In two dimensions, a Rouleau triangle is an equilateral triangle with curved arcs connecting each corner, creating a shape with a constant width but a smaller area than a circle. Now scientists from the Norwegian University of Science and Technology They extended the shape to the third dimension and beyond, and discovered that it solved a mathematical problem that had been underdeveloped since 1988. It was introduced by Oded Sharam, who thought about the possibility of objects with a fixed width smaller than a sphere of higher dimensions. Details can be read in the published work On the preprint server arXiv.
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Dr. Andre Bondarenko from the Norwegian University of Science and Technology says:
What’s even cooler is that the volume of each shape can be easily calculated. So we can compare the size n of a shape with the size n of a unit sphere and see mathematically that the sizes of our shapes are significantly smaller.
The new format doesn’t have a name yet
Rollo’s triangle, named after a 19th-century engineer but used much earlier by scientists such as Euler and Leonardo da Vinci, can be created by constructing three interlocking circles. The area at their center is a Rollo triangle.
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The Blaschke-Liebig theorem, published independently by mathematicians in 1914 and 1915, states that a triangle has the smallest area of all curves of a given fixed width. In simple terms, this means that its width has the same value no matter where we draw two parallel lines along the outside of the shape.
In two dimensions, the shape is a Rouleau triangle. The shape we see in three-dimensional space is rectangular, but our mind can visualize it. Beyond the third dimension, the team can mathematically predict the fixed width of the shape in more dimensions.
Dr. Andre Primark from the University of Manitoba and co-author of the study explains:
Perhaps one of the reasons we have such success with this design is that our bodies are “unbalanced” in some way, with too much volume pushed in one direction. This makes the object less sphere-like, allowing it to achieve a smaller volume for the same width.
In higher dimensions, the shape will be relatively smaller than a sphere of equivalent dimension. It is also worth noting that the shape can roll smoothly like a wheel even though it is not round.
The format doesn’t have a name yet, though it seems like it’s only a matter of time. Last year, a 13-sided aperiodic monolith was named “Einstein” — not after the Austrian physicist, but the German expression for a single stone (ein stein).
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