Think of a notebook, which exists in three dimensions. It has height, width and depth. What happens if you hold that notebook above a flat surface and shine a light down on the book? “This will cast a rectangular shadow underneath,” notes Steven Schaefer. He took part in Bray’s study. (Now, he is a PhD student in computer science at the University of Michigan in Ann Arbor.)
The notebook’s shadow is two-dimensional, or 2-D. It has only height and width. So that shadow does not provide a direct image of the 3-D notebook. But looking at the notebook’s 2-D shadow from different directions can reveal information about the book’s third dimension, Schaefer says.
Likewise, looking at 3-D “shadows” of imagined 4-D objects can offer insight into what objects in a fourth dimension might look like. The students on Bray’s team created a virtual 4-D shape on a computer using code. Then, they projected their 4-D shape into three dimensions. They printed out the resulting structure with a 3-D printer. This required writing computer code, or software, to build their projection.
That code converted thousands of points from the 4-D structure into a format that the 3-D printer could understand. The code essentially “massaged” those data in a way that allowed the printer to churn out a 3-D representation of the 4-D structure, Schaefer explains.
“A great thing about modern computing is that you have all these visualization tools,” says Bray. But to use them, the students had to understand the math at a really deep level. Without it, they couldn’t write the code that told the printer what to do.
“When I was 15, I thought there were no physical [math] experiments,” Schaefer says. It seemed that in math, “all one cares about is proofs.” (
