600-cell (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "600-cell" in English language version.

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Borovik 2006; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases." Borovik, Alexandre (2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich (eds.). The Coxeter Legacy. Providence, Rhode Island: American Mathematical Society. pp. 17–43. ISBN 978-0821837221.

ams.org

Stillwell 2001, p. 24. Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
Stillwell 2001, pp. 22–23, The Poincaré Homology Sphere. Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.

mathscinet.ams.org

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

Tyrrell & Semple 1971, pp. 6–7, §4. Isoclinic planes in Euclidean space E₄. Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
Banchoff 2013, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the Clifford torus, showed how the honeycombs correspond to Hopf fibrations, and made decompositions composed of meridian and equatorial cell rings with illustrations. Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
Tyrrell & Semple 1971, pp. 34–57, Linear Systems of Clifford Parallels. Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.

arxiv.org

Waegell & Aravind 2009, pp. 3–4, §3.2 The 75 bases of the 600-cell; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Denney et al. 2020. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, p. 438. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Zamboj 2021, pp. 10–11, §Hopf coordinates. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Dechant 2021, pp. 18–20, §6. The Coxeter Plane. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Denney et al. 2020, p. 434. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, pp. 437–439, §4 The planes of the 600-cell. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Waegell & Aravind 2009, p. 5, §3.4. The 24-cell: points, lines, and Reye's configuration; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon planes are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Copher 2019, p. 6, §3.2 Theorem 3.4. Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv:1903.06971 [math.MG].
Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)" Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Dechant 2021, §1. Introduction. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Zamboj 2021. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Dechant 2021, pp. 20–22, §7. The Grand Antiprism and H₂ × H₂. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Zamboj 2021, pp. 6–12, §2 Mathematical background. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Dechant 2017, pp. 410–419, §6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections." Dechant, Pierre-Philippe (2017). "The E8 Geometry from a Clifford Perspective". Advances in Applied Clifford Algebras. 27: 397–421. arXiv:1603.04805. doi:10.1007/s00006-016-0675-9. S2CID 253595386.
Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
Waegell & Aravind 2009, pp. 2–5, §3. The 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Kim & Rote 2016, pp. 13–14, §8.2 Equivalence of an Invariant Family and a Hopf Bundle. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Kim & Rote 2016, p. 12-16, §8 The Construction of Hopf Fibrations; see §8.3. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Kim & Rote 2016, p. 14, §8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Kim & Rote 2016, pp. 14–16, §8.3 Properties of the Hopf Fibration. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Kim & Rote 2016, p. 8, Left and Right Pairs of Isoclinic Planes. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
Denney et al. 2020, §2 The Labeling of H₄. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework." Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Dechant 2021, pp. 22–24, §8. Snub 24-cell. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Sikiric, Mathieu; Myrvold, Wendy (2007). "The special cuts of 600-cell". Beiträge zur Algebra und Geometrie. 49 (1). arXiv:0708.3443.

books.google.com

Oss 1899; van Oss does not mention the arc distances between vertices of the 600-cell. Oss, Salomon Levi van (1899). "Das regelmässige Sechshundertzell und seine selbstdeckenden Bewegungen". Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde). 7 (1). Amsterdam: 1–18.
Oss 1899, pp. 1–18. Oss, Salomon Levi van (1899). "Das regelmässige Sechshundertzell und seine selbstdeckenden Bewegungen". Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde). 7 (1). Amsterdam: 1–18.

doi.org

Waegell & Aravind 2009, pp. 3–4, §3.2 The 75 bases of the 600-cell; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Denney et al. 2020. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, p. 438. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Zamboj 2021, pp. 10–11, §Hopf coordinates. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Buekenhout & Parker 1998. Buekenhout, F.; Parker, M. (15 May 1998). "The number of nets of the regular convex polytopes in dimension <= 4". Discrete Mathematics. 186 (1–3): 69–94. doi:10.1016/S0012-365X(97)00225-2.
Dechant 2021, pp. 18–20, §6. The Coxeter Plane. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Steinbach 1997, p. 23, Figure 3; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here. Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.
Denney et al. 2020, p. 434. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, pp. 437–439, §4 The planes of the 600-cell. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Sadoc 2001, p. 576, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Waegell & Aravind 2009, p. 5, §3.4. The 24-cell: points, lines, and Reye's configuration; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon planes are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, p. 577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Lemmens & Seidel 1973. Lemmens, P.W.H.; Seidel, J.J. (1973). "Equi-isoclinic subspaces of Euclidean spaces". Indagationes Mathematicae (Proceedings). 76 (2): 98–107. doi:10.1016/1385-7258(73)90042-5. ISSN 1385-7258.
Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors $\left(w,x,y,z\right)_{1}$ and $\left(w,x,y,z\right)_{2}$ according to
${\begin{pmatrix}w_{2}\\x_{2}\\y_{2}\\z_{2}\end{pmatrix}}*{\begin{pmatrix}w_{1}\\x_{1}\\y_{1}\\z_{1}\end{pmatrix}}={\begin{pmatrix}{w_{2}w_{1}-x_{2}x_{1}-y_{2}y_{1}-z_{2}z_{1}}\\{w_{2}x_{1}+x_{2}w_{1}+y_{2}z_{1}-z_{2}y_{1}}\\{w_{2}y_{1}-x_{2}z_{1}+y_{2}w_{1}+z_{2}x_{1}}\\{w_{2}z_{1}+x_{2}y_{1}-y_{2}x_{1}+z_{2}w_{1}}\end{pmatrix}}$ Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
Sadoc 2001, pp. 575–578, §2 Geometry of the {3,3,5}-polytope in S₃; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Banchoff 2013, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the Clifford torus, showed how the honeycombs correspond to Hopf fibrations, and made decompositions composed of meridian and equatorial cell rings with illustrations. Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
Sadoc 2001, p. 578, §2.6 The {3, 3, 5} polytope: a set of four helices. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Dechant 2021, §1. Introduction. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Zamboj 2021. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Sadoc & Charvolin 2009, §1.2 The curved space approach; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the 3-sphere S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,^[af] along which the molecules can be aligned without any conflict between compactness and torsion.... The fibres of this Hopf fibration are great circles of S³, the whole family of which is also called the Clifford parallels. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.^[bf] These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.^[bg] They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint." Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A. 42 (46): 465209. Bibcode:2009JPhA...42T5209S. doi:10.1088/1751-8113/42/46/465209. S2CID 120065066.
Miyazaki 1990; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes). Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes". International Journal of Space Structures. 5 (3–4): 309–323. doi:10.1177/026635119000500312. S2CID 113846838.
Itoh & Nara 2021, §4. From the 24-cell onto an octahedron; "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.
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.
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.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope." Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column; in caption (sic) dodecagons should be decagons. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Dechant 2021, pp. 20–22, §7. The Grand Antiprism and H₂ × H₂. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Zamboj 2021, pp. 6–12, §2 Mathematical background. Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
Dechant 2017, pp. 410–419, §6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections." Dechant, Pierre-Philippe (2017). "The E8 Geometry from a Clifford Perspective". Advances in Applied Clifford Algebras. 27: 397–421. arXiv:1603.04805. doi:10.1007/s00006-016-0675-9. S2CID 253595386.
Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.^[co] Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
Perez-Gracia & Thomas 2017. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut." Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). doi:10.1007/s00006-019-0960-5. S2CID 253592159.
Mamone, Pileio & Levitt 2010, §4.5 Regular Convex 4-Polytopes, Table 2. Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
Waegell & Aravind 2009, pp. 2–5, §3. The 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, §1. Introduction; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."^[cp] Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Steinbach 1997, p. 24. Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.
Mebius 2015, p. 1, "Quaternion algebra is the tool par excellence for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the theory of 4D rotations turns out to offer the easiest road to the representation of 3D rotations by quaternions.". Mebius, Johan (July 2015) [11 Jan 1994]. Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics (Thesis). Delft University of Technology. doi:10.13140/RG.2.1.3310.3205.
Denney et al. 2020, §2 The Labeling of H₄. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework." Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Dechant 2021, pp. 22–24, §8. Snub 24-cell. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.

dx.doi.org

Lemmens & Seidel 1973. Lemmens, P.W.H.; Seidel, J.J. (1973). "Equi-isoclinic subspaces of Euclidean spaces". Indagationes Mathematicae (Proceedings). 76 (2): 98–107. doi:10.1016/1385-7258(73)90042-5. ISSN 1385-7258.

handle.net

hdl.handle.net

Perez-Gracia & Thomas 2017. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, §1. Introduction; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."^[cp] Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.

harvard.edu

ui.adsabs.harvard.edu

Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors $\left(w,x,y,z\right)_{1}$ and $\left(w,x,y,z\right)_{2}$ according to
${\begin{pmatrix}w_{2}\\x_{2}\\y_{2}\\z_{2}\end{pmatrix}}*{\begin{pmatrix}w_{1}\\x_{1}\\y_{1}\\z_{1}\end{pmatrix}}={\begin{pmatrix}{w_{2}w_{1}-x_{2}x_{1}-y_{2}y_{1}-z_{2}z_{1}}\\{w_{2}x_{1}+x_{2}w_{1}+y_{2}z_{1}-z_{2}y_{1}}\\{w_{2}y_{1}-x_{2}z_{1}+y_{2}w_{1}+z_{2}x_{1}}\\{w_{2}z_{1}+x_{2}y_{1}-y_{2}x_{1}+z_{2}w_{1}}\end{pmatrix}}$ Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
Sadoc & Charvolin 2009, §1.2 The curved space approach; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the 3-sphere S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,^[af] along which the molecules can be aligned without any conflict between compactness and torsion.... The fibres of this Hopf fibration are great circles of S³, the whole family of which is also called the Clifford parallels. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.^[bf] These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.^[bg] They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint." Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A. 42 (46): 465209. Bibcode:2009JPhA...42T5209S. doi:10.1088/1751-8113/42/46/465209. S2CID 120065066.
Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.^[co] Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
Mamone, Pileio & Levitt 2010, §4.5 Regular Convex 4-Polytopes, Table 2. Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.

iop.org

stacks.iop.org

Sadoc & Charvolin 2009, §1.2 The curved space approach; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the 3-sphere S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,^[af] along which the molecules can be aligned without any conflict between compactness and torsion.... The fibres of this Hopf fibration are great circles of S³, the whole family of which is also called the Clifford parallels. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.^[bf] These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.^[bg] They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint." Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A. 42 (46): 465209. Bibcode:2009JPhA...42T5209S. doi:10.1088/1751-8113/42/46/465209. S2CID 120065066.

johncarlosbaez.wordpress.com

Baez, John (7 March 2017). "Pi and the Golden Ratio". Azimuth. Retrieved 10 October 2022.

jstor.org

Steinbach 1997, p. 23, Figure 3; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here. Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.
Steinbach 1997, p. 24. Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.

researchgate.net

Sadoc 2001, p. 576, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, p. 577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 575–578, §2 Geometry of the {3,3,5}-polytope in S₃; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, p. 578, §2.6 The {3, 3, 5} polytope: a set of four helices. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope." Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column; in caption (sic) dodecagons should be decagons. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.

semanticscholar.org

api.semanticscholar.org

Waegell & Aravind 2009, pp. 3–4, §3.2 The 75 bases of the 600-cell; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Denney et al. 2020. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, p. 438. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Dechant 2021, pp. 18–20, §6. The Coxeter Plane. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Denney et al. 2020, p. 434. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Denney et al. 2020, pp. 437–439, §4 The planes of the 600-cell. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Sadoc 2001, p. 576, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Waegell & Aravind 2009, p. 5, §3.4. The 24-cell: points, lines, and Reye's configuration; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon planes are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, p. 577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 575–578, §2 Geometry of the {3,3,5}-polytope in S₃; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, p. 578, §2.6 The {3, 3, 5} polytope: a set of four helices. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Dechant 2021, §1. Introduction. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Sadoc & Charvolin 2009, §1.2 The curved space approach; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the 3-sphere S³, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,^[af] along which the molecules can be aligned without any conflict between compactness and torsion.... The fibres of this Hopf fibration are great circles of S³, the whole family of which is also called the Clifford parallels. Two of these fibers are C_∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.^[bf] These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S³.^[bg] They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint." Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A. 42 (46): 465209. Bibcode:2009JPhA...42T5209S. doi:10.1088/1751-8113/42/46/465209. S2CID 120065066.
Miyazaki 1990; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes). Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes". International Journal of Space Structures. 5 (3–4): 309–323. doi:10.1177/026635119000500312. S2CID 113846838.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope." Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column; in caption (sic) dodecagons should be decagons. Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
Dechant 2021, pp. 20–22, §7. The Grand Antiprism and H₂ × H₂. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Dechant 2017, pp. 410–419, §6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections." Dechant, Pierre-Philippe (2017). "The E8 Geometry from a Clifford Perspective". Advances in Applied Clifford Algebras. 27: 397–421. arXiv:1603.04805. doi:10.1007/s00006-016-0675-9. S2CID 253595386.
Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
Perez-Gracia & Thomas 2017. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut." Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). doi:10.1007/s00006-019-0960-5. S2CID 253592159.
Waegell & Aravind 2009, pp. 2–5, §3. The 600-cell. Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, §1. Introduction; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."^[cp] Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Denney et al. 2020, §2 The Labeling of H₄. Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework." Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.
Dechant 2021, pp. 22–24, §8. Snub 24-cell. Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. 31 (3). Springer Science and Business Media. arXiv:2103.07817. doi:10.1007/s00006-021-01139-2. S2CID 232232920.

tudelft.nl

resolver.tudelft.nl

van Ittersum 2020, pp. 80–95, §4.3. van Ittersum, Clara (2020). Symmetry groups of regular polytopes in three and four dimensions (Thesis). Delft University of Technology.

uni-goettingen.de

resolver.sub.uni-goettingen.de

Schläfli 1858; this paper of Schläfli's describing the double six configuration was one of the only fragments of his discovery of the regular polytopes in higher dimensions to be published during his lifetime.^[35] Schläfli, Ludwig (1858), Cayley, Arthur (ed.), "An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface", Quarterly Journal of Pure and Applied Mathematics, 2: 55–65, 110–120

upc.edu

upcommons.upc.edu

Perez-Gracia & Thomas 2017. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
Perez-Gracia & Thomas 2017, §1. Introduction; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."^[cp] Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.

worldcat.org

search.worldcat.org

Lemmens & Seidel 1973. Lemmens, P.W.H.; Seidel, J.J. (1973). "Equi-isoclinic subspaces of Euclidean spaces". Indagationes Mathematicae (Proceedings). 76 (2): 98–107. doi:10.1016/1385-7258(73)90042-5. ISSN 1385-7258.
Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.