Navigating the Eightfold Way

Concept: Plotting baryons by isospin projection I₃ (x) and strangeness S (y) reveals the SU(3) "Eightfold Way" multiplets. Charge obeys the Gell-Mann–Nishijima rule Q = I₃ + (B+S)/2, and the one empty corner of the spin-3/2 decuplet predicted the Ω⁻ (1962) before its discovery at Brookhaven (1964) — with the equal-spacing mass ladder even predicting its mass (~1680 MeV vs. measured 1672 MeV).
Serves: Modern-physics and intro particle units: quark model, quantum numbers, and how symmetry patterns drive prediction (like Mendeleev's gaps in the periodic table).
Inquiry: Sort tiles onto the (I₃, S) chart; a symmetric pattern emerges; one corner stays empty. Students predict the missing particle's charge from the pattern, then click the gap to check against history.
Misconception: "I₃ = 0 means electrically neutral." Charge is constant along diagonals, not columns: the I₃ = 0, S = −3 corner demands Q = −1. This wrong intuition is one of the guided-inquiry choices, rebutted only when chosen.

Principle	Where it appears
Predict before reveal	Gated inquiry cards make students commit the gap's charge and mass before the reveal; feedback compares prediction to the 1964 measurement.
Misconception as a choice	The "Q = 0" choice encodes the column-equals-charge intuition; its rebuttal appears only when selected.
Invariants made visible	Rows share S (tile color), dashed diagonals share Q, and the mass ladder shares ≈147 MeV spacing.
Operational quantities	Every tile carries Q, I₃, S computed from its quark content — readable in the particle card.
Formal layer on demand	Gell-Mann–Nishijima and the equal-spacing mass rule live behind the ⚛︎ The Physics button, which opens a full revision sheet.

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Speed

Eightfold Way — hadrons on the (I₃, S) chart

click any tile for its story

Guided Inquiry

1 · A map of baryons

Let’s start with the map. Spin-½ baryons sit on a chart of isospin projection I₃ (across) and strangeness S (down), and together they make a hexagon with a doubled centre. The colour of each tile tracks S.

2 · Where is charge constant?

Now let’s find the charge. Electric charge Q isn’t the same number as I₃. So on this chart, along which line do you think Q stays the same?

3 · The empty corner

Now meet the spin-3/2 family. It’s a triangle with one empty corner, at I₃ = 0, S = −3. What charge do you think belongs there?

4 · Price the missing baryon

One last step: let’s put a price on it. In the decuplet, each extra unit of strangeness adds about 150 MeV (1232 → 1385 → 1533). What mass do you think belongs at S = −3?

Listen

Multiplet

Mode

Layout

Charge diagonals Quark content on tiles

Particle card

—

Quantum numbers

Q— I₃— S— mass—

Click any tile.

Q = I₃ + (B + S)/2 (B = 1 for baryons)

m: 1232 → 1385 → 1533 → predicts m(Ω) ≈ 1680 MeV

Gell-Mann–Nishijima rule, equal-spacing mass ladder (Ω⁻ measured: 1672 MeV).

The Physics

The Eightfold Way · what this sim shows, the physics behind it, and every control

At a glance

The Eightfold Way sorts the hadrons by two quantum numbers, isospin projection I₃ and strangeness S, and the particles fall into regular shapes. This sim lets you build those charts tile by tile, read each particle's numbers, and see what an empty seat in the pattern means.

What you'll learn

How a baryon's place on the chart follows from the quarks inside it.
How electric charge Q relates to the two chart coordinates, I₃ and S.
How to read a particle's quantum numbers off its tile.
What a gap in the pattern tells you about a particle nobody has seen.

The spin-½ baryon octet as the sim draws it. Σ⁰ and Λ⁰ share one point.

What's on screen

Tiles: one disc per particle, placed by I₃ (across) and S (down). The colour is the S row: blue 0, teal −1, purple −2, gold −3, and pink +1 for kaons.
Empty seat (dashed ring): the decuplet position at I₃ = 0, S = −3, when the nine known tiles are placed and no tenth tile is. Clicking it fills it.
Particle card (sidebar): Q, I₃, S, mass and quark content of the selected tile.

See itOpen the spin-½ baryon octet with every tile placed. Watch for: three rows of different colours, with two tiles sharing the middle of the second row.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Mode	Which family of particles is on the chart: baryon octet (spin ½), baryon decuplet (spin 3/2), or the meson nonet with spin 0 or spin 1. Switching clears the selected tile.	4 buttons
Scatter tiles (Layout)	Throws the current family's tiles onto a shelf under the chart, to be dragged back by hand. A tile dropped on its own seat gets a green ring. On the wrong seat it gets a red cross and returns to the shelf.	button
Place all (Layout)	Puts every tile of the current family on its seat. In the decuplet, after Scatter tiles, the tenth seat stays empty and shows as a dashed ring.	button
Charge diagonals (Overlays)	Shows or hides the dashed lines labelled Q = …, one for each charge on the chart.	on / off
Quark content on tiles (Overlays)	Writes each tile's quark letters on its face, such as uud for the proton.	on / off
Tiles on the chart	Click a tile to select it and fill the Particle card. After Scatter tiles, drag tiles from the shelf onto the chart.	click / drag
Particle card	Shows Q, I₃, S, mass and quark content of the selected tile, with a one-line note on its discovery.	panel
Speed	Paces the ring that flashes when a tile lands. The chart itself does not change.	0.25×–4× opens at 1×
↻ Reset	Puts the tiles back and restarts from the opening state: in free exploration, everything placed, Ω⁻ shown and the diagonals on.	top bar

Fixed: mass never moves a tile. It appears only in the Particle card.

Quarks set each tile's place

A baryon is three quarks drawn from u, d and s. Counting them gives the numbers the chart uses:

n_u, n_d, n_s are the numbers of u, d and s quarks in the baryon.

S counts strange quarks. Each s quark lowers S by 1. The colour of a tile is its S row, so colour is a count of s quarks.
I₃ counts u against d. Each u adds +½, each d adds −½, and s adds nothing. The proton, uud, has I₃ = +½ and S = 0.
B, the baryon number, is 1 for every baryon, so it never moves a tile.
Spin chooses the chart. The spin-½ baryons fill the octet, a hexagon with two tiles at its centre. The spin-3/2 baryons fill the decuplet, a triangle with rows of 4, 3, 2 and 1 seats.

See itWrite the quark content on every tile of the octet. Watch for: the number of s in the letters going up by one from each row to the next.

Charge runs along diagonals

Electric charge Q is not a third axis. I₃, S and B fix it through the Gell-Mann–Nishijima formula:

B = 1 on the baryon charts and B = 0 on the meson nonets.

A row shares S and a column shares I₃, but neither shares Q. Holding Q fixed and solving for I₃ gives a line:

Each row down moves the line half a column to the right.

Charge is constant along lines that slope down and to the right, not along rows or columns.
The decuplet has four such lines (Q = +2, +1, 0, −1). The octet and each meson nonet have three (Q = +1, 0, −1).

See itSwitch on the Charge diagonals in the octet. Watch for: dashed lines leaning down to the right, with n, Σ⁰, Λ⁰ and Ξ⁰ all on the Q = 0 line.

Reading a tile with the formula

To read a tile, take B, S and I₃ from its quark content, then apply the formula. Pick a particle below, or click a tile on the chart.

Multiplet Particle Charge diagonals Quark content on tiles

These are the sim's own controls. Changing them here changes the chart.

Mesons are a quark and an antiquark, so B = 0. An antiquark has the opposite S and I₃ to its quark, so for a meson Q = I₃ + S/2.

The meson nonets: an antiquark flips the sign of S and I₃.

A check: adding the quark charges (u +⅔, d −⅓, s −⅓) gives the same Q. The proton, uud, has ⅔ + ⅔ − ⅓ = 1.
Only B, S and I₃ reach the chart. Spin chooses which chart, and mass appears only in the Particle card.

See itSelect Ξ⁻ (dss) in the octet, with quark content on. Watch for: the Particle card reading I₃ = −1/2, S = −2 and Q = −1, as the formula gives.

Where the shapes come from

The shapes are not chosen. Three quark flavours can combine only into a few fixed families:

Three quarks make baryons; a quark and an antiquark make mesons.

The sim shows the 10 with spin 3/2 (the decuplet) and one of the 8s with spin ½ (the octet). A meson nonet is an 8 plus a 1, nine tiles in all.
One point can hold several tiles. Σ⁰ and Λ⁰ are both uds, so both sit at I₃ = 0, S = −1. They differ in total isospin (Σ⁰ is in the I = 1 triplet with Σ⁺ and Σ⁻, Λ⁰ is alone) and in mass, 1193 and 1116 MeV. The nonet's centre holds π⁰, η and η′ in the same way.
The pattern is approximate. The s quark is heavier than u and d, so the masses climb as you go down the rows.

See itOpen the spin-0 meson nonet. Watch for: six tiles around the edge and π⁰, η and η′ crowded at the centre.

The empty seat became the Ω⁻

By 1962 nine of the decuplet's ten seats were filled. The tenth, at I₃ = 0 and S = −3, already had a charge and a rough mass.

Charge. With B = 1, S = −3 and I₃ = 0, Q = 0 + (1 − 3)/2 = −1. That is the line through Δ⁻, Σ*⁻ and Ξ*⁻.
Mass. Down the decuplet each added unit of strangeness costs a similar step. One more step gives about 1680 MeV.

Masses of the decuplet rows, with one more equal step.

Gell-Mann announced this in 1962. The Ω⁻, made of three strange quarks (sss), was found at Brookhaven in 1964, and its mass is now measured as 1672 MeV. The steps are only roughly equal: 153 and 148 MeV between the first rows, then 139 MeV from Ξ* (1533) to the real Ω⁻.

See itStart with the tenth seat empty, then fill it. Watch for: the Particle card reading Q = ? and mass ?, then Q = −1 and 1672 MeV once Ω⁻ takes the seat.

Where it matters

The quark model. The octets, decuplet and nonets are exactly the families that three quark flavours allow. Gell-Mann and Zweig proposed quarks in 1964 to explain this pattern.
Strangeness as bookkeeping. The strong interaction conserves S and the weak interaction does not. Strange particles are therefore made in pairs and decay slowly, typically after about 10⁻¹⁰ s instead of 10⁻²³ s.
Why colour was needed. The Δ⁺⁺ is uuu with all three spins aligned, which looked like a violation of the exclusion principle. A new three-valued quantum number, colour, resolved it.

Go deeper: why only a 10 and one 8 for the lightest baryons

In the lightest baryons the quarks share the same spatial state, and colour is antisymmetric under exchange. The remaining spin and flavour part must then be symmetric.
That allows only the 10 with spin 3/2 and one 8 with spin ½. The other 8 and the 1 appear only among excited baryons.

Summary

In one line. Quark content places each hadron by I₃ and S, charge follows as Q = I₃ + (B+S)/2, and a gap in the pattern fixes the charge of a particle that has not been seen.

Key points

S = −n_s and I₃ = (n_u − n_d)/2 come from counting quarks.
Lines of constant Q slope down to the right.
Mass is not geometry. In the decuplet it rises by roughly 150 MeV per row.

Common mix-ups

"I₃ = 0 means the particle is neutral." Only when B + S = 0. At I₃ = 0, Q = (B+S)/2: Σ⁰ is neutral, but Ω⁻ has Q = −1.
"A row of one colour is a row of one charge." Colour marks S. The Σ⁻, Σ⁰ and Σ⁺ in one octet row have charges −1, 0 and +1.
"Σ⁰ and Λ⁰ are one particle drawn twice." They share quarks and a point, but differ in total isospin and mass.

Things to try on your own

In the octet, switch on Charge diagonals and list the tiles on each line. Repeat in the decuplet.
In the decuplet, read each row's mass from the Particle card and work out the step between rows.
In the spin-0 nonet, select K⁺ and K⁻. Compare their I₃ and S, and check Q = I₃ + S/2 for each.