In the world of mathematics, a significant breakthrough has been made as a team of researchers has successfully created a three-dimensional version of the renowned Rylo Triangle, along with unveiling similar figures in higher-dimensional spaces. This groundbreaking work has allowed the scientists to address a problem initially posed by a mathematician back in 1988. The study detailing their findings has been published on the Preprints Arxiv server, showcasing the unique properties of these figures (source).
The Rylo Triangle is an extraordinary figure with a constant width, formed by the intersection of three circles with centers positioned at the vertices of an equilateral triangle. Its width remains uniform, irrespective of the direction in which the distance between its parallel boundary lines is measured. The Blashka-Libe theorem from 1914 and 1915 asserts that the Rylo Triangle has the smallest area among all figures with a consistent width.
By extending the concept of the Rylo Triangle into a three-dimensional space, the researchers have created a solid shape that maintains a constant width and boasts a significantly smaller volume compared to a sphere of similar dimensions. Andrei Bondarenko from the Norwegian University of Science and Technology highlighted the significance of this achievement, emphasizing the ease of calculating the volume of these unique figures. This ability allows for a mathematical comparison of their volume with that of a sphere, proving their volumes are exponentially smaller.
Expanding their research, the team simulated these figures in spaces of higher dimensions, with Andrei Primark from the University of Manitoba attributing their success to the asymmetrical shape of the figures. This asymmetry enables a reduction in volume while preserving the constant width, making these figures distinct from spheres and aiding in volume reduction.
In the highest dimensions, the newly discovered figures are proportionally smaller than a sphere of the same dimension yet retain the ability to roll like a wheel, despite not being circular. This unique rolling capability has garnered attention from the New Scientist publication, suggesting potential applications in various technical fields.
This mathematical breakthrough not only resolves a longstanding problem but also paves the way for future advancements and explorations. The figure derived from the Rylo Triangle, with its constant width and reduced volume compared to spheres, holds promise for diverse applications in science and technology. The research team continues to delve deeper into their project, aiming to uncover novel uses for their discoveries across various disciplines.