Equally circumscribed cyclic polyhedra generalize Platonic solids

This paper presents a methodological improvement of an approach which was recently published in this journal (Wohlhart, 2017) [1]. In this paper “equally circumscribed irregular cyclic polyhedra" were defined and a synthesizing method was proposed for them. The subject of the present paper is a more straightforward new synthesizing method for such polyhedra, based on the following facts. First, two neighbouring face centers form a pair of reflection points via a reflection plane defined by the center of the circumsphere and the intersection line of the two face planes. Second, two neighbouring vertices form a pair of reflection points via the reflection plane defined by the center of the circumsphere and the two face centers. By inserting appropriate sub-mechanisms into the faces of such a polyhedron and interconnecting them properly by gussets, many different mechanisms mobilizing the cyclic polyhedra can be obtained (Wohlhart, 2007) [2]. With the new method the search for cyclic polyhedra with partially identical faces is opened. Two of such semiregular cyclic polyhedra are presented together with the mechanisms which transform them into their duals.