Related papers: The Integer Cuboid Table
By examining the 3 surface angles which exist at any of the 8 vertices of a Diophantine parallelepiped, and classifying them by the appearance of a right angle, it is discovered that 5 unique classes of Diophantine parallelepipeds exist. It…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges, with integer face diagonals, and with integer space diagonal as well. Finding such parallelepipeds or proving their non-existence is an old unsolved mathematical…
There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths all positive integers. In particular, there is a parallelepiped with edge lengths 271, 106, 103, minor face diagonal lengths 101, 266, 255, major…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The problem of finding such parallelepipeds or proving their non-existence is an old unsolved…
One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids…
A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second…
A rectangular parallelepiped is called a cuboid (standing box). It is called perfect if its edges, face diagonals and body diagonal all have integer length. Euler gave an example where only the body diagonal failed to be an integer (Euler…
An Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals. An Euler cuboid is called perfect if its space diagonal is also integer. Some Euler cuboids are already discovered. As for perfect cuboids, none…
A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural…
A perfect cuboid is a rectangular parallelepiped with integer edges, integer face diagonals, and integer space diagonal. Such cuboids have not yet been found, but nor has their existence been disproved. Perfect cuboids are described by a…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all…
A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their…
A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E_{1,s}: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number $s \neq 0, \pm 1$, and define $\tilde{A}$…
A perfect cuboid is formed when an Euler brick whose edges and face diagonals are all integers also has an integer internal diagonal. It is known that if a perfect cuboid exists the internal diagonal is odd. No perfect cuboid has been…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four equations with respect to six variables.…
Using a quartic surface and its rational curves we can give an infinite number of integer hexahedra; these are 6 sided 3d solids, each face a trapezoid, with all sides and diagonals having intger lengths.
Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on…
Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and $p_2+p_3=i$. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are…
The original tables of body cuboids by Maurice Kraitchik are corrected, restoring 159 missing cuboids. His table range is then extended for all odd sides less than 1,000,000 to a new limit of 4,294,967,295. Over this new range, 12,517…