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Summary: ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS
Volume 31, Number 1, Spring 2001
A QUARTIC SURFACE OF
INTEGER HEXAHEDRA
ROGER C. ALPERIN
ABSTRACT. We prove that there are infinitely many six-
sided polyhedra in R3, each with four congruent trapezoidal
faces and two congruent rectangular faces, so that the faces
have integer sides and diagonals, and also the solid has integer
length diagonals. The solutions are obtained from the integer
points on a particular quartic surface.
A long standing unsolved problem asks whether or not there can be a
parallelepiped in R3
whose sides and diagonals have integer length. If
one weakens the requirement and just asks for a six-sided polyhedron
with quadrilateral faces, then one can find examples with integer length
sides and diagonals. Peterson and Jordan [1] described a method for
making these `perfect' hexahedra. We review their method.
Take two congruent rectangles positioned as if they formed the
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