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         Some people look at an equation and see a bunch of complicated numbers and symbols while others see beauty. Now, thanks to a new tool created at Carnegie Mellon University, anyone can now translate the abstractions of mathematics into beautiful and instructive illustrations.

        This exciting new tool is named Penrose after the mathematician Roger Penrose, who is famous for using diagrams and other drawings to communicate complicated mathematical ideas. Penrose enables users to create diagrams simply by typing mathematical expressions and letting the software do the drawing. Unlike a graphing calculator, these aren’t restricted to basic functions, but can be complex relationships from any area of mathematics. ”Some mathematicians have a talent for drawing beautiful diagrams by hand, but they vanish as soon as the chalkboard is erased,” said Keenan Crane, an assistant professor of computer science and robotics. “We want to make this expressive power available to anyone.”

        Diagrams are often underused in mathematical communication, since producing high-quality illustrations is beyond the skill of many researchers and requires a great deal of time and effort. Penrose addresses these challenges by letting diagram-drawing experts turn their knowledge about creating diagrams into computer codes so that other users can access this capability using familiar mathematical language and a computer. “We started off by asking: ‘How do people translate mathematical ideas into pictures in their head?’” said Katherine Ye, a Ph.D. student in the Computer Science Department who is involved in the development of Penrose. “The secret sauce of our system is to empower people to easily ‘explain’ this translation process to the computer, so the computer can do all the hard work of actually making the picture.”  

        Once the computer learns how the user wants to see a mathematical object visualized – a vector represented by a little arrow, for instance, or a point represented as a dot – it uses these rules to draw several candidate diagrams. Users can then select and edit the diagrams they want from a gallery of possibilities. A special, simple-to-learn programming language was also developed so that they can easily convey the ideas in their minds to the Penrose system, Crane said. “Mathematicians can get very picky about notations,” he explained. “We let them define whatever notation they want, so they can express themselves naturally.”

        The researchers will present Penrose at the SIGGRAPH 2020 Conference on Computer Graphics and Interactive Techniques, which will be held this July. “Our vision is to be able to dust off an old math textbook from the library, drop it into the computer and get a beautifully illustrated book -- that way more people understand,” Crane said, noting that Penrose is a first step toward this goal.

 (Adapted from sciencedaily.com)

Question 36: Which best serves as the title for the passage?

        A. Penrose: The Tool Making Textbooks Outdated                  

        B. Visualizing Mathematics With Penrose

        C. The Age of Computer Mathematicians            

        D. Why Mathematicians Cannot Draw Diagrams

Question 37: The word “restricted” in paragraph 2 can be best replaced by ___________.

        A. confined              B. enclosed           C. included            D. contained

Question 38: In paragraph 2, what do we learn about the tool Penrose?            

        A. It was named after the famous mathematician who developed it.

        B. It helps users to create mathematical diagrams by themselves.

        C. It is much more powerful than a typical graphing calculator.

        D. It was created so that anyone can draw diagrams by hand.

Question 39: The phrase “secret sauce” in paragraph 3 is closest in meaning to __________.

        A. unexpected benefit                     B. mysterious feature

        C. unnatural ability                      D. special element

Question 40: Penrose address the challenges of creating diagrams in mathematical communication by allowing users to _________.            

        A. make mathematical diagrams from computer codes with the help of diagram-drawing experts. 

        B. describe how they visualize mathematical ideas to the computer so that it can create diagrams. 

        C. learn about how experts translate mathematical language into pictures created with computers. 

        D. utilize codes developed by diagram experts to figure out how to visualize mathematical ideas.

Question 41: The word “they” in paragraph 4 refers to ___________.                  

        A. users           B. diagrams           C. possibilities          D. ideas

Question 42: Which of the following can be inferred from the passage?

        A. Hand-drawn diagrams are not as beautiful and easy to understand as digital diagrams.

        B. In general, mathematicians are not very good at remembering and defining notations.

        C. How a mathematical object should be visualized varies from person to person.

        D. Almost all people consider traditional math textbooks to be extremely boring.

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