three-asset frontier · simplex → bullet → (σ, μ) projection · drag the center panel to rotate
1 · Weight simplex
corners = 100% in one asset · color = barycentric mix
A
B
C
2 · The "bullet" — (σ, μ, w₃)
drag to rotate · pull the flatten slider to collapse the 3rd axis
3 · Projection → frontier
looking straight down the 3rd axis · curve = efficient frontier
Every dot is the same portfolio across all three panels (matched by color). The real map is
simplex → (σ, μ), a 2-D→2-D map that's many-to-one — which is why differently-colored dots
pile onto the same spot in panel 3 and the region fills in. The bullet in panel 2 only looks
3-D because we re-attached one weight coordinate to un-stack those collisions. Drag the flatten
slider to 0 and the bullet squashes into exactly panel 3: no risk-return information was ever lost,
because the third axis carried none. The efficient frontier is the left edge.
1 · Weight simplex (tetrahedron)
4 corners = 100% in one asset · 3-D, drag to rotate
A
B
C
D
2 · The bullet — (σ, μ, w₄)
a filled volume now — one extra axis can't fully un-stack a 3-D domain
3 · Projection → frontier
same (σ, μ) target · curve = efficient frontier
With four assets the simplex is a tetrahedron (3-D), so the real map is now
3-D → (σ, μ) — even more many-to-one. Re-attaching one weight coordinate (panel 2)
no longer separates everything: the cloud is a genuinely filled solid, not a thin surface,
because you'd need two extra axes to fully un-stack a 3-D domain. Flatten it and it still
collapses to the same 2-D frontier in panel 3 — driving the point home that the projected-away
dimensions are bookkeeping coordinates of weight space, never risk-return content.