Tag Archives: M. C. Escher

Illustrating math at the University of Saskatchewan (Canada)

Art and math intersect in Dr. Steven Rayan’s work on quantum materials at the University of Saskatchewan (USask).

An illustration by Elliot Kienzle (undergraduate research assistant, quanTA Centre, USask) of a hyperbolic crystal in action

A May 2, 2022 USask news release (also received via email) describes Rayan’s work in more detail,

Art and mathematics may go hand-in-hand when building new and better materials for use in quantum computing and other quantum applications, according to University of Saskatchewan (USask) mathematician Dr. Steven Rayan (PhD).

Quantum materials are what futuristic dreams are made of. Such materials are able to efficiently conduct and insulate electric currents – the everyday equivalent of never having a lightbulb flicker. Quantum materials may be the fabric of tomorrow’s supercomputers, ones that can quickly and accurately analyze and solve problems to a degree far beyond what was previously thought possible.

“Before the 1700s, people were amazed that metals could be melted down and reshaped to suit their needs, be it the need for building materials or for tools. There was no thought that, perhaps, metals were capable of something much more — such as conducting electricity,” said Rayan, an associate professor of mathematics and statistics in the USask College of Arts and Science who also serves as the director of the USask Centre for Quantum Topology and its Applications (quanTA).

“Today, we’re at a similar juncture. We may be impressed with what materials are capable of right now, but tomorrow’s materials will redefine our expectations. We are standing at a doorway and on the other side of it is a whole new world of materials capable of things that we previously could not imagine.”

Many conducting materials exhibit a crystal-like structure that consists of tiny cells repeating over and over. Previous research published in Science Advances had highlighted Rayan and University of Alberta physicist Dr. Joseph Maciejko’s (PhD) success in defining a new type of quantum material that does not follow a typical crystal structure but instead consists of “hyperbolic” crystals that are warped and curved. 

“This is an immense paradigm shift in the understanding of what it means to be a ‘material’,” said Rayan.

It is expected that hyperbolic materials will exhibit the perfect conductivity of current quantum materials, but at slightly higher temperatures. Today’s quantum materials often need to be supercooled to extremely low temperatures to reach their full potential. Maintaining such temperatures is an obstacle to implementing widespread quantum computing, which has the potential to impact information security, drug design, vaccine development, and other crucial tasks. Hyperbolic materials may be part of the solution to this problem.

Hyperbolic materials may also be the key to new types of sensors and medical imaging devices, such as magnetic resonance imaging (MRI) machines that take advantage of quantum effects in order to be more lightweight for use in rural or remote environments.

USask recently named Quantum Innovation as one of its three new signature areas of research [Note: Link removed] to respond to emerging questions and needs in the pursuit of new knowledge.

“All of this comes at the right time, as new technologies like quantum computers, quantum sensors, and next-generation fuel cells are putting new demands on materials and exposing the limits of existing components,” said Rayan.

This year has seen two new articles by Rayan together with co-authors extending previous research of hyperbolic materials. The first is written with Maciejko and appears in the prestigious journal Proceedings of the National Academy of Sciences (PNAS). The second has been written with University of Maryland undergraduate student Elliot Kienzle, who served as a USask quanTA research assistant under Rayan’s supervision in summer of 2021.

In these two articles, the power of mathematics used to study quantum and hyperbolic crystals is significantly extended through the use of tools from geometry. These tools have not typically been applied to the study of materials. The results will make it much easier for scientists experimenting with hyperbolic materials to make accurate predictions about how they will behave as electrical conductors.

Reflecting on the initial breakthrough of considering hyperbolic geometry rather than ordinary geometry, Rayan said, “What is interesting is that these warped crystals have appeared in mathematics for over 100 years as well as in art – for instance, in the beautiful woodcuts of M.C. Escher – and it is very satisfying to see these ideas practically applied in science.”

The work also intersects with art in another way. The article with Kienzle, which was released in pre-publication form on February 1, 2022 [sic], was accompanied by exclusive hand drawings provided by Kienzle. With concepts in mathematics and physics often being difficult to visualize, the artwork helps the work to come to life and invites everyone to learn about the function and power of quantum materials. 

The artwork, which is unusual for mathematics or physics papers, has garnered a lot of positive attention on social media.

“Elliot is tremendously talented not only as an emerging researcher in mathematics and physics, but also as an artist,” said Rayan. “His illustrations have added a new dimension to our work, and I hope that this is the start of a new trend in these types of papers where the quality and creativity of illustrations are as important as the correctness of equations.”

Here are links to and citations for both of Rayan’s most recent papers,

Hyperbolic band theory through Higgs bundles by Elliot Kienzle and Steven Rayan. arXiv:2201.12689 (or arXiv:2201.12689v1 [math-ph] for this version) DOI: https://doi.org/10.48550/arXiv.2201.1268 Submitted on 30 Jan 2022

This paper is open access and open for peer review.

Automorphic Bloch theorems for hyperbolic lattices by Joseph Maciejko and Steven Rayan. PNAS February 25, 2022 | 119 (9) e2116869119 DOI: https://doi.org/10.1073/pnas.2116869119

This peer-reviewed paper is behind a paywall.

New ‘Star of David’-shaped molecule from University of Manchester

It sounds like the scientists took their inspiration from Maurits Cornelius Escher (M. C. Escher) when they created their ‘Star of David’ molecule. From a Sept. 22, 2014 news item on Nanowerk,

Scientists at The University of Manchester have generated a new star-shaped molecule made up of interlocking rings, which is the most complex of its kind ever created.

Here’s a representation of the molecule,

Atoms in the Star of David molecule. Image credit: University of Manchester

Atoms in the Star of David molecule. Image credit: University of Manchester

Here’s a ‘star’ sculpture based on Escher’s work,

Sculpture of the small stellated dodecahedron that appears in Escher's Gravitation. It can be found in front of the "Mesa+" building on the Campus of the University of Twente.

Sculpture of the small stellated dodecahedron that appears in Escher’s Gravitation. It can be found in front of the “Mesa+” building on the Campus of the University of Twente (Netherlands)

If you get a chance to see the Escher ‘star’, you’ll be able to see more plainly how the planes of the ‘star’ interlock. (I had the opportunity when visiting the University of Twente in Oct. 2012.)

Getting back to Manchester, a Sept. 22, 2014 University of Manchester press release (also on EurekAlert but dated Sept. 21, 2014), which originated the news item, describes the decades-long effort to create this molecule and provides a few technical details,

Known as a ‘Star of David’ molecule, scientists have been trying to create one for over a quarter of a century and the team’s findings are published at 1800 London time / 1300 US Eastern Time on 21 September 2014 in the journal Nature Chemistry.

Consisting of two molecular triangles, entwined about each other three times into a hexagram, the structure’s interlocked molecules are tiny – each triangle is 114 atoms in length around the perimeter. The molecular triangles are threaded around each other at the same time that the triangles are formed, by a process called ‘self-assembly’, similar to how the DNA double helix is formed in biology.

The molecule was created at The University of Manchester by PhD student Alex Stephens.

Professor David Leigh, in Manchester’s School of Chemistry, said: “It was a great day when Alex finally got it in the lab.  In nature, biology already uses molecular chainmail to make the tough, light shells of certain viruses and now we are on the path towards being able to reproduce its remarkable properties.

“It’s the next step on the road to man-made molecular chainmail, which could lead to the development of new materials which are light, flexible and very strong.  Just as chainmail was a breakthrough over heavy suits of armour in medieval times, this could be a big step towards materials created using nanotechnology. I hope this will lead to many exciting developments in the future.”

The team’s next step will be to make larger, more elaborate, interlocked structures.

Here’s a link to and a citation for the paper,

A Star of David catenane by David A. Leigh, Robin G. Pritchard, & Alexander J. Stephens. Nature Chemistry (2014) doi:10.1038/nchem.2056
Published online 21 September 2014

This paper is behind a paywall.