A team of researchers from CMU and Technion recently introduced a new system, Penrose, that can turn complex mathematical notations into various styles of simple diagrams. The novel system rapidly attracted attention on social media as a promising visualization tool for effectively communicating complex mathematical ideas and concepts.
The researchers noted that it is relatively rare to find diagrams in mathematical writings, which mostly contain formal and technical language. While high-quality visuals can be useful for effectively conveying mathematical ideas, the graphic tools required to produce such diagrams are not always available or user-friendly, and this has limited their usage in mathematical writing. “There are a ton of great tools for making diagrams, but we found ourselves always either spending a long time drawing them in a graphical interface, or meticulously tweaking coordinates in code,” first author on the Penrose paper and fourth-year CMU PhD student Katherine Ye tweeted.
Other popular professional typesetting systems such as LaTex convert mathematical input into a visual representation through algorithmic codifying. The CMU team similarly designed Penrose to codify the best practices of mathematical illustrators in a way that is reusable and widely accessible. Ye says Penrose enables users to create diagrams by simply typing in mathematical expressions that describe relationships, whereupon “the tool automatically takes care of laying everything out.”
But unlike other visualization tools that generate diagrams through direct manipulation or low-level graphics programming, Penrose offers more than just illustrations. With a visual representation that is user-defined in a constraint-based specification language, Penrose’s capability to separate abstract mathematical objects and their visual representations can also provide insights on the inspection and debugging of user-defined data structures. The mappings from mathematical objects to visual icons make Penrose a flexible and scalable system.
Although the system enables even novice users to create diagrams by simply typing mathematical statements in familiar notations, users are expected to have the necessary knowledge to confirm whether the diagrams correctly communicate the intended mathematical meaning.
The team named the system after renowned mathematician and physicist Roger Penrose, whose eponymous Penrose Diagram is a two-dimensional illustration of the causal relations between different points in spacetime. The team hopes Penrose can be developed into a broader platform for automatic illustration that could, for example, automatically illustrate math textbooks or webpages. They will present the work at the SIGGRAPH 2020 Conference on Computer Graphics and Interactive Techniques.
The paper Penrose: From Mathematical Notation to Beautiful Diagrams and the associated code can be found on the project website.
Journalist: Fangyu Cai | Editor: Michael Sarazen