## 1 Introduction  
While the assembly of excavation fragments from their remaining thermal magnetism is suggested as a theoretically sound method in the bibliography [1], there are only a few older (1975) references [2] regarding corresponding research. Research on thermoremanent magnetism in the archaeometry field since the 1980s mainly focused on the chronological dating of fired clay walls of kilns. At the same time, as reported in numerous publications [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33], the reconstruction of vases is attempted almost solely using algorithms for optical data processing based on fragment morphology. Despite the intense scientific interest in resolving the matter using the capabilities of the new digital technology, it remains an exclusively empirical process due to the time requirements and sheer volume and complexity of the proposed methods, in order for them to be applicable in practice. A central prerequisite before fragment assembly is the selection of the fragments that belong to the same vase from among the numerous ceramic findings of an archaeological excavation. This is performed empirically and with great difficulty, especially in excavations conducted over long periods of time or ones with disturbed excavation layers. However, the criterion for finding excavation fragments of the same vase should be the similarity of the values of the remanent magnetism, since these belong to vases that are made of the same clay, fired under the same conditions. Therefore, the natural way of assembling ceramic shards is not the processing of their imagery but rather the study of the directionality of their oriented thermoremanent magnetism along the geomagnetic field, which happened at the same place and time, and was configured by their rotational symmetry. In the present work, magnetic field measurements were performed using fluxgate sensors. All experiments were conducted in the Experimental Physics Laboratory, School of Electrical and Computer Engineering, Technical University of Crete. Application of the proposed method to six vases from the last two centuries fired in traditional kilns gave excellent results. We describe the detection methodology of the fragments' field by a fluxgate sensor in Section 2. The easy and unexpected assembly of base fragments and neighboring body fragments in six vases from their magnetic field directivity determined by a single fluxgate sensor, despite the irregular shape of fragments and the magnetic anisotropy of the fired clay, was the motivation for further investigation. For the systematic investigation and the comparison of the magnetic field, base and body fragment measurements were taken at the vertices of a lattice formed by the grooves and the vertical lines to them, with a specific orientation of the sensor. The measurements in the above positions were repeated in a less tedious and more accurate way, by using a sensor/magnet apparatus forming a three-dimensional orthogonal system for the magnetic field, which is described in Section 3. A brief review of clay magnetic compounds is provided in Appendix A, which is suggested background reading material for non-experts in mineralogy. A series of experiments, mainly in base specimens of vases 4, 5 and 6 were carried out (see Appendices B.1-B.5) to investigate how the sensors on the fragments' surface were excited and the region of the fragment that is detected by the sensors was identified, for the calculation of thermoremanent magnetization (Appendix B.6) from the sensor readings. We propose the experiments to be read in order to provide a clear focus on their results and the theoretical approach to determine the sample area detected by the sensor, which are summarised in Section 4, and which were used for calculation of magnetization (Appendix B.7) in irregular fragments of the base vases 1, 2 and 3, with the described methodology presented in Section 5.