References analysis of the Wikipedia article "120-cell" in the English version

arxiv.org

Dechant 2021, p. 18, Remark 5.7, explains why not.^[e] 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.

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.

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, Günter (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].

Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism. Kim, Heuna; Rote, Günter (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].

Schleimer & Segerman 2013. Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.

Denney et al. 2020, p. 5, §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:1912.06156v1. doi:10.1515/advgeom-2020-0005. S2CID 220367622.

Waegell & Aravind 2014, pp. 3–4, §2 Geometry of the 120-cell: rays and bases. Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell". Foundations of Physics. 44 (10): 1085–1095. arXiv:1309.7530v3. Bibcode:2014FoPh...44.1085W. doi:10.1007/s10701-014-9830-0. S2CID 254504443.

Waegell & Aravind 2014, pp. 5–6. Waegell, Mordecai; Aravind, P.K. (10 Sep 2014). "Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell". Foundations of Physics. 44 (10): 1085–1095. arXiv:1309.7530v3. Bibcode:2014FoPh...44.1085W. doi:10.1007/s10701-014-9830-0. S2CID 254504443.

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.

Schleimer & Segerman 2013, p. 16, §6.1. Layers of dodecahedra. Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.

Schleimer & Segerman 2013, pp. 16–18, §6.2. Rings of dodecahedra. Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.

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:2003.09236v2. doi:10.1093/jcde/qwab018.

Zamboj 2021, pp. 23–29, §5 Hopf tori corresponding to circles on B². 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:2003.09236v2. doi:10.1093/jcde/qwab018.

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:1912.06156v1. 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.

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:1912.06156v1. doi:10.1515/advgeom-2020-0005. S2CID 220367622.

doi.org

Mamone, Pileio & Levitt 2010, p. 1442, Table 3. 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, 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.

Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of pentagram isoclinic rotations of an individual 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell, the 120-cell's characteristic rotation. 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, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes, Table 2, Symmetry group 𝛨₄; the 120-cell has 7200 distinct rotational displacements (and 7200 reflections), which can be grouped as 25 distinct isoclinic rotations. 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. 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.

Schleimer & Segerman 2013. Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.

Banchoff 2013. 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.

Chilton 1964. Chilton, B. L. (September 1964). "On the projection of the regular polytope {5,3,3} into a regular triacontagon". Canadian Mathematical Bulletin. 7 (3): 385–398. doi:10.4153/CMB-1964-037-9.

harvard.edu

ui.adsabs.harvard.edu

\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}}

isama.org

Sullivan 1991, pp. 4–5, The Dodecahedron. Sullivan, John M. (1991). "Generating and Rendering Four-Dimensional Polytopes". Mathematica Journal. 1 (3): 76–85.

Sullivan 1991, p. 15, Other Properties of the 120-cell. Sullivan, John M. (1991). "Generating and Rendering Four-Dimensional Polytopes". Mathematica Journal. 1 (3): 76–85.

semanticscholar.org

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Schleimer & Segerman 2013. Schleimer, Saul; Segerman, Henry (2013). "Puzzling the 120-cell". Notices Amer. Math. Soc. 62 (11): 1309–1316. arXiv:1310.3549. doi:10.1090/noti1297. S2CID 117636740.

120-cell (English Wikipedia)

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