Exploring the configurations \((12_{4}, 16_{3})) of Václav Metelka in a cubic structure

In this paper, we investigate four configurations \((12_{4}, 16_{3})) in cubic structures, introduced by Václav Metelka, and discover new unexpected extra collinearities in two Metelka’s configurations that arise from our realizations in cubic structures. Next, we establish the existence of various geometric concepts in four Metelka’s configuration including tangential points, inflection points, sextactic points, and quadrilaterals, and find and study significant relationships among these geometric concepts.