This doesn’t happen very often, so brace yourselves: mathematicians have declared a new class of shape – but it’s not like your typical circle, triangle or square.
So what is it? The ‘shape’ is one seen throughout nature, which the scientists have named the ‘soft cell’. If you’re not getting a clear image of this just yet, we don’t blame you. Rather confusingly, the shape can take different forms, so long as it has rounded edges and fits together in a tessellated grid – known as ‘tiling’ in maths.
In 2D, tessellating fully rounded shapes isn't possible, unless there are ‘cusps’ – the sharp points between curves (like the top of a teardrop). An example of this in 2D is the cross section of an onion.
But the researchers behind the new study have discovered it is possible to tesselate a fully rounded shape in 3D – such as the chambers of a nautilus shell (the spiralling mollusc with orange stripes). These chambers look angular in 2D, but the researchers were amazed to see that, when modelled in 3D, there were no edges at all.
While these shapes have been known for centuries, no-one has formalised the notion of soft cells until now.
“Simply, no one has done this before”, Dr Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics in New York City, who was not involved in the work, told journal Nature. “It’s really amazing how many basic things there are to consider.”
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In 2023 Goodman-Strauss was part of a team that discovered they could create an irregular tiling pattern using just a single shape: in other words, tiles that are all the same, but don’t line up in a predictable and repeating grid.
The Hungarian team behind the newer paper, published in the journal PNAS Nexus, considered what happens if you give this tile, known as an ‘einstein’, rounded corners. Using algorithms to convert geometric shapes into soft cells, they discovered that in 3D, soft cells can fill all the gaps without having any corners at all.
The team then tried to work out the maximum ‘softness’ a shape can have, and realised that the softest shapes are not compact and simple but actually flare out at the sides like wings (like the shape of a horse saddle).
In nature, the researchers think, corners are points of structural weakness. Bending around corners may also cost energy and build tension at edges, so natural shapes tend to avoid them (think of the soft outline of an island formed by rivers bending around land).
The discovery could inspire architecture: already, since finishing writing the paper, the researchers have collaborated with architects at the California College of Arts in San Francisco, USA, to design buildings comprised of soft cells.
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