Integral regular truncated pyramids with rectangular bases and the diophantine equation x^2+y^2+z^2= t^2

TL;DR

This paper characterizes all integral regular truncated pyramids with rectangular bases by deriving key geometric conditions and solving related Diophantine equations, providing parametric descriptions and special cases.

Contribution

It introduces a complete parametric framework for classifying integral regular truncated pyramids with rectangular bases, linking geometry with Diophantine equations.

Findings

Derived key geometric conditions for integral pyramids.

Reduced the problem to solving a sum of squares Diophantine equation.

Provided parametric solutions and special cases for the pyramids.

Abstract

A regular truncated pyramid with rectangular bases;consists of two rectangular bases whose centers are orthogonally aligned with respect to the parallel planes containing their bases; and two pairs of congruent isosceles trapezoids(the four lateral faces). Thrre are six lengths involved:the larger base dimensions a and b; a>(or=)b. The smaller base dimensions c and d; c>(or=d). The height H, and the common length t of the four lateral faces. When a,b,c,d,H,t, and the volume V are all positive integers; we have an integral regular truncated pyramid with rectangular bases(see Definition 1 in the introduction). The two key geometric conditions that the above six lengths must satisfy are, a/b=c/d(see Section 3) and the equation, 4t^2= 4H^2+(a-c)^2+(b-d)^2 (*), derived in Section4. When H,a,c,b,d,t; are all positive integers. A modulo4 congruence shows that both the positive integers a-c and b-d; must be even. Consequently, equation (*) reduces to the equation, t^2= H^2+x^2+y^2 (**). All the positive integer solutions to (**) can be found, parametrically described, in reference [1]. Using the general positive integer solution to (**), we Proposition 1(Section7); which gives precise conditions that describe the set of all integral regular truncated pyramids with rectangular bases. In Proposition 2, we describe a 3-parameter family of such pyramids. In Proposition 3, we describe the square case: a=b> c=d.