Making math more Lego-like—3-D picture language has far-reaching potential, including in physics

March 3, 2017 by Peter Reuell, Harvard University
Making math more Lego-like—3-D picture language has far-reaching potential, including in physics
"[A] picture is worth 1,000 symbols,” quips Professor Arthur Jaffe (left). Jaffe and postdoctoral fellow Zhengwei Liu have developed a pictorial mathematical language that can convey pages of algebraic equations in a single 3-D drawing. Credit: Rose Lincoln/Harvard Staff Photographer

Galileo called mathematics the "language with which God wrote the universe." He described a picture-language, and now that language has a new dimension.

The Harvard trio of Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski has developed a 3-D picture- for mathematics with potential as a tool across a range of topics, from pure math to physics.

Though not the first pictorial language of mathematics, the new one, called quon, holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images. The language is described in a February 2017 paper published in the Proceedings of the National Academy of Sciences.

"It's a big deal," said Jacob Biamonte of the Quantum Complexity Science Initiative after reading the research. "The paper will set a new foundation for a vast topic."

"This paper is the result of work we've been doing for the past year and a half, and we regard this as the start of something new and exciting," Jaffe said. "It seems to be the tip of an iceberg. We invented our language to solve a problem in , but we have already found that this language led us to the discovery of new mathematical results in other areas of mathematics. We expect that it will also have interesting applications in physics."

When it comes to the "language" of mathematics, humans start with the basics—by learning their numbers. As we get older, however, things become more complex.

"We learn to use algebra, and we use letters to represent variables or other values that might be altered," Liu said. "Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace 'symbol proof' by 'picture proof.'"

The new language relies on images to convey the same information that is found in traditional algebraic equations—and in some cases, even more.

"An image can contain information that is very hard to describe algebraically," Liu said. "It is very easy to transmit meaning through an image, and easy for people to understand what they see in an image, so we visualize these concepts and instead of words or letters can communicate via pictures."

"So this pictorial language for mathematics can give you insights and a way of thinking that you don't see in the usual, algebraic way of approaching mathematics," Jaffe said. "For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension."

In their most recent work, the researchers moved their language into a more literal realm, creating 3-D images that, when manipulated, can trigger mathematical insights.

"Where before we had been working in two dimensions, we now see that it's valuable to have a language that's Lego-like, and in three dimensions," Jaffe said. "By pushing these pictures around, or working with them like an object you can deform, the images can have different mathematical meanings, and in that way we can create equations."

Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.

"It turns out one picture is worth 1,000 symbols," Jaffe said.

"We could describe this algebraically, and it might require an entire page of equations," Liu added. "But we can do that in one picture, so it can capture a lot of information."

Having found a fit with quantum information, the researchers are now exploring how their language might also be useful in a number of other subjects in and physics.

"We don't want to make claims at this point," Jaffe said, "but we believe and are thinking about quite a few other areas where this picture-language could be important."

Explore further: Brain study suggests neural networks related to mathematics are different from those used for language

More information: Zhengwei Liu et al. Quon 3D language for quantum information, Proceedings of the National Academy of Sciences (2017). DOI: 10.1073/pnas.1621345114

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8 comments

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Semmster
5 / 5 (2) Mar 03, 2017
Is there a way I could get to try this? I instinctively feel the power of math but something happened to my ability to digest it.
Whydening Gyre
5 / 5 (1) Mar 03, 2017
Is there a way I could get to try this? I instinctively feel the power of math but something happened to my ability to digest it.

This will be deleted for excess verbiage, but....
Sip some Crown, then 1 forward somersault with a half twist to the right. Then, 1 backward somersault with another 1/2 twist to the right.
You'll feel the force...
antialias_physorg
5 / 5 (2) Mar 03, 2017
Is there a way I could get to try this?

Here's a paper detailing how it works
http://www.pnas.o...114.full

But if you have an issue with some advanced math then this probably won't help, either.
Captain Stumpy
5 / 5 (1) Mar 03, 2017
Is there a way I could get to try this? I instinctively feel the power of math but something happened to my ability to digest it.
@Semmster
math is like a muscle - if you don't use it, it will atrophy, and to get better you have to use it, experience it and work your arse off

here is a suggestion: start small - start with the "idiots guide" or "math for dummies" books as they lay it out simply and easily, and you can work in sections

then buy these books: https://www.amazo...0KMZL49E

from there you can work into things like physics or advanced math topics

more importantly, like working out, it is easier to do with a friend or partner, so i suggest a tutor or similar training method
andylasttry
5 / 5 (1) Mar 04, 2017
The Human brain can visualize vortices in space, or vortices in a liquid easily. Liquid dynamics is horrendous to model mathematically. ???? . A new pictorial way may be the way forward.

If the greeks and egyptians could hack the pictograms, why not a new mathematical pictogram language to model space,and particles. All particles are waves, all forces are waves, gravity is a vibration. Are you guys talking ether theory, or qauntum mechanics, or are both one and the same.
ski137
not rated yet Mar 05, 2017
Feynman diagrams for mathematics?
Ranbir
not rated yet Mar 09, 2017
Will be nice to see Egyptian hieroglyphics describe mathematics of 21st century AD and beyond !!!
BubbaNicholson
1 / 5 (1) Mar 18, 2017
A simplified device might convey Callish, a language of words and sounds that convey the positions and movements of objects in space.

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