Equally circumscribed cyclic polyhedra generalize Platonic solids

• Published in: Mechanism and machine theory Vol. 133; pp. 150 - 163
• Main Author: Wohlhart, Karl
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2019
• Subjects: Cyclic polyhedra; Double dual cyclic polyhedra; Dual cyclic polyhedra; Reflection points via reflection plane (mirror); Double dual cyclic polyhedra; Cyclic polyhedra; Dual cyclic polyhedra; Reflection points via reflection plane (mirror)
• ISSN: 0094-114X; 1873-3999
• DOI: 10.1016/j.mechmachtheory.2018.10.004

•The subject of this paper is already given expression in its title “equally circumscribed cyclic polyhedra”, since these represent a generalization of classical “Platonic solids” and it has become apparent in our research that a generalization of this kind has never yet been tried and/or reported.•A new method for the synthesis of “equally circumscribed cyclic polyhedra”, based on two simple reflection theorems is presented and applied in this work to construct five “equally circumscribed irregular cyclic polyhedra”. One of them, the irregular tetrahedron turns out to be a tetrahedron with four equal irregular triangular faces. No trace whatsoever of this very special irregular tetrahedron could be found in the course of exhaustive research in the literature on geometry under the keyword “tetrahedron”.•New mechanisms are designed that allow a change of an equally circumscribed polyhedron into its dual polyhedron.•Open questions are formulated that show directions for future research, namely the question, whether the unlimited variety of “equally circumscribed irregular cyclic polyhedra” contains not only an irregular tetrahedron with equal faces, but also an irregular hexahedron with equal faces or an irregular dodecahedron with equal faces. An answer to this question together with an accompanying. 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.