We present an Evolutionary Visual Exploration (EVE) system that combines visual analytics with stochastic optimisation to aid the exploration of multidimensional datasets characterised by a large number of possible views or projections. Starting from dimensions whose values are automatically calculated by a PCA, an interactive evolutionary algorithm progressively builds (or evolves) non-trivial viewpoints in the form of linear and non-linear dimension combinations, to help users discover new interesting views and relationships in their data. The criteria for evolving new dimensions is not known a priori and are partially specified by the user via an interactive interface: (i) The user selects views with meaningful or interesting visual patterns and provides a satisfaction score. (ii) The system calibrates a fitness function (optimised by the evolutionary algorithm) to take into account the user input, and then calculates new views. Our method leverages automatic tools to detect interesting visual features and human interpretation to derive meaning, validate the findings and guide the exploration without having to grasp advanced statistical concepts. To validate our method, we built a prototype tool (EvoGraphDice) as an extension of an existing scatterplot matrix inspection tool, and conducted an observational study with five domain experts. Our results show that EvoGraphDice can help users quantify qualitative hypotheses and try out different scenarios to dynamically transform their data. Importantly, it allowed our experts to think laterally, better formulate their research questions and build new hypotheses for further investigation.