Some people look at an equation and see a lot of numbers and symbols; others see beauty. Thanks to a new tool created at Carnegie Mellon University, anyone can translate math abstractions into beautiful and instructive illustrations.
The tool allows users to create diagrams simply by writing an ordinary mathematical expression and letting the software do the drawing. Unlike a graphing calculator, these expressions are not limited to basic functions, but can be complex relationships from any area of mathematics.
The researchers named it Penrose after the well-known mathematician and physicist Roger Penrose, famous for using diagrams and other drawings to communicate complicated mathematical and scientific ideas.
“Some mathematicians have a talent for drawing beautiful diagrams by hand, but they disappear as soon as the board is erased,” said Keenan Crane, assistant professor of computer science and robotics. “We want to make this expressive power available to anyone.”
Diagrams are often underutilized in technical communication, since the production of high-quality digital illustrations is beyond the skill of many researchers and requires a lot of tedious work.
Penrose addresses these challenges by allowing diagram drawing experts to code how they would in the system. Other users can access this ability using familiar mathematical language, letting the computer do most of the hard work.
The researchers will present Penrose at the SIGGRAPH 2020 Conference on Computer Graphics and Interactive Techniques, which will take place virtually this July due to the COVID-19 pandemic.
A new software tool from Carnegie Mellon University converts abstract mathematical expressions into images that can be more easily understood. Credit: Carnegie Mellon University.
“We start by asking, ‘How do people translate mathematical ideas into images in their heads?'” Said Katherine Ye, Ph.D. student in the computer department. “The secret sauce of our system is empowering people to easily ‘explain’ this translation process to the computer, so that the computer can do all the hard work of making the image.”
Once the computer learns how the user wants to see the displayed mathematical objects, a vector represented by a small arrow, for example, or a point represented as a point, use these rules to draw various candidate diagrams. The user can select and edit the diagrams they want from a gallery of possibilities.
The research team developed a special programming language for this purpose that mathematicians should have no problem learning, Crane said.
“Mathematicians can be very picky about notation,” he explained. “We allow them to define the notation they want, so that they can express themselves naturally.”
An interdisciplinary team developed Penrose. In addition to Ye and Crane, the team included Nimo Ni and Jenna Wise, both PhD students at CMU’s Software Research Institute (ISR); Jonathan Aldrich, professor of ISR; Joshua Sunshine, ISR Senior Investigator; cognitive science student Max Krieger; and Dor Ma’ayan, a former master’s student at the Technion-Israel Institute of Technology.
“Our vision is to be able to dust off an old library math textbook, put it on the computer and get a beautifully illustrated book, that way more people understand it,” said Crane, noting that Penrose is a first step toward this goal. . .
Reference: ACM SIGGRAPH International Conference on Computer Graphics and Interactive Techniques
DOI: 10.1145 / 3386569.3392375
The National Science Foundation, the Defense Advanced Research Projects Agency, the Sloan Foundation, Microsoft Research, and the Packard Foundation supported this research.
