Kinematics of the cuboctahedron (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Kinematics of the cuboctahedron" in English language version.

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Uberti, R.; Janse van Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.), Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89–100, doi:10.1007/978-1-4612-1712-1_9, ISBN 978-0-387-98580-0, MR 1655039; see Table 2, p. 97
Verheyen 1989, p. 203; "As Clinton observed in his paper on expanding rigid structures,^[7] each triangle is subject to a translation-rotation along its symmetry axis. When starting from the position in the octahedron, these axes are the four triangular symmetry axes of the octahedron. When describing cylinders about the triangles along the axes, each vertex common to two triangles moves along the intersecting [helical] curve of the two cylinders." Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers and Mathematics with Applications. 17 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0. MR 0994201.
Verheyen 1989. Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers and Mathematics with Applications. 17 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0. MR 0994201.

archive.org

Kenner 1976, pp. 11–19, §2. Spherical tensegrities. Kenner, Hugh (1976). Geodesic Math and How to Use It. University of California Press. ISBN 978-0520029248.
Kenner 1976, p. 14, Equilibrium. Kenner, Hugh (1976). Geodesic Math and How to Use It. University of California Press. ISBN 978-0520029248.
Kenner 1976, pp. 16–17, Elasticity Multiplication. Kenner, Hugh (1976). Geodesic Math and How to Use It. University of California Press. ISBN 978-0520029248.
Kenner 1976, p. 12, Equilibrium. Kenner, Hugh (1976). Geodesic Math and How to Use It. University of California Press. ISBN 978-0520029248.

bridgesmathart.org

archive.bridgesmathart.org

Gunn & Sullivan 2008, §3. Pyritohedral Symmetry; "The pyritohedral 3D symmetry group is the unique polyhedral point group that is neither a rotation group nor a reflection group." Gunn, Charles; Sullivan, John M. (2008), "The Borromean Rings: A video about the New IMU logo", in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN 978-0-9665201-9-4; see the video itself at "The Borromean Rings: A new logo for the IMU", International Mathematical Union

doi.org

Koca et al. 2016, p. 145, 4. Pyritohedral Group and Related Polyhedra; see Table 1. Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01). "Symmetry of the Pyritohedron and Lattices". Sultan Qaboos University Journal for Science [SQUJS]. 21 (2): 139. doi:10.24200/squjs.vol21iss2pp139-149.
Koca et al. 2016, 4.1 Construction of the vertices of the pseudoicosahedron. Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01). "Symmetry of the Pyritohedron and Lattices". Sultan Qaboos University Journal for Science [SQUJS]. 21 (2): 139. doi:10.24200/squjs.vol21iss2pp139-149.
Uberti, R.; Janse van Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", in Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (eds.), Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications, vol. 103, New York: Springer, pp. 89–100, doi:10.1007/978-1-4612-1712-1_9, ISBN 978-0-387-98580-0, MR 1655039; see Table 2, p. 97
Verheyen 1989, p. 203; "As Clinton observed in his paper on expanding rigid structures,^[7] each triangle is subject to a translation-rotation along its symmetry axis. When starting from the position in the octahedron, these axes are the four triangular symmetry axes of the octahedron. When describing cylinders about the triangles along the axes, each vertex common to two triangles moves along the intersecting [helical] curve of the two cylinders." Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers and Mathematics with Applications. 17 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0. MR 0994201.
Itoh & Nara 2021, p. 13, §4. From the 24-cell onto an octahedron; "Lemma 4.2. There is a continuous motion of Q (the cuboctahedron without square faces) shown in Fig. 5a onto the octahedron W⁰ satisfying the following conditions for each face F of Q, e.g. F = 𝚫a₁a₂a₃. (1) F is rotated and moved toward along the line l joining the centroids of F and 𝚫v₁v₂v₃. (2) F always touches the cylinder T(F), that is, F is always orthogonal to l." Itoh, Jin-ichi; Nara, Chie (2021). "Continuous flattening of the 2-dimensional skeleton of a regular 24-cell". Journal of Geometry. 112 (13). doi:10.1007/s00022-021-00575-6.
Verheyen 1989. Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers and Mathematics with Applications. 17 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0. MR 0994201.
Itoh & Nara 2021, Abstract; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller." Itoh, Jin-ichi; Nara, Chie (2021). "Continuous flattening of the 2-dimensional skeleton of a regular 24-cell". Journal of Geometry. 112 (13). doi:10.1007/s00022-021-00575-6.

uiuc.edu

torus.math.uiuc.edu

Gunn & Sullivan 2008, §3. Pyritohedral Symmetry; "The pyritohedral 3D symmetry group is the unique polyhedral point group that is neither a rotation group nor a reflection group." Gunn, Charles; Sullivan, John M. (2008), "The Borromean Rings: A video about the New IMU logo", in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN 978-0-9665201-9-4; see the video itself at "The Borromean Rings: A new logo for the IMU", International Mathematical Union

youtube.com

Fuller 1975, Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire rigid-edge transformation cycle; in this film, he does not demonstrate the elastic-edge transformation (which he observed in the tensegrity icosahedron), but he does show how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times; actually, Fuller could have rotated any of the kinematic polyhedra in an inscribing cube in this way: the entire cuboctahedron transformation cycle takes place inside an inscribing cube of varying edge length, with the 12 vertices always on the surface of the cube. Fuller, R. Buckminster (1975). "Vector Equilibrium". Everything I Know Sessions. Philadelphia.