In this paper we present a theoretical method, together with its experimental confirmation, to obtain structures of light by connecting diffraction-resistant cylindrical beams of finite lengths and different radii. The resulting "Lego-beams" can assume, on demand, various unprecendent spatial configurations. We also experimentally generate some of them on using a computational holographic technique and a spatial light modulator. Our new, interesting method of linking together various pieces of light can find applications in all fields where structured light beams are needed, in particular such as optical tweezers, e.g. for biological manipulations, optical guiding of atoms, light orbital angular momentum control, holography, lithography, non-linear-optics, interaction of electromagnetic radiation with Bose-Einstein condensates, and so on, besides in general the field of Localized Waves (non-diffracting beams and pulses). [MZR] ( * ) Visiting c/o Decom, Unicamp, by a PVE fellowship of CAPES (Brazil) arXiv:1712.01118v2 [physics.optics] 5 Apr 2018Structured Light [1,2,3, 4,5,6] has been more and more studied, and applied in various sectors, like optical tweezers [7,8,9,10,11,12,13,14,15,16,17], optical guiding of atoms [18,19,20,21,22,23,24], imaging [25], light orbital angular momentum control and applications [26,27,28,29,30,31], and photonics in general.A rather efficient method to model longitudinally the intensity of non-diffracting beams is by the so-called Frozen Waves (FWs) [1,2,3,32,33,34,35,36], obtained from superpositions of co-propagating Bessel beams, endowed with the same frequency and order. The resulting diffraction resistant beam, with a longitudinal intensity shape freely chosen a priori, may then propagate, remaining confined, along the propagation axis z, or over a cylindrical surface (depending on the order of the constituting Bessel beams), while its "spot" size, and the cylindrical surface radius, can be as well chosen a priori. In this way, it is possible to construct, e.g., cylindrical beams whose static envelopes possess non-negligible energy density in finite, well-defined spatial intervals only: so that they can be regarded as segments or cylindrical pieces of light.Aiming also at a greater control on the beam transverse shape, another method was recently proposed [24,26], where different-order FW-type beams are superposed, which possess appreciable intensities along different, but consecutive, space intervals: So that one ends with cylindrical structures of light endowed with different radii and located in different positions along the z axis. This new method resulted efficient, incidentally, also for controlling the orbital angular momentum along the propagation axis [26].Anyway, and interestingly enough, it is possible to join together in the same way even two FW-type beams bearing the same order, by getting again a structure with two different-radius cylinders, each one in its own space interval. To this aim, it is sufficient that each equal-order FW possesses a different value of ...