How to Python soft cell infinite polyhedral tilings?

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"Soft cells and the geometry of seashells" (2024)
https://arxiv.org/abs/2402.04190 :

> A central problem of geometry is the tiling of space with simple
structures. The classical solutions, such as triangles, squares, and
hexagons in the plane and cubes and other polyhedra in three-dimensional
space are built with sharp corners and flat faces. However, many tilings in
Nature are characterized by shapes with curved edges, non-flat faces, and
few, if any, sharp corners. An important question is then to relate
prototypical sharp tilings to softer natural shapes. Here, we solve this
problem by introducing a new class of shapes, the \textit{soft cells},
minimizing the number of sharp corners and filling space as \emph{soft
tilings}. We prove that an infinite class of polyhedral tilings can be
smoothly deformed into soft tilings and we construct the soft versions of
all Dirichlet-Voronoi cells associated with point lattices in two and three
dimensions. Remarkably, these ideal soft shapes, born out of geometry, are
found abundantly in nature, from cells to shells.

Schema:NewsArticles about said schema.org/ScholarlyArticle:

-
https://www.popularmechanics.com/science/math/a46973545/soft-cells-secret-g=
eometry-of-life/

https://www.aol.com/lifestyle/mathematicians-discovered-secret-geometry-lif=
e-174500998.html
:

  > The team believes that they=E2=80=99ve solved the problem of dimensions=
 with
this new =E2=80=9Cinfinite class of polyhedral tilings=E2=80=9D that can sm=
oothly deform
into soft tiles and construct soft versions of cells generally associated
with point lattices in both two and three dimensions. [...]
  >
  > In two dimensions, these soft shell shapes are pretty easy to
describe=E2=80=94according to the paper, they are =E2=80=9Ccells with curve=
d boundaries
which have only two corners.=E2=80=9D In the three-dimensional space, thing=
s get a
little more complicated, but the goal is the same: let things be bendy and
minimize the amount of =E2=80=9Ccorners=E2=80=9D present. In 3D, a soft cel=
l shape can have
no corners at all.
  >
  > =E2=80=9CWe found that architects have found these kinds of shapes intu=
itively
when they wanted to avoid corners,=E2=80=9D Domokos said.

A math thing to be Python'd.

What [Python,] geometry software could do or does 2D, 3D, and N-D infinite
polyhedral tilings like this?

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<div dir=3D"auto"><div dir=3D"auto">&quot;Soft cells and the geometry of se=
ashells&quot; (2024)=C2=A0</div><a href=3D"https://arxiv.org/abs/2402.04190=
">https://arxiv.org/abs/2402.04190</a> :<div dir=3D"auto"><br></div><div di=
r=3D"auto">&gt; A central problem of geometry is the tiling of space with s=
imple structures. The classical solutions, such as triangles, squares, and =
hexagons in the plane and cubes and other polyhedra in three-dimensional sp=
ace are built with sharp corners and flat faces. However, many tilings in N=
ature are characterized by shapes with curved edges, non-flat faces, and fe=
w, if any, sharp corners. An important question is then to relate prototypi=
cal sharp tilings to softer natural shapes. Here, we solve this problem by =
introducing a new class of shapes, the \textit{soft cells}, minimizing the =
number of sharp corners and filling space as \emph{soft tilings}. We prove =
that an infinite class of polyhedral tilings can be smoothly deformed into =
soft tilings and we construct the soft versions of all Dirichlet-Voronoi ce=
lls associated with point lattices in two and three dimensions. Remarkably,=
 these ideal soft shapes, born out of geometry, are found abundantly in nat=
ure, from cells to shells.</div><div dir=3D"auto"><br></div><div dir=3D"aut=
o">Schema:NewsArticles about said <a href=3D"http://schema.org/ScholarlyArt=
icle">schema.org/ScholarlyArticle</a>:</div><div dir=3D"auto"><br></div><di=
v dir=3D"auto">- <a href=3D"https://www.popularmechanics.com/science/math/a=
46973545/soft-cells-secret-geometry-of-life/">https://www.popularmechanics.=
com/science/math/a46973545/soft-cells-secret-geometry-of-life/</a></div><di=
v dir=3D"auto">=C2=A0 <a href=3D"https://www.aol.com/lifestyle/mathematicia=
ns-discovered-secret-geometry-life-174500998.html">https://www.aol.com/life=
style/mathematicians-discovered-secret-geometry-life-174500998.html</a> :</=
div><div dir=3D"auto"><br></div><div dir=3D"auto">=C2=A0 &gt; The team beli=
eves that they=E2=80=99ve solved the problem of dimensions with this new =
=E2=80=9Cinfinite class of polyhedral tilings=E2=80=9D that can smoothly de=
form into soft tiles and construct soft versions of cells generally associa=
ted with point lattices in both two and three dimensions. [...]</div><div d=
ir=3D"auto">=C2=A0 &gt;</div><div dir=3D"auto">=C2=A0 &gt; In two dimension=
s, these soft shell shapes are pretty easy to describe=E2=80=94according to=
 the paper, they are =E2=80=9Ccells with curved boundaries which have only =
two corners.=E2=80=9D In the three-dimensional space, things get a little m=
ore complicated, but the goal is the same: let things be bendy and minimize=
 the amount of =E2=80=9Ccorners=E2=80=9D present. In 3D, a soft cell shape =
can have no corners at all.</div><div dir=3D"auto">=C2=A0 &gt;=C2=A0</div><=
div dir=3D"auto">=C2=A0 &gt; =E2=80=9CWe found that architects have found t=
hese kinds of shapes intuitively when they wanted to avoid corners,=E2=80=
=9D Domokos said.</div><div dir=3D"auto"><br></div><div dir=3D"auto">A math=
 thing to be Python&#39;d.</div><div dir=3D"auto"><br></div><div dir=3D"aut=
o">What [Python,] geometry software could do or does 2D, 3D, and N-D infini=
te polyhedral tilings like this?</div><div dir=3D"auto"><br></div></div>

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