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The properties of DNA that make it an effective genetic material also allow it to be ideal for programmed self-assembly. Such DNA-programmed assembly has been utilized to construct responsive DNA origami and wireframe nanoassemblies, yet replicating these hybrid nanomaterials remains challenging. Here we report a strategy for replicating DNA wireframe nanoassemblies using the isothermal ligase chain reaction lesion-induced DNA amplification (LIDA). We designed a triangle wireframe structure that can be formed in one step by ring-closing of its linear analog. Introducing a small amount of the wireframe triangle to an excess of the linear analog and complementary fragments, one of which contains a destabilizing abasic lesion, leads to rapid, sigmoidal self-replication of the wireframe triangle via cross-catalysis. Using the same cross-catalytic strategy we also demonstrate rapid self-replication of a hybrid wireframe triangle containing synthetic vertices as well as the self-replication of circular DNA. This work reveals the suitability of isothermal ligase chain reactions such as LIDA to self-replicate complex DNA architectures, opening the door to incorporating self-replication, a hallmark of life, into biomimetic DNA nanotechnology.