References analysis of the Wikipedia article "24-cell" in the English version

ams.org

Stillwell 2001, p. 17. Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.

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, p. 18-21. Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.

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

archive.org

Coxeter 1995, p. 25, (Paper 3) Two aspects of the regular 24-cell in four dimensions. Coxeter, H.S.M. (1995), Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.), Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.), Wiley-Interscience Publication, ISBN 978-0-471-01003-6

Two aspects of the regular 24-cell in four dimensions

Tyrrell & Semple 1971, pp. 5–6, §3. Clifford's original definition of parallelism. Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
Coxeter 1995, p. 29, (Paper 3) Two aspects of the regular 24-cell in four dimensions; "The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±⁠1/2⁠, ±⁠1/2⁠, 0, 0)]". Coxeter, H.S.M. (1995), Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.), Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.), Wiley-Interscience Publication, ISBN 978-0-471-01003-6
- Coxeter 1995, pp. 30–32, (Paper 3) Two aspects of the regular 24-cell in four dimensions; §3. The Dodecagonal Aspect;^[ce] Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the 120-cell, a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells. Coxeter, H.S.M. (1995), Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.), Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.), Wiley-Interscience Publication, ISBN 978-0-471-01003-6
  - Tyrrell & Semple 1971, pp. 1–9, §1. Introduction. Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
  - Tyrrell & Semple 1971, pp. 20–33, Clifford Parallel Spaces and Clifford Reguli. Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
  - 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.
  - 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 a particular study of the 24-cell's 4 rings of 6 octahedral cells 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.
  - Banchoff 2013, pp. 265–266. 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.

arxiv.org

Waegell & Aravind 2009, pp. 4–5, §3.4 The 24-cell: points, lines and Reye's configuration; In the 24-cell Reye's "points" and "lines" are axes and hexagons, 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: Mathematical and Theoretical. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.

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].

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].

Kim & Rote 2016, p. 6, §5. Four-Dimensional Rotations. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

Kim & Rote 2016, pp. 7–10, §6. Angles between two Planes in 4-Space. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

Kim & Rote 2016, pp. 8–9, Relations to Clifford parallelism. 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].

Kim & Rote 2016, pp. 14–16, §8.3 Properties of the Hopf Fibration; 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, p. 12, §8 The Construction of Hopf Fibrations; 3. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

Kim & Rote 2016, pp. 17–20, §10 The Coxeter Classification of Four-Dimensional Point Groups. Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

doi.org

Koca, Al-Ajmi & Koc 2007. Koca, Mehmet; Al-Ajmi, Mudhahir; Koc, Ramazan (November 2007). "Polyhedra obtained from Coxeter groups and quaternions". Journal of Mathematical Physics. 48 (11): 113514. Bibcode:2007JMP....48k3514K. doi:10.1063/1.2809467.

Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄. 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, §7. Conclusions; "Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change." 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. 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.

Mebius 2015, pp. 2–3, Motivation; "This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the Philippine wine dance or Binasuan and performed by physicist Richard P. Feynman during his Dirac memorial lecture 1986^[53] to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is." 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.

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, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations. 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.

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 a particular study of the 24-cell's 4 rings of 6 octahedral cells 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.

Banchoff 2013, pp. 265–266. 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.

gregegan.net

Egan 2021, animation of a rotating 24-cell: red half-integer vertices (tesseract), yellow and black integer vertices (16-cell). Egan, Greg (23 December 2021). "Symmetries and the 24-cell". gregegan.net. Retrieved 10 October 2022.

Egan 2021; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations. Egan, Greg (23 December 2021). "Symmetries and the 24-cell". gregegan.net. Retrieved 10 October 2022.

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24-cell (English Wikipedia)

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