The Reuleaux triangle, a curious shape with constant width, has intrigued mathematicians for centuries. This equilateral triangle with curved sides, boasting the same width regardless of orientation, has a smaller area than a circle. Now, mathematicians have extended this concept to higher dimensions, potentially solving a mathematical problem posed in 1988.
This problem, originally proposed by mathematician Oded Schramm, questioned the existence of constant-width objects smaller than a sphere in higher dimensions. A team of researchers has recently addressed this question in a preprint paper hosted on arXiv, presenting a novel shape with the desired properties.
The Reuleaux triangle: A shape with constant width and minimal area.
The most remarkable aspect is the ease of calculating the volume of each shape,” explains Andriy Bondarenko, a mathematician at the Norwegian University of Science and Technology and co-author of the study. “This allows us to rigorously compare the n-volume of our shape with the n-volume of a unit ball, demonstrating mathematically that our shapes’ volumes are exponentially smaller.”
The Reuleaux triangle, though named after a 19th-century engineer, has roots in the work of prominent figures like Euler and Leonardo da Vinci. It can be constructed by the intersection of three circles. The Blaschke-Lebesgue theorem, independently published in 1914 and 1915, establishes that the Reuleaux triangle has the smallest area among all curves of a given constant width. This means its width remains constant regardless of the placement of parallel lines tangent to its exterior.
Extending the Concept to Higher Dimensions
In two dimensions, the shape is the familiar Reuleaux triangle. In three dimensions, it becomes an oblong form, still visually comprehensible. Beyond the third dimension, the team mathematically projects the shape’s constant width into increasingly higher dimensions.
The two-dimensional representation of the shape.
“Our success with this construction may be partially attributed to the inherent ‘imbalance’ of our shapes, with a significant volume concentrated in a specific direction,” suggests Andriy Prymark, a mathematician at the University of Manitoba and co-author of the research. “This characteristic differentiates it from a sphere, allowing for smaller volume while maintaining constant width.”
As reported by New Scientist, the shape becomes proportionally smaller than a sphere of equivalent dimension as the dimensions increase. Furthermore, despite its non-circular form, this novel shape can roll smoothly like a wheel.
A Shape in Search of a Name
Unlike the recently discovered 13-sided “hat” or the “Spectre,” a nickname for a specific configuration of the “vampire Einstein” problem, this new shape awaits a fitting name. Given its constant width, always smaller than a sphere of the same dimension, perhaps a name like “the Svelte” would be appropriate.
This discovery contributes significantly to our understanding of constant-width shapes in higher dimensions and offers a potential solution to a long-standing mathematical problem. The unique properties of this shape, including its rolling motion and smaller volume compared to a sphere, open up exciting avenues for further research and potential applications.
