###### Abstract  
In this work, the complex geometry of beams obtained from topology optimization is characterized through the fractal dimension (\(F_{D}\)). The fractal dimension is employed as an efficiency measure of the mass distribution in the beams, that is, the capacity of the optimized solutions to be efficiently distributed in the design space. Furthermore, the possible relationships between the fractal dimension and beams' mechanical properties are explored. First, a set of theoretical beams are studied based on their well-known fractal dimension. A 3D fractal called Menger sponge is reproduced on a Michell's beam (cantilever with a single force applied at the end). The programming codes that generate those beams are created in Matlab software, as are the algorithms for estimating the fractal dimension (box-counting method). Subsequently, identical beams are modelled in the software Inspire in order to apply the topology optimization and determine the mechanical parameters from the static analysis. Results indicate that the fractal dimension is affected by the design geometry and proposed optimized solutions. In addition, several relationships among fractal dimension and some mechanical resistance parameters could be established. The obtained relations depended on the objectives that were initially defined in the topology optimization. fractal dimension; topology optimization; static analysis; box-counting; Michell's beam 4.0.0000-0001-0001-0001-0001-0001-0001-001-0001-0001-0001-001-On the other hand, topology optimization is a design method to obtain optimized structures. Topology optimization aims to determine the optimal distribution of a material within a design domain by considering both a set of boundary conditions and performance objectives [12]. This procedure arises as a solution for defining geometries in order to minimize the cost of the product, while maintaining the constraints and fitting the boundary conditions [13]. Topology optimization is based on iterative mathematical algorithms for reaching a design goal. In general, this process aims to minimize the amount of material for designing and manufacturing a product, while meeting a set of restrictions (stresses, limitation of displacement, stiffness, maximum deviation, etc.). Generally, topology optimization methods employ finite element methods (FEMs) and are common add-ons in computer-aided design programs such as Inspire, Solidworks, Fusion 360, etc. The main advantage of product design based on topology optimization is to obtain pieces with reduced weight and volume without compromising their mechanical specifications. However, the complex geometry of these solutions can make them difficult to manufacture. This disadvantage has been overcome thanks to advances in additive manufacturing (e.g., 3D printing). Inspired on the so-called Michell's beam [14], this work is conducted using a cantilever beam as a case study. This is the first mention of structural topology optimization and dates back to 1904 [15]. It is based on a geometrically unconstrained problem, in which the mechanical compliance of the optimized object is optimal with respect to stress [16]. The pioneer topology optimization developed by [14] was based on truss structures, and later extended to grillages (beam systems) in [17; 18]. In this context, the application of fractal geometry for generating tensegrity structures has been explored by means of self-similar subdivisions of finite or infinite basic modules [19; 20]. This makes it possible to generate tensegrity fractals that optimize the values of suitable complexity parameters under mechanical performance criteria [21]. Topology optimization has also focused on other objectives, such as the minimization of the dynamic components in the frequency range of vibration [22].