Some Elementary Aspects of 4-dimensional Geometry
J. Scott Carter (University of South Alabama), David A. Mullens, (University of South Alabama)
TL;DR
This paper explores a 4-dimensional geometric interpretation of Heron's formula, linking it to scissors congruence, and investigates decompositions of various elementary 4D solids.
Contribution
It introduces a novel 4D perspective on Heron's formula and analyzes decompositions of hypercubes and related solids.
Findings
Heron's formula can be viewed as a scissors congruence in 4D.
Decompositions of hypercubes and hyper-parallelograms are examined.
New geometric interpretations of 4D solids are proposed.
Abstract
We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
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