In January 2026 a workshop at the Institut Henri Poincaré brought together mathematicians, historians, artists and educators to ask what it would mean for an illustration to be rigorous in mathematical research. As described in Terence Tao’s guest post, the discussion arose from a special trimester on the use of illustration in mathematical research and led to an article now accepted in the Notices of the American Mathematical Society. Participants noted that while theorem production has become the dominant measure of success, it is only a proxy for deeper understanding, a proxy that AI has disrupted by making proofs abundant.
One way illustration contributes rigor is by revealing mathematical structure that symbolic reasoning alone might miss. The post describes the authors’ work on algebraic number starscapes—images of the complex roots of rational cubic polynomials—where moving to a shader‑based rendering turned expected circles into ellipses. This unintended distortion exposed an extra dimension of meaningful structure that had not been previously explored. Similarly, the post references the discovery of the three‑dimensional aperiodic monotile Chair44: while the original paper presented a dense geometric argument, a diagram of matching arrows made the supertile structure visible and enabled a classical hierarchical proof of aperiodicity. In both cases, the illustration served as an exploratory tool that uncovered new insights.
Beyond revealing structure, illustration functions as a tool for thought and a focal point for community engagement. The workshop adopted a ‘crit’ framework from art foundations, in which a work is presented with minimal introduction and discussed to reveal how it is received by the audience versus the creator’s intent. This practice echoes the informal conversations mathematicians already have about proofs and diagrams, but places them in a communal space that makes value and understanding explicit. The post links this to Tim Ingold’s idea of thinking through making, noting that, just as Monge’s students learned through the act of drawing, value and understanding often arise from the effort of creating an illustration or a proof, not solely from viewing the finished product.