The growth of science and technology is a recombinative process, wherein new discoveries and inventions are built from prior knowledge. Yet relatively little is known about the manner in which scientific and technological knowledge develop and coalesce into larger structures that enable or constrain future breakthroughs. Network science has recently emerged as a framework for measuring the structure and dynamics of knowledge. While helpful, existing approaches struggle to capture the global properties of the underlying networks, leading to conflicting observations about the nature of scientific and technological progress. We bridge this methodological gap using tools from algebraic topology to characterize the higher-order structure of knowledge networks in science and technology across scale. We observe rapid growth in the higher-order structure of knowledge in many scientific and technological fields. This growth is not observable using traditional network measures. We further demonstrate that the emergence of higher-order structure coincides with decline in lower-order structure, and has historically far outpaced the corresponding emergence of higher-order structure in scientific and technological collaboration networks. Up to a point, increases in higher-order structure are associated with better outcomes, as measured by the novelty and impact of papers and patents. However, the nature of science and technology produced under higher-order regimes also appears to be qualitatively different from that produced under lower-order ones, with the former exhibiting greater linguistic abstractness and greater tendencies for building upon prior streams of knowledge.