Let $n$ be a positive integer. We provide an explicit geometrically motivated 1-Lipschitz map from the space of persistence diagrams on n points (equipped with the Bottleneck distance) into Hilbert space. Such maps are a crucial step in topological data analysis, allowing the use of statistics (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such transformations is that they are coarse and uniform embeddings with explicit distortion functions. Furthermore, we provide an explicit 1-Lipschitz map from the space of persistence diagrams on $n$ points on a bounded domain into a Euclidean space with an explicit distortion function. Our ideas come from geometric topology and dimension theory, and our methods are best described as quantitative dimension theory. This is joint work with Ziga Virk.