Data structures for similarity search are commonly evaluated on data in vector spaces, but distance-based data structures are also applicable to non-vector spaces with no natural concept of dimensionality. The intrinsic dimensionality statistic of Chavez and Navarro provides a way to compare the performance of similarity indexing and search algorithms across different spaces, and predict the performance of index data structures on non-vector spaces by relating them to equivalent vector spaces. We characterise its asymptotic behaviour, and give experimental results to calibrate these comparisons.