# Irregular Quadrilateral Pyramids (Edge-Based Formula)

1. Jan 14, 2013

### Poggio

I have developed a formula that, given eight edge-lengths of an irregular quadrilateral pyramid, calculates the pyramid's base diagonals, altitude, etc. The formula converts the eight edge-lengths into five quartic coefficients. The four quartic roots yield candidate values for the square of one of the pyramid's base diagonals. The pyramid's altitude can then be easily determined given the eight edge-lengths and the one base diagonal.

I'm an amateur. The proof is based on simple solid geometry and basic algebraic manipulation. Using Excel I've studied the permutations of consecutive-integer irregular pyramids and have applied the formula across very large sets of data. Beyond yielding a precise method, the work shows that any eight edge-lengths in a stipulated arrangement will yield from zero to four viable constructable pyramids (in either convex, concave or complex form).

A quick example: Of all permutations of the set of edge-lengths (3,4,5,6,7,8,9,10) there are many viable pyramids that can be produced. Many stipulated arrangements have no constructable solutions at all. But two stipulated arrangements produce four constructions. Arrangement (3,6,8,9,5,7,10,4) has two convex, one concave and one complex construction. Arrangement (3,7,10,8,5,6,9,4) had three convex and one complex construction.

My two questions: Is this work significant at all? Can I find a place to present/publish it?

Thanks for any feedback.

Last edited: Jan 14, 2013