Altermagnets are a recently discovered class of magnetic materials that combine a collinear, zero-magnetization spin structure, characteristic of antiferromagnets, with spin-split electronic bands, a hallmark of ferromagnets. This unique behavior arises from the breaking of combined time-reversal and spatial symmetries (such as inversion or lattice translation), which are preserved in conventional antiferromagnets. To date, research has mainly focused on altermagnetic phases in periodic crystals, where the order is linked to rotational symmetries compatible with translational periodicity. In this Letter, we demonstrate that quasicrystals, which possess rotational symmetries incompatible with periodicity, can host exotic altermagnetic orders. Using symmetry analysis and self-consistent mean-field theory, we predict stable $g$-wave and $i$-wave altermagnetism in octagonal and dodecagonal quasicrystals, respectively. These phases are characterized by global $C_{8}T$ and $C_{12}T$ symmetries and exhibit anisotropic spin-splittings in their spectral functions and spin conductance, with characteristic eight and twelvefold nodal structures that establish a theoretical framework for identifying these phases in future experiments. Our findings establish quasicrystals as a versatile platform for realizing unconventional altermagnetic orders beyond the constraints of periodicity.

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• Altermagnetism
• Electronic structure
• Quantum transport
• Quasicrystals
• Landauer formula
• Mean field theory