Archimedes proves that the volume of the solid so cut off is one sixth part of the volume of the parallelepiped.
"Archimedes" by Thomas Little Heath
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
The same may be said for the proposition about the diagonal plane of a parallelepiped.
"The Teaching of Geometry" by David Eugene Smith
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THE HUES OF LIFE ARE DULL AND GRAY,
THE SWEETS OF LIFE INSIPID,
WHEN thou, MY CHARMER, ART AWAY -
OLD BRICK, OR RATHER, LET ME SAY,
OLD PARALLELEPIPED!'
"Phantasmagoria Canto VII ( Sad Souvenaunce )" by Lewis Carroll
This formula remains correct if applied to a rectangular parallelepiped and a derivative ∂a paral lel to the edges of the domain.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
While the curvature matrix becomes singular along the corners and edges of the parallelepiped one may argue that these singularities do not contribute to the right hand side of (45) because the derivatives of u0 vanish in the directions in which K is singular. A direct argument for this case is provided in .
Understanding the Random Displacement Model: From Ground-State Properties to Localization
In this case no curvature term appears as the faces of the parallelepiped are ﬂat.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
Nevertheless, by a reﬁned analysis we can show that these statements remain true for the case where the domain is a rectangular parallelepiped.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
The second term in (45) does not vanish unless the domain is a rectangular parallelepiped! Of course, in this case one would assume that the potential shares the symmetries of the domain.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
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