## 1 Introduction
This possibility is unprecedented in structural work with XFEL diffraction patterns. It is with the aim of further developing this idea that the work here is undertaken. Initially, following the basic theory presented here, the aim is only to reconstruct from an ensemble of randomly oriented particles of simple form, its low resolution 3D structure. Indeed, there has already been experimental work on \"nanorice\" particles of the type we describe here. While it may be true that such a simple structure may be obtained by SAXS alone, the demonstration of such a method even for a simple structure paves the way to its application to more complex structures that may not be accessible to SAXS. Furthermore, \"machine learning\" algorithms based on manifold embedding [9] have been used to perform an approximate sorting of the experimental diffraction patterns. Following this paper, we have begun to work with the authors of that paper on going all the way from measured diffraction patterns to structure. We describe here how the structure of \"nanorice\" particles on which experimental data is already available at the public web site cidb.org can be reconstructed in principle. There has also been some work reported on the reconstruction of prolate and oblate spheroids by the method of cryptotomography [2]. However, this does require diffraction patterns from single particles to be identified beforehand from an experimental data set [10]. In the present paper, we describe an algorithm that works on either prolate or oblate spheroids and yet allows the possibility of working with diffraction patterns of multiple randomly oriented particles. To do this, we employ a method [11] of reconstructing the structure of a particle from XFEL diffraction patterns of random unknown orientations of the object. This method has close similarities to a method proposed much earlier [3] for recovering the structure of particles randomly oriented, as in the method of small angle X-ray scattering (SAXS). Although, of course, in the 1970s, it was not possible to focus an X-ray beam to hit just a single particle, this has now become possible with the recent advent of the X-ray free electron laser (XFEL). It should be stressed that the method we describe in this paper is flexible enough to reconstruct a structure from \"single-particle\" experiments such as in the recent Single Particle Initiative (SPI) at the Linac Coherent Light Source (LCLS) in Stanford, CA, USA. What is used is the average over measured diffraction patterns of the angular correlations on each diffraction pattern. Since this uses diffraction data of perhaps millions of diffraction patterns, its expected shot noise would be significantly reduced by this averaging process [12]. Such methods are very different from those of traditional crystallography on crystals, as the orientation of each particle in generally unknown a priori and there are not even Bragg spots that enable their orientation to be determined via indexing [13]. We describe our method next. One of its advantages is that it has been shown to be equally applicable to ensembles of molecules of random orientations [8]. This could be an advantage experimentally, as it will allow a fuller use of available experimental data, and obviates the need for methods [10] that eliminate diffraction patterns from multiple particle hits from the analysis.