We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal, and explicitly computable, bond percolation thresholds. Fur- thermore for both bond percolation and the random-cluster model all large scale properties, such as the values of the percolation threshold and the critical exponents, of the graphs are equal. This equivalence holds for all values of p and all q ∈ [0,∞] for the random-cluster model. The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice. The exact bond percolation threshold may be explicitly determined as the root of a polynomial by using a generalised star-triangle transformation. Related randomly oriented models share the same bond percolation threshold value...