Agate geodes contain spheroidal patterns characterized by spectacularly coloured and circularly concentric laminations with radially aligned quartz crystals, yet the origin of these geometric patterns has remained enigmatic. Here, detailed comparisons are documented between these kinds of patterns in a selection of geodes and concretions and those produced by abiotic chemically oscillating reactions. We find strikingly comparable self-similar, fractal patterns in both natural volcanogenic geodes and sedimentary concretions as well as in these benchtop experiments. In addition, the mineralogical composition of patterns and associated organic matter point to the oxidation of organic compounds in both geodes and concretions. This process occurred during diagenetic or supergene alteration, and it is consistent with spontaneous and abiotic chemically oscillating reactions. It is concluded that the oxidation of organic acids was involved in the formation of these patterns and that these rocks indicate oxidation–reduction reactions involving organic carbon, which itself may be abiotic or biological in origin. Hence, agate geodes and concretions represent the abiotic biosignatures of possible biological origin in volcanic and sedimentary rocks.