For a compact set X ⊆ C n X \subseteq {{\mathbf {C}}^n} , let h r ( X ) {h_r}(X) denote the rationally convex hull of X; let Δ {\mathbf {\Delta }} denote the closed unit disk in C; and, following Wermer, for a compact set S such that ∂ Δ ⊆ S ⊆ Δ \partial {\mathbf {\Delta }} \subseteq S \subseteq {\mathbf {\Delta }} let X S = S × S ∩ ∂ Δ 2 {X_S} = S \times S \cap \partial {{\mathbf {\Delta }}^2} . It is shown that \[ h r ( X S ) = { ( z , w ) ∈ S × S | u S ( z ) + u S ( w ) ≤ 1 } {h_r}({X_S}) = \{ (z,w) \in S \times S|{u_S}{(z)^ + }{u_S}(w) \leq 1\} \] where u S {u_S} is a function on S which, in the case when S is smoothly bounded, is specified by requiring u S | ∂ Δ = 0 , u S | ∂ S ∖ ∂ Δ = 1 {u_S}{|_{\partial {\mathbf {\Delta }}}} = 0,{u_S}{|_{\partial S\backslash \partial {\mathbf {\Delta }}}} = 1 and u S | int ⁡ S {u_S}{|_{\operatorname {int} S}} harmonic. In particular this provides a precise description of h r ( X ) {h_r}(X) for certain sets X ⊆ C 2 X \subseteq {{\mathbf {C}}^2} with the property that h r ( X ) ≠ X {h_r}(X) \ne X , but h r ( X ) {h_r}(X) does not contain analytic structure (as Wermer demonstrated, there are S for which X = X S X = {X_S} has these properties). Furthermore, it follows that whenever h r ( X S ) ≠ X S {h_r}({X_S}) \ne {X_S} then there is a Gleason part of h r ( X S ) {h_r}({X_S}) for the algebra R ( X S ) R({X_S}) with positive four-dimensional measure. In fact, the Gleason part of any point ( z , w ) ∈ h r ( X S ) ∩ int ⁡ Δ 2 (z,w) \in {h_r}({X_S}) \cap \operatorname {int} {{\mathbf {\Delta }}^2} such that u S ( z ) + u S ( w ) > 1 {u_S}(z) + {u_S}(w) > 1 has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set Y ⊆ C 2 Y \subseteq {{\mathbf {C}}^2} such that each point of Y is a peak point for R ( Y ) R(Y) even though R ( Y ) ≠ C ( Y ) R(Y) \ne C(Y) ; namely, Y = X ~ T = { ( z , w ) ∈ C 2 | z ∈ T , | w | = 1 − | z | 2 } Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf {C}}^2}|z \in T,|w| = \sqrt {1 - |z{|^2}} \} where T is any compact subset of int ⁡ Δ \operatorname {int} {\mathbf {\Delta }} having the property that R ( T ) ≠ C ( T ) R(T) \ne C(T) even though there are no nontrivial Jensen measures for R ( T ) R(T) . This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that R ( X ~ T ) R({\tilde X_T}) is not even in general locally dense in C ( X ~ T ) C({\tilde X_T}) , a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in C 6 {{\mathbf {C}}^6} ) with the same pathological properties are obtained from X S {X_S} and X ~ T {\tilde X_T} .