New dense superball packings in three dimensions
-
Maria Dostert
Abstract
We construct a new family of lattice packings for superballs in three dimensions (unit balls for the
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1439786
Funding statement: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the first author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” semester program. The second author was partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the DFG. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie agreement number 764759.
Acknowledgements
We thank Cristóbal Guzmán and Philippe Moustrou for helpful discussions. We thank the referee for useful suggestions and corrections.
Communicated by: M. Henk
A Source code for the proof of Theorem 4.2
We used the following program written in Sage in the proof of Theorem 4.2.
def f(p,x,y,z):
return vector([x^p+y^p+z^p-1, (x–y)^p+(z–x)^p+(y+z)^p–1, 3*(–x+y+z)^p–1])
def Df(p,x,y,z):
pm = p–1
return p*Matrix([[x^pm, y^pm, z^pm],
[(x–y)^pm–(z–x)^pm, –(x–y)^pm+(y+z)^pm, (z–x)^pm+(y+z)^pm],
[–3*(–x+y+z)^pm, 3*(–x+y+z)^pm, 3*(–x+y+z)^pm]])
def linfinitynorm(A):
return max([A.row(0).norm(1),A.row(1).norm(1),A.row(2).norm(1)])
def verify(p0,x0,y0,z0,eps,peps):
p = RIF(p0,p0+peps)
x = RIF(x0-eps,x0+eps)
y = RIF(y0-eps,y0+eps)
z = RIF(z0-eps,z0+eps)
T = Df(p0,x0,y0,z0)^(–1)
A = Df(p,x,y,z)*T – identity_matrix(3)
lhs = linfinitynorm(A)
rhs = 1 – linfinitynorm(T)*f(p,x0,y0,z0).norm(infinity)/eps
print(lhs.str(style=’brackets’)+’<’+rhs.str(style=’brackets’)+’:’+str(lhs < rhs))
B Examples of choices made in the proof of Theorem 4.2
verify(1.0, 0.333333333333, 0.166666666667, 0.5, 0.03, 0.01)
verify(1.01, 0.336543320255, 0.169227330456, 0.504294897412, 0.03, 0.01)
verify(1.02, 0.339721855623, 0.171809715243, 0.508503843298, 0.03, 0.01)
⋮
verify(1.5, 0.475292821919, 0.375983627555, 0.580059051165, 0.03, 0.01)
verify(1.51, 0.47822053429, 0.384961182567, 0.576346694842, 0.03, 0.01)
verify(1.52, 0.481163698665, 0.394556223383, 0.572012690078, 0.006, 0.001)
verify(1.521, 0.48145875646, 0.395553814361, 0.571540873724, 0.006, 0.001)
verify(1.522, 0.481753934423, 0.396558835694, 0.571061553436, 0.006, 0.001)
verify(1.523, 0.482049228267, 0.39757142775, 0.570574584849, 0.006, 0.001)
⋮
verify(1.577, 0.497880292399, 0.472696125604, 0.523437325276, 0.006, 0.001)
verify(1.578, 0.498157887988, 0.475000219764, 0.521630841401, 0.006, 0.001)
verify(1.579, 0.498433446144, 0.477421354522, 0.519705097786, 0.006, 0.001)
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli of weighted stable curves
- Classification of slant surfaces in 𝕊3 × ℝ
- New dense superball packings in three dimensions
- Explicit computation of some families of Hurwitz numbers, II
- An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property
- Moduli of stable sheaves supported on curves of genus three contained in a quadric surface
- Exceptional points for finitely generated Fuchsian groups of the first kind
- Tropical superelliptic curves
- Differentiability of projective transformations in dimension 2
- On pseudo-Einstein real hypersurfaces
- On the moduli spaces of singular principal bundles on stable curves
- Three-dimensional connected groups of automorphisms of toroidal circle planes
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli of weighted stable curves
- Classification of slant surfaces in 𝕊3 × ℝ
- New dense superball packings in three dimensions
- Explicit computation of some families of Hurwitz numbers, II
- An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property
- Moduli of stable sheaves supported on curves of genus three contained in a quadric surface
- Exceptional points for finitely generated Fuchsian groups of the first kind
- Tropical superelliptic curves
- Differentiability of projective transformations in dimension 2
- On pseudo-Einstein real hypersurfaces
- On the moduli spaces of singular principal bundles on stable curves
- Three-dimensional connected groups of automorphisms of toroidal circle planes
