The mechanics of frictional attachment between surfaces with pillars, inspired by the head fixation system of dragonflies, is analyzed. The system consists of two surfaces of interdigitating pillars held together through friction, as by the densely packed bristles of two brushes when pressed together. The adhesive strength of the system is promoted by high elastic modulus, high friction coefficient, large aspect ratio, and dense packing of the fibers. However, the design is limited by the compressive buckling, the compressive indentation or cracking of the contacting pillars, yielding in shear or similar mechanisms that limit the achievable friction stress, and tensile failure of the pillars upon pull-out. Maps, which summarize the strength of the adhesive system and the failure limits and illustrate the trade-off among the design parameters, are presented. Case studies for steel, nylon, and ceramic pillars show that useful strength can be achieved in such attachments; when buckling during assembly and contact failure can be avoided, adhesive performance as high as 30% of the tensile strength of the pillar material may be possible.