In two dimensions, it is a Rouleau triangle: an equilateral triangle with curved arcs connecting each corner, creating a shape of constant width but with a smaller area than a circle. Now, a team of mathematicians says they have scaled the shape into the third dimension and beyond, and found that it solves a mathematical problem that has been tumbling since 1988.
the The original problem It was proposed by Oded Schramm, a mathematician who thought about the possibility of the existence of objects with a fixed width smaller than a sphere of higher dimension. The team’s research is Currently hosted On the arXiv preprint server.
“The most surprising thing is that the volume of each shape can be easily calculated,” study co-author Andre Bondarenko, a mathematician at the Norwegian University of Science and Technology, said in an email to Gizmodo. “So we can compare n– Shape size with n– The volume of a unit sphere, and we see with mathematical precision that the volumes of our shapes are significantly smaller.
Rollo’s triangle (named after a 19th-century engineer, but popularized long before that by scientists such as Euler and Leonardo da Vinci) can be formed by constructing three interlocking circles; That space in the middle is Rouleau’s triangle. the Blaschke-Liebig theory, published independently by mathematicians in 1914 and 1915, stated that a triangle has the least area of all curves of a given fixed width. Simply put, this means that its width is the same value no matter where you draw two parallel lines along the outside of the shape. get it?
In two dimensions, the shape is a Rouleau triangle. When looking at 3D space, the shape is rectangular, but it is something our brains can visualize. Beyond the third dimension, the team can mathematically calculate the constant width of the shape even in increasing dimensions.
“Perhaps one of the reasons we build so successfully is that our bodies are somehow unbalanced, with too much volume pushed in one direction,” said Andre Primark, a mathematician at the University of Manitoba and co-author. of the research, in an email to Gizmodo. “In this way, the body becomes less ball-like, which allows for this [it] To achieve a smaller size with the same width.”
As reported new world, in higher dimensions the shape will be relatively smaller than the sphere of equivalent dimension. As New Scientist also noted, the shape can roll smoothly like a wheel even though it is not round.
This format has yet to have a cool name, consider discovering it last year The 13-sided shape is called the “hat” And the vampire Einstein (real name) is called “the ghost.“The new shape has a fixed width that is always smaller than its dimensional domain – perhaps ‘Svelte?’
more: The updated “Vampire Einstein” shape finally solves the annoying mathematical pattern problem