We introduce the medial axis as a tool in the analysis of geometric structure of void space in porous media. The medial axis traces the fundamental geometry of the void pathways. We describe an algorithm for generating the medial axis of the void structure from digitized three dimensional images of porous media obtained from X ray CAT scans. The medial axis is constructed during an iterative erosion procedure which, at each step, replaces the image of the void structure with a smaller version obtained by eroding its surface layer of voxels. The algorithm is applied to high (5 μm) resolution microtomographic images of two rock chips (Berea sandstone and Danish chalk) and a sample of uniform (100 μm) diameter, packed glass beads. We statistically investigate several geometrical properties of the structure of the medial axes obtained. The first is the distribution of relative volumes in each erosion layer of the void space. We find the distributions to be exponential for the two real rock samples and normal for the packed glass beads. The second property investigated is the distribution of volumes of disconnected segments of the medial axis which are in one-to-one correspondence with disconnected void segments of the sample. We find indications for a universal power law behavior governing the distribution of volumes of the smallest disconnected pieces. The final behavior studied is a geometric tortuosity as measured by shortest paths through the medial axis. This tortuosity distribution appears well described by a gamma distribution.