enough to watch, slow enough to follow. Nothing is replayed from a script — each frequency is the outcome of its own binomial draw.

The three results the panels are drawing. Write M for the gene copies sampled each generation: M = N under haploidy, M = 2N under diploidy.

Var(Δp) = p(1 − p)/Mhow far one generation of sampling can move the frequency: largest at p = 0.5, and smaller in every larger population — this is the curve the variance panel plots

u(p) = pprobability of fixation: with nothing favoured, an allele's chance of ending up as the only one is simply how common it is now

H(t) = H₀(1 − 1/M)^t ≈ H₀e^−t/Mheterozygosity — the variation still left — decays by a factor 1/M per generation, so half of it is gone after about 0.69M. This is the dashed line on the Run 10 Simulations heterozygosity chart

E[T_fix] ≈ 2M generationshow long an allele that does fix takes to get there — 4N under diploidy, 2N under haploidy

Read together they say that fixation is not a special event but the end every run has: drift removes variation at a rate set by M and nothing else, and a locus started at p = 0.5 reaches one edge or the other after about 1.4M generations on average.

What the mutation slider does. Each gene copy has a probability μ of turning into the other allele as it is passed on, the same rate in both directions — so mutation on its own pulls the frequency towards 0.5 rather than towards either allele. It is applied to the frequency the next generation is drawn from, after the wheel has been sized, which is why the wheel grows a small tick at the un-mutated frequency once μ is above 0: the gap between tick and slice is what mutation did. Two things follow. Sampling is now from p′ rather than p, which is what the binomial panel's label changes to say; and 0 and 1 stop being absorbing, so a run does not end when it reaches one.

Composing mutation with sampling gives one exact recursion for the expected heterozygosity, and it is what the dashed line is drawn from at every setting:

E[H_t+1] = (1 − 1/M)·[(1 − 2μ)²H_t + 2μ(1 − μ)]at μ = 0 this is the decay above; above 0 it climbs or falls to a balance instead of to zero. At M = 50 and μ = 0.005 the balance works out at H = 0.247, and 4,000 simulated runs put it at 0.249

Where the default G comes from. The generations slider opens at 1.5 × E[T_fix], which is 3M, so a run at the default settings has room to finish instead of stopping part-way through the wander. It follows N and the ploidy as you change them, and stops following the moment you move the slider yourself. It is a generous figure at p = 0.5: 2M is the time for an allele starting from a single copy, and one starting at half takes closer to 1.4M, so most default runs reach an edge with generations to spare.

Each generation the raw frequency p is reweighted to p′ = p(1+s) / (p(1+s) + (1−p)) before the usual binomial sampling — so p′, not p, is what the wheel spins on and what the binomial panel draws from. In Diploid mode the weights are genotypic — w(A₁A₁) = 1+s, w(A₁A₂) = 1+h·s, w(A₂A₂) = 1 — and p′ is the allele frequency computed from those weighted genotypes. The dashed line on the chart is the path that same reweighting would take with no sampling noise at all.

Where that leads. Without sampling, the haploid recursion has a closed form — the allele's odds multiply by (1 + s) every generation:

p_t/(1 − p_t) = [p₀/(1 − p₀)]·(1 + s)^tthe dashed line: an S-shaped sweep, slow while the allele is rare and slow again while the last few of the other allele are mopped up

Sampling turns that certainty into a bet. The fixation probability from frequency p, for the haploid model, is Kimura's diffusion result:

u(p) = (1 − e^−2Nsp)/(1 − e^−2Ns)for one new copy, p = 1/N, this is about 2s whenever Ns ≫ 1 — an allele with a 10% advantage is still lost about four times in five

How long that sweep takes, once it is under way, is the other figure the room uses:

E[T_fix] ≈ (2/s)·ln M generationswith M the gene copies — (2/s)·ln 2N under diploidy — so a strong advantage sweeps in a few dozen generations and a weak one takes thousands

Where the default G comes from. The generations slider opens at 1.5 × that, and follows N, the ploidy and s until you move it yourself. It is capped at the neutral 3M, because (2/s)·ln M runs away as s approaches zero while the truth does not: with s that small it is drift that decides the outcome, on drift's timescale. That cap is also what the slider does at s = 0, where the room is the Drift Room with an extra control.

Two limits are worth setting the sliders to. When |Ns| is around 1 or less the exponentials nearly cancel and u(p) → p, the neutral answer: the advantage is real, but the population is too small for it to be the thing that decides. When Ns ≫ 1 the runs hug the dashed line. In Diploid mode the same approximation for a new mutation is 2hs rather than 2s, which is why lowering h towards 0 makes a genuinely advantageous allele so easy to lose: while it is rare it is almost only ever in heterozygotes, and a recessive advantage is invisible there.

One spin makes four derived alleles at four independent loci, each entering a population of N as a single copy at frequency 1/N. Every run tracks at least G = 2.5·N generations and extends automatically until all four have fixed or been lost, which is when the four lines come to share one axis. With selection on, each allele is reweighted by its fitness exactly as in the Selection Room — and no value of s ever resizes a wheel slice.

What the four lines should do. Each allele enters a haploid population at p₀ = 1/N, so with selection off:

u = 1/N for each · E[number of the four that fix] = 4/Nat N = 50 that is 0.08 alleles per spin, so most spins end with all four lines on the floor

E[T_fix | it fixes] ≈ 2N generationswhere the 2.5·N default for G comes from: long enough for a typical winner to finish, short enough that the chart is not mostly flat line

Switching selection on replaces the first of those with u(1/N) = (1 − e^−2s)/(1 − e^−2Ns) ≈ 2s for a beneficial allele — a different number, arrived at without touching a single wheel. That is the room's whole point written as two equations: the wheel sets how often a variant is made, s sets what becomes of it afterwards, and they are separate parameters of the same run.

One generation, in order. Each population independently draws N individuals from its own current gene pool — a binomial draw at its own frequency, the Drift Room's recursion run twice — and only then are the migrants exchanged. So the frequencies the chart plots are post-migration frequencies, and the drift that happened in a generation is always drift within a population, never drift on a pool the two share.

Fractional migrants. Settings below one migrant are real settings, not rounded ones: at M = 0.3 a single individual crosses in three generations out of ten, chosen at random, and none in the other seven. The number that crosses in a given generation is shown on the panel, so a reader can see that "0.3 migrants" is a statement about the long run and never about a generation.

What the exchange actually does. M individuals are picked at random from A and M from B, and they swap places. Both populations therefore stay at exactly N, and the expected change in A's frequency in one exchange is

E[Δp_A] = m(p_B − p_A), with m = M/Ngene flow always pulls a population towards the other one, by a fraction m of the gap between them — so it can close a difference but never create one

How differentiation is measured. F_ST compares the variation within the two populations against the variation in the two taken together. Writing p̄ for the average of p_A and p_B:

H_S = ½[2p_A(1−p_A) + 2p_B(1−p_B)] · H_T = 2p̄(1−p̄) · F_ST = 1 − H_S/H_T0 when the two populations have the same frequency, 1 when each is fixed for a different allele, and undefined once neither carries any variation at all

What the sweep measures, and why no curve is drawn over it. Each point is a hundred runs at that setting. The teal series is the mean of each run's own mean F_ST; the gold series is the share of those hundred runs in which F_ST reached 0.5 at any point, which is this room's operational reading of "completely differentiated".

Every sweep run starts at p = 0.5 in both populations, whatever the deck says, and that is not a detail. The sweep asks how much differentiation drift manages to build against a given amount of gene flow, which can only be measured from populations that begin with none. Started at the default p = 0.90 and p = 0.10 the two are already past the 0.5 mark before the first generation is drawn, and the panel would report every setting as complete differentiation and mean nothing by it.

The result every textbook quotes is F_ST ≈ 1/(1 + 4Nm), and it does not describe this room directly. It is the many-island result, and with two populations the coefficient is about four times larger — closer to 1/(1 + 16Nm) — because each population's immigrants all come from the only other place there is. It is also an equilibrium between migration and mutation, and with the mutation slider at its default of 0 — which is where every figure below was measured — nothing here replaces what drift takes away.

Given both of those, the measured means come out closer to the two-population formula than you might expect. At N = 50 they run about 0.31, 0.20, 0.15, 0.09, 0.05 and 0.03 as Nm goes 0.1, 0.3, 0.5, 1, 2, 4, against 0.38, 0.17, 0.11, 0.06, 0.03 and 0.015 from 1/(1 + 16Nm). The same shape throughout, crossing from a little below the formula where migration is weakest to about double it where migration is strong — the excess a pair of populations on their way to losing all their variation would be expected to carry. The room plots what it measured and leaves the formula in this paragraph, where its assumptions can be stated alongside it, rather than drawing it across the panel as though it were the answer.

Turning the mutation slider up does not close that gap — it opens a different one, and the reason is worth having. Mutation does the job it was added for: the variation stops running out, so a run can be carried as long as you like and F_ST settles at a value rather than being a quantity measured on the way to nothing. But it does not push the numbers up towards the formula. Over a thousand runs the no-migration end falls — mean F_ST of 0.44, 0.32 and 0.24 at μ = 0, 0.002 and 0.005 — and carried all the way to stationarity it falls further still. The reason is that this room has two alleles, and symmetric mutation between two alleles pulls both populations towards p = 0.5. It is a homogenising force in its own right: a second thing making the two more alike, working alongside the migrants. The formula assumes the opposite arrangement — infinitely many alleles, so mutation only ever makes variants that are new, adding diversity inside a population without making two populations any more similar to each other. What survives is the part that matters: the share of runs reaching 0.5 at Nm = 1 is 27%, 25% and 24% across those same three settings. Gene flow's cliff does not move.

Where the default G comes from. 200 generations is 4N at the default N = 50 — long enough for a run with no migration to reach an edge, and for a run with migration to settle and stay settled long enough to be worth averaging. Like the Drift Room, it follows N until you move it yourself.

Each of the five traits is its own independent locus. While it is monomorphic it has a chance μ per generation of producing a new variant, which then runs a real diploid Wright–Fisher trajectory — selection and drift, with the beneficial allele fully dominant (h = 1) — until it fixes or is lost. A mutation's selection coefficient is signed at the moment it arises, positive if the step it makes happens to move the trait the way its habitat prefers and negative if it does not, and it does not change afterwards.

Two rates set the pace, and only one of them involves N. Supply first: a monomorphic locus waits a geometric number of generations for its next mutation.

E[wait for a mutation] = 1/μ generationsabout 100 generations per locus at the default μ = 0.010 — and N does not appear, because μ here is per locus per generation, not per gene copy

Then fate. The derived allele is fully dominant, so its advantage is exposed from the first heterozygote and Haldane's approximation applies directly:

u ≈ 2s when the step went the way the habitat prefers · u = 1/(2N) when the habitat cannot see the trait at alla mutation's step points the way the habitat prefers about half the time, so adaptive substitutions arrive at about μ · ½ · 2s = μs per generation, and the expected wait for one at a locus is about 1/(μs) — roughly 500 generations at the defaults μ = 0.010, s = 0.20

Together they say what a larger N is and is not doing here. It buys no extra mutations at all; what it buys is sharper sorting of the ones that arrive, because the neutral route to fixation shrinks as 1/(2N) while the selected route stays at 2s. The ratio between the two is 4Ns, so a small population fixes a great deal of what its habitat never asked for.

The ancestral population. 1,013 independent diploid loci. Each locus's minor-allele frequency is drawn from the mutation–drift stationary distribution Beta(θ, θ) and rounded to a whole number of copies out of 2N, so about a third of the loci come out monomorphic and the variation at the rest is mostly rare — a median minor allele of about 7% at the default θ; the polymorphic count and the heterozygosity on the ancestral card are results, not settings. θ is a control of its own rather than 4Nμ computed from the run, because it describes the ancestor's long-term history and its N_e need not equal the N the daughters get.

The three kinds of locus. 13 build the five traits (shape 3, colour 4, eye size 1, fin size 2, tail size 3), each allele carrying an effect vector that its habitat scores. 40 are physiological: no effect vector, only a position on one axis of demand with the ancestral allele at 0, still water asking for −1 and fast water for +1, and an allele favoured when it brings the locus closer to what its habitat needs. The remaining 960 have no effect a habitat can see.

Each generation. Every locus runs its own Wright–Fisher trajectory in each population: binomial sampling of the 2N gene pool, plus selection where the habitat has a preference, with the beneficial allele dominant (h = 1) so a new mutation is exposed from its first heterozygote. A locus carries at most two alleles at a time; one that has gone monomorphic can mutate, producing an allele that has never existed before at one copy in 2N. Mutations are local to the population they arise in.

What is counted. A fixed difference is a locus where the two populations are each fixed, for different alleles. A mutation is a new allele that arose after the split and a substitution is one that reached fixation; standing variants sorting are counted as neither. Phenotypes are additive in allele dosage, so a population's mean fish is the baseline plus each allele frequency times its effect. Speciation is declared at X fixed differences, or, with body colour acting as a mating signal, at Y degrees of divergence in mean body hue.

The standard results behind the numbers on the cards. The ancestor's frequencies come from Beta(θ, θ), whose mean heterozygosity has a closed form:

E[H] = 2·E[p(1 − p)] = θ/(2θ + 1)0.12 at the default θ = 0.16 — which is the figure the ancestral card reports, up to the rounding to whole gene copies

After the split, what separates the three kinds of locus is not how often they mutate — that is μ for all of them — but what happens to a mutation once it exists. Each enters at 1/(2N), and:

invisible locus: u = 1/(2N) · scored locus: u ≈ 2s · ratio = 4Nsat N = 200 and s = 0.20, a locus the habitat can see substitutes about 160 times as readily as one it cannot

That ratio is why the 53 loci a habitat scores diverge visibly over a run while the 960 it ignores mostly sit still, and it is also why the two daughters are not symmetric: population B at N = 50 fixes neutral mutations four times as readily as population A at 200, so some of what separates them is drift in the smaller one rather than adaptation in either. The neutral substitution rate itself is μ/(2N) per locus per generation — but only for a run long compared with the ≈4N generations one neutral fixation takes, and at the defaults G = 1000 and N = 200 it is not, which is part of why so much of the early divergence is standing variation sorting rather than new mutation.

Two founders and four generations, run twice from the same start. There is no mutation and no selection here, and an allele is only an ancestry label: each individual is a single chromosome, drawn as its two copies. On the left every individual makes two exact clones of itself. On the right the two parents make two offspring, each receiving one gamete from each parent, and a gamete is built by a single crossover whose position is drawn uniformly from the middle 64% of the chromosome, with a coin deciding which copy contributes the top piece. Two individuals count as the same genotype when they carry the same pair of copies — crossover positions included, which is what the “distinct genotypes” readout is counting.

Numbers on one side.

asexual N(t) = 2^t+1 · sexual N(t) = 2 · ratio = 2^tthe two-fold cost of sex is charged every generation, so it compounds: 2, 4, 8, 16 against a flat 2, and the eight-fold gap on screen at generation 3 would be a thousand-fold by generation 10

Variety on the other.

asexual G(t) = 2, for every tcloning is the identity operation: it can copy a genotype but it cannot make one, so the count on the left never leaves the two the founders brought, however large the head count grows

sexual G(t) = 2, but never the same two twicethe crossover position is drawn from a continuous range, so two meioses coincide with probability zero and every individual on the right is a genotype that has never existed

What the drawing leaves out is worth a number too. One chromosome with one crossover is the smallest thing that shows the mechanism, and much less than meiosis really does: independent assortment alone, before any crossover at all, gives 2ⁿ distinct gametes from n chromosome pairs — 2²³ ≈ 8.4 million in a human, and merely 2 here, since n = 1. Nearly all the variety you can see on the right is coming from where the crossover happened to fall.

And one thing the short run hides. From generation 2 the only mates available are siblings, and continued full-sib mating removes heterozygosity at a fixed rate: H_t = ½H_t−1 + ¼H_t−2, which settles to a decline by a factor (1 + √5)/4 ≈ 0.809 each generation — about a fifth of what is left, every generation, for as long as the line continues. Four generations are far too few for that to bite — the run is short so that every genotype can still be drawn — but it does mean the right-hand panel understates what sex generates in a population where mates are not siblings.

One autosomal locus with two alleles, in a diploid population of N. G0 is whatever you build: with the random-mating panel closed the slider sets p and the parents are laid out in Hardy–Weinberg proportions at that p; with it open the two handles set the genotype mix directly and the allele frequency becomes a consequence of it, p = f(A₁A₁) + ½·f(A₁A₂). Mating then draws 2N gametes independently from that gene pool — each one A₁ with probability p — and pairs them two at a time. Nothing else happens to them: no selection, no mutation, no migration, and no second generation.

The theorem, and three things that follow from it.

f(A₁A₁) = p² · f(A₁A₂) = 2pq · f(A₂A₂) = q², with q = 1 − pthe probabilities of the two independent gametes an offspring gets, which is all "random mating" means — reached in a single generation from any starting mix, and unchanged in every generation after it

max 2pq = ½ at p = q = 0.5the peak of the middle curve: no population can have more than half its individuals heterozygous at a locus with two alleles

share of A₁ copies sitting in heterozygotes = 2pq/2p = qa rare allele hides in heterozygotes, in proportion to how rare it is: at p = 0.1, nine copies in ten are in an A₁A₂, and only p² = 1% of individuals are A₁A₁

E[p₁] = p₀ · Var(p₁) = p₀q₀/2Nthe assumption this room breaks on purpose: the theorem wants an infinite population, and drawing a real 2N gametes leaves this much wobble — a standard deviation of 0.05 at N = 50, p = 0.5

The rare-allele line is the answer to the Try this above, and it is also why the Selection Room's h carries so much weight: an allele that is rare is an allele selection almost only ever meets in a heterozygote, so a recessive one is nearly invisible exactly when it is scarce. The variance line is why Expected and After one random mating never land quite on top of each other. Because the gametes are drawn independently and then paired, each offspring is A₁A₁, A₁A₂ or A₂A₂ with exactly the probabilities above, so G1's genotype counts are a multinomial sample of size N from them and miss the expected bars by around √[f(1 − f)/N]. Press Mate randomly a few times at N = 10 and then a few more at N = 200: the same law both times, sampled coarsely or finely.

Gamete frequencies are written x(A₁B₁), x(A₁B₂), x(A₂B₁), x(A₂B₂), and the linkage disequilibrium between the two loci is D = x(A₁B₁)·x(A₂B₂) − x(A₁B₂)·x(A₂B₁), zero when the two are inherited independently. Recombination at rate r per generation erodes it geometrically: D(t) = D₀(1 − r)^t. Part 2 runs a two-locus Wright–Fisher model of a diploid population of N, bookkept as its 2N chromosomes. Fitness is multiplicative across the two loci and across an individual’s two chromosomes, which means there is no dominance and the gamete frequencies on their own are enough: selection reweights the four gamete classes, recombination reshuffles them at rate r, and 2N chromosomes are then drawn at random to make the next generation. Both beneficial alleles start at the frequency f₀ you set, on different backgrounds — at the defaults that is five copies of each in 2N = 100, not one — so the chromosome carrying both can only be made by recombination.

Two numbers to read off the sliders. The geometric decay above has a half-life:

t_½ = ln 2 / [−ln(1 − r)] ≈ 0.69/r generationsabout 69 generations to lose half the association at r = 0.01, and barely one at r = 0.5, which is what unlinked means

In Part 2 the quantity that decides the run is how fast the missing chromosome gets made. Both mutations start at f₀ on opposite backgrounds, so x(A₁B₁) = 0 and D = −f₀², and recombination moves −rD into the coupling classes each generation:

E[copies of A₁B₁ in the next generation] ≈ 2N·r·f₀²0.0025 at the defaults N = 50, r = 0.01, f₀ = 0.05 — some 400 generations of waiting for the first copy of it to appear

Meanwhile, by the closed form in the Selection Room, either mutation on its own would climb from 5% to 95% in about 30 generations at s = 0.20. The race is not close, and losing it is exactly what Hill–Robertson interference is: the two alleles settle their contest between them long before recombination can put them on the same chromosome.