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The design process of 3D mechanical metamaterials is still an emerging field and in this paper, we propose for the first time, a new design and optimisation approach based on 3D projections of 4D geometries (4-polytopes) and evolutionary algorithms. We find that through iterative parametric optimisation, 4-polytope projected mechanical metamaterials can be optimised to achieve both high specific stiffness and high specific yield strengths. Samples manufactured using a low-stereolithography method were tested in compression. We find that optimised tesseracts (8-cell structures) had a higher specific yield strength (22.8 kNm/kg) than that of honeycomb structures tested out-ofplane (19.4 kNm/kg) and a specific stiffness of (0.68 MNm/kg) which is more than 3-fold that of gyroid structures. The compressive strength to solid-modulus ratio of the 8-cell tesseract is very high (3×10 −3 ), exceeding that of out-ofplane honeycombs, which are themselves closer in value to 5-cell pentatopes (2×10 −3 ). 8-cell and 5-cell structures are in the region of one order of magnitude higher than 16-cell and 24-cell structures (∼ 2 × 10 −4 − 8 × 10 −4 ) and are hence comparable to nanostructured metamaterials. The 8-cell tesseracts are 18% stiffer, 43% stronger, and 19% tougher in compression than out-of-plane honeycomb structures, but unlike honeycombs, 8-cell tesseracts are 3D structures with cubic symmetry. Architecture has a profound effect on the relative consistency of properties with cubically symmetric structures displaying the greatest levels of consistency in terms of both strength and stiffness reduction as a function of pore space. The results presented in this paper showcase the potential of this new class of mechanical metamaterial based on 3D projected 4-polytopes.

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In this paper, we explore the mechanical behavior of a new class of mechanical metamaterials based on the 3D projections of 4-dimensional geometries (4-polytopes) subjected to loading in tension. We demonstrate that the specific properties of mechanical metamaterials can be enhanced by more than 4-fold when optimized within a framework powered by an evolutionary algorithm. Optimized metamaterial structures were manufactured using the low-forcestereolithography prototyping technique and mechanically tested in tension. The experimental results show that the best-performing metamaterial structure, the 8-cell (tesseract), has specific yield strength and specific stiffness values in a similar range to those of hexagonal honeycombs tested out-of-plane. Nevertheless, the 8-cell structures are also cubically symmetrical and have the same mechanical properties in three orthogonal axes. The effect of structure is quantified by comparing the tensile strength against the Young's modulus of bulk solid material. We find that the final value of the 8-cell structures exceeds that of the hexagonal honeycomb by 76%. The 5-cell (pentatope) and 16-cell (orthoplex) metamaterials are shown to be more effective in bearing tensile loads than the gyroid structures, while the 24-cell (octaplex) structures exhibit the lowest ratio and possess the least optimal structure-properties relationships. The findings presented in this paper showcase the importance of macro-scale architecture and highlight the potential of 3D projections of 4-polytopes as the basis for a new class of mechanical metamaterials.