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For a normal split toric variety $X_{\Sigma }$ of dimension $n$, the orbit-cone correspondence assigns an orbit $O(\sigma )$ of the algebraic torus to each toric cone $\sigma \in \Sigma$, withThe zero toric cone gives the dense algebraic torus; a ray of a fan gives a codimension-one orbit. The torus-fixed points correspond exactly to full-dimensional toric cones. A maximal toric cone need not be full-dimensional: the toric fan containing only the zero toric cone gives the algebraic torus itself and has no fixed point when $n>0$.

For affine space ${\mathbb{A}}^2$, the first-quadrant toric fan has a two-dimensional zero-cone orbit $(k^{\ast }{)}^2$, two one-dimensional toric ray orbits on the punctured axes, and one zero-dimensional orbit at the origin. The face-to-orbit reversal also explains the formula for an orbit closure in a toric variety. ^[ref]