Dehn invariant (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Dehn invariant" in English language version.

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26doi.org

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1archive.org

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mathscinet.ams.org

Giovannini, Eduardo N. (2021), "David Hilbert and the foundations of the theory of plane area", Archive for History of Exact Sciences, 75 (6): 649–698, doi:10.1007/s00407-021-00278-z, MR 4324749
Gruber, Peter M. (2007), "Chapter 16: Volume of Polytopes and Hilbert's Third Problem", Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, pp. 280–291, doi:10.1007/978-3-540-71133-9, ISBN 978-3-540-71132-2, MR 2335496.
Benko, David (2007), "A new approach to Hilbert's third problem", American Mathematical Monthly, 114 (8): 665–676, doi:10.1080/00029890.2007.11920458, JSTOR 27642302, MR 2354437, S2CID 7213930.
Dupont, Johan L.; Sah, Chih-Han (2000), "Three questions about simplices in spherical and hyperbolic 3-space", The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, pp. 49–76, doi:10.1007/978-1-4612-1340-6_3, ISBN 978-1-4612-7102-4, MR 1731633. See in particular p. 61.
Jessen, Børge (1968), "The algebra of polyhedra and the Dehn–Sydler theorem", Mathematica Scandinavica, 22 (2): 241–256, doi:10.7146/math.scand.a-10888, JSTOR 24489773, MR 0251633.
Conway, J. H.; Radin, C.; Sadun, L. (1999), "On angles whose squared trigonometric functions are rational", Discrete and Computational Geometry, 22 (3): 321–332, arXiv:math-ph/9812019, doi:10.1007/PL00009463, MR 1706614, S2CID 563915, Table 3, p. 331.
Akiyama, Jin; Matsunaga, Kiyoko (2015), "15.3 Hilbert's Third Problem and Dehn Theorem", Treks Into Intuitive Geometry, Springer, Tokyo, pp. 382–388, doi:10.1007/978-4-431-55843-9, ISBN 978-4-431-55841-5, MR 3380801.
Shephard, G. C. (1974), "Combinatorial properties of associated zonotopes", Canadian Journal of Mathematics, 26 (2): 302–321, doi:10.4153/CJM-1974-032-5, MR 0362054; see in particular section 5, "cubical dissections of zonotopes"
Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions", Commentarii Mathematici Helvetici (in French), 40: 43–80, doi:10.1007/bf02564364, MR 0192407, S2CID 123317371
Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln", Archiv der Mathematik (in German), 35 (6): 583–587, doi:10.1007/BF01235384, MR 0604258, S2CID 121301319.
Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill $\mathbb {R} ^{n}$ and scissors congruence", Discrete & Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797.
Goncharov, Alexander (1999), "Volumes of hyperbolic manifolds and mixed Tate motives", Journal of the American Mathematical Society, 12 (2): 569–618, doi:10.1090/S0894-0347-99-00293-3, MR 1649192.
Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry, 99 (1–2): 1–13, arXiv:0901.2989, doi:10.1007/s00022-011-0061-7, MR 2823098, S2CID 17515249.
Hartshorne, Robin (2000), Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 232–234, doi:10.1007/978-0-387-22676-7, ISBN 0-387-98650-2, MR 1761093.
Stillwell, John (1998), Numbers and geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 164, doi:10.1007/978-1-4612-0687-3, ISBN 0-387-98289-2, MR 1479640.
Dupont, Johan L. (2001), Scissors congruences, group homology and characteristic classes, Nankai Tracts in Mathematics, vol. 1, River Edge, New Jersey: World Scientific, p. 4, doi:10.1142/9789812810335, ISBN 981-02-4507-6, MR 1832859, archived from the original on 2016-04-29.
Fuchs, Dmitry; Tabachnikov, Serge (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 312, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979. This source uses the Hamel basis formulation of the Dehn invariant, but with tensor notation used for the unit vectors.
Coulson, David; Goodman, Oliver A.; Hodgson, Craig D.; Neumann, Walter D. (2000), "Computing arithmetic invariants of 3-manifolds", Experimental Mathematics, 9 (1): 127–152, doi:10.1080/10586458.2000.10504641, MR 1758805, S2CID 1313215
Dupont, Johan L.; Sah, Chih-Han (1990), "Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences", Acta Mathematica, 164 (1–2): 1–27, doi:10.1007/BF02392750, MR 1037596
Spandaw, Jeroen (2004), "Dissecting cuboids into cuboids", The American Mathematical Monthly, 111 (5): 425–429, doi:10.2307/4145269, JSTOR 4145269, MR 2057392
Gaifullin, Alexander A.; Ignashchenko, Leonid S. (August 2018), "Dehn invariant and scissors congruence of flexible polyhedra", Proceedings of the Steklov Institute of Mathematics, 302 (1): 130–145, arXiv:1710.11247, doi:10.1134/s0081543818060068, MR 3894642
Alexandrov, Victor; Connelly, Robert (2011), "Flexible suspensions with a hexagonal equator", Illinois Journal of Mathematics, 55 (1): 127–155, arXiv:0905.3683, doi:10.1215/ijm/1355927031, MR 3006683, S2CID 12302514.
Alexander, Ralph (1985), "Lipschitzian mappings and total mean curvature of polyhedral surfaces. I", Transactions of the American Mathematical Society, 288 (2): 661–678, doi:10.2307/1999957, JSTOR 1999957, MR 0776397.

archive.org

scholar.archive.org

Benko, David (2007), "A new approach to Hilbert's third problem", American Mathematical Monthly, 114 (8): 665–676, doi:10.1080/00029890.2007.11920458, JSTOR 27642302, MR 2354437, S2CID 7213930.

arxiv.org

Conway, J. H.; Radin, C.; Sadun, L. (1999), "On angles whose squared trigonometric functions are rational", Discrete and Computational Geometry, 22 (3): 321–332, arXiv:math-ph/9812019, doi:10.1007/PL00009463, MR 1706614, S2CID 563915, Table 3, p. 331.
Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry, 99 (1–2): 1–13, arXiv:0901.2989, doi:10.1007/s00022-011-0061-7, MR 2823098, S2CID 17515249.
Eppstein, David (2022), "Orthogonal dissection into few rectangles", Proceedings of the 34th Canadian Conference on Computational Geometry, arXiv:2206.10675
Gaifullin, Alexander A.; Ignashchenko, Leonid S. (August 2018), "Dehn invariant and scissors congruence of flexible polyhedra", Proceedings of the Steklov Institute of Mathematics, 302 (1): 130–145, arXiv:1710.11247, doi:10.1134/s0081543818060068, MR 3894642
Alexandrov, Victor; Connelly, Robert (2011), "Flexible suspensions with a hexagonal equator", Illinois Journal of Mathematics, 55 (1): 127–155, arXiv:0905.3683, doi:10.1215/ijm/1355927031, MR 3006683, S2CID 12302514.

au.dk

home.math.au.dk

Dupont, Johan L. (2001), Scissors congruences, group homology and characteristic classes, Nankai Tracts in Mathematics, vol. 1, River Edge, New Jersey: World Scientific, p. 4, doi:10.1142/9789812810335, ISBN 981-02-4507-6, MR 1832859, archived from the original on 2016-04-29.

books.google.com

Dupont, Johan L.; Sah, Chih-Han (2000), "Three questions about simplices in spherical and hyperbolic 3-space", The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, pp. 49–76, doi:10.1007/978-1-4612-1340-6_3, ISBN 978-1-4612-7102-4, MR 1731633. See in particular p. 61.
Hartshorne, Robin (2000), Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 232–234, doi:10.1007/978-0-387-22676-7, ISBN 0-387-98650-2, MR 1761093.
Stillwell, John (1998), Numbers and geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 164, doi:10.1007/978-1-4612-0687-3, ISBN 0-387-98289-2, MR 1479640.
Fuchs, Dmitry; Tabachnikov, Serge (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 312, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979. This source uses the Hamel basis formulation of the Dehn invariant, but with tensor notation used for the unit vectors.

brown.edu

math.brown.edu

Schwartz, Rich (June 10, 2013), "The Dehn–Sydler theorem explained" (PDF), Math Notes, Brown University Department of Mathematics, retrieved 2023-03-13

doi.org

Giovannini, Eduardo N. (2021), "David Hilbert and the foundations of the theory of plane area", Archive for History of Exact Sciences, 75 (6): 649–698, doi:10.1007/s00407-021-00278-z, MR 4324749
Zeeman, E. C. (July 2002), "On Hilbert's third problem", The Mathematical Gazette, 86 (506): 241–247, doi:10.2307/3621846, JSTOR 3621846
Gruber, Peter M. (2007), "Chapter 16: Volume of Polytopes and Hilbert's Third Problem", Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, pp. 280–291, doi:10.1007/978-3-540-71133-9, ISBN 978-3-540-71132-2, MR 2335496.
Benko, David (2007), "A new approach to Hilbert's third problem", American Mathematical Monthly, 114 (8): 665–676, doi:10.1080/00029890.2007.11920458, JSTOR 27642302, MR 2354437, S2CID 7213930.
Dupont, Johan L.; Sah, Chih-Han (2000), "Three questions about simplices in spherical and hyperbolic 3-space", The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, pp. 49–76, doi:10.1007/978-1-4612-1340-6_3, ISBN 978-1-4612-7102-4, MR 1731633. See in particular p. 61.
Jessen, Børge (1968), "The algebra of polyhedra and the Dehn–Sydler theorem", Mathematica Scandinavica, 22 (2): 241–256, doi:10.7146/math.scand.a-10888, JSTOR 24489773, MR 0251633.
Conway, J. H.; Radin, C.; Sadun, L. (1999), "On angles whose squared trigonometric functions are rational", Discrete and Computational Geometry, 22 (3): 321–332, arXiv:math-ph/9812019, doi:10.1007/PL00009463, MR 1706614, S2CID 563915, Table 3, p. 331.
Akiyama, Jin; Matsunaga, Kiyoko (2015), "15.3 Hilbert's Third Problem and Dehn Theorem", Treks Into Intuitive Geometry, Springer, Tokyo, pp. 382–388, doi:10.1007/978-4-431-55843-9, ISBN 978-4-431-55841-5, MR 3380801.
Shephard, G. C. (1974), "Combinatorial properties of associated zonotopes", Canadian Journal of Mathematics, 26 (2): 302–321, doi:10.4153/CJM-1974-032-5, MR 0362054; see in particular section 5, "cubical dissections of zonotopes"
Dehn, Max (1901), "Ueber den Rauminhalt", Mathematische Annalen (in German), 55 (3): 465–478, doi:10.1007/BF01448001, S2CID 120068465
Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions", Commentarii Mathematici Helvetici (in French), 40: 43–80, doi:10.1007/bf02564364, MR 0192407, S2CID 123317371
Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln", Archiv der Mathematik (in German), 35 (6): 583–587, doi:10.1007/BF01235384, MR 0604258, S2CID 121301319.
Lagarias, J. C.; Moews, D. (1995), "Polytopes that fill $\mathbb {R} ^{n}$ and scissors congruence", Discrete & Computational Geometry, 13 (3–4): 573–583, doi:10.1007/BF02574064, MR 1318797.
Goncharov, Alexander (1999), "Volumes of hyperbolic manifolds and mixed Tate motives", Journal of the American Mathematical Society, 12 (2): 569–618, doi:10.1090/S0894-0347-99-00293-3, MR 1649192.
Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry, 99 (1–2): 1–13, arXiv:0901.2989, doi:10.1007/s00022-011-0061-7, MR 2823098, S2CID 17515249.
Hartshorne, Robin (2000), Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 232–234, doi:10.1007/978-0-387-22676-7, ISBN 0-387-98650-2, MR 1761093.
Stillwell, John (1998), Numbers and geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, p. 164, doi:10.1007/978-1-4612-0687-3, ISBN 0-387-98289-2, MR 1479640.
Dupont, Johan L. (2001), Scissors congruences, group homology and characteristic classes, Nankai Tracts in Mathematics, vol. 1, River Edge, New Jersey: World Scientific, p. 4, doi:10.1142/9789812810335, ISBN 981-02-4507-6, MR 1832859, archived from the original on 2016-04-29.
Fuchs, Dmitry; Tabachnikov, Serge (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 312, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979. This source uses the Hamel basis formulation of the Dehn invariant, but with tensor notation used for the unit vectors.
Coulson, David; Goodman, Oliver A.; Hodgson, Craig D.; Neumann, Walter D. (2000), "Computing arithmetic invariants of 3-manifolds", Experimental Mathematics, 9 (1): 127–152, doi:10.1080/10586458.2000.10504641, MR 1758805, S2CID 1313215
Dupont, Johan L.; Sah, Chih-Han (1990), "Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences", Acta Mathematica, 164 (1–2): 1–27, doi:10.1007/BF02392750, MR 1037596
Dehn, Max (1903), "Über Zerlegung von Rechtecken in Rechtecke", Mathematische Annalen, 57 (3): 314–332, doi:10.1007/BF01444289
Spandaw, Jeroen (2004), "Dissecting cuboids into cuboids", The American Mathematical Monthly, 111 (5): 425–429, doi:10.2307/4145269, JSTOR 4145269, MR 2057392
Gaifullin, Alexander A.; Ignashchenko, Leonid S. (August 2018), "Dehn invariant and scissors congruence of flexible polyhedra", Proceedings of the Steklov Institute of Mathematics, 302 (1): 130–145, arXiv:1710.11247, doi:10.1134/s0081543818060068, MR 3894642
Alexandrov, Victor; Connelly, Robert (2011), "Flexible suspensions with a hexagonal equator", Illinois Journal of Mathematics, 55 (1): 127–155, arXiv:0905.3683, doi:10.1215/ijm/1355927031, MR 3006683, S2CID 12302514.
Alexander, Ralph (1985), "Lipschitzian mappings and total mean curvature of polyhedral surfaces. I", Transactions of the American Mathematical Society, 288 (2): 661–678, doi:10.2307/1999957, JSTOR 1999957, MR 0776397.

encyclopediaofmath.org

Hazewinkel, M. (2001) [1994], "Dehn invariant", Encyclopedia of Mathematics, EMS Press

eudml.org

Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions", Commentarii Mathematici Helvetici (in French), 40: 43–80, doi:10.1007/bf02564364, MR 0192407, S2CID 123317371

jstor.org

Zeeman, E. C. (July 2002), "On Hilbert's third problem", The Mathematical Gazette, 86 (506): 241–247, doi:10.2307/3621846, JSTOR 3621846
Benko, David (2007), "A new approach to Hilbert's third problem", American Mathematical Monthly, 114 (8): 665–676, doi:10.1080/00029890.2007.11920458, JSTOR 27642302, MR 2354437, S2CID 7213930.
Jessen, Børge (1968), "The algebra of polyhedra and the Dehn–Sydler theorem", Mathematica Scandinavica, 22 (2): 241–256, doi:10.7146/math.scand.a-10888, JSTOR 24489773, MR 0251633.
Spandaw, Jeroen (2004), "Dissecting cuboids into cuboids", The American Mathematical Monthly, 111 (5): 425–429, doi:10.2307/4145269, JSTOR 4145269, MR 2057392
Alexander, Ralph (1985), "Lipschitzian mappings and total mean curvature of polyhedral surfaces. I", Transactions of the American Mathematical Society, 288 (2): 661–678, doi:10.2307/1999957, JSTOR 1999957, MR 0776397.

projecteuclid.org

Coulson, David; Goodman, Oliver A.; Hodgson, Craig D.; Neumann, Walter D. (2000), "Computing arithmetic invariants of 3-manifolds", Experimental Mathematics, 9 (1): 127–152, doi:10.1080/10586458.2000.10504641, MR 1758805, S2CID 1313215

semanticscholar.org

api.semanticscholar.org

Benko, David (2007), "A new approach to Hilbert's third problem", American Mathematical Monthly, 114 (8): 665–676, doi:10.1080/00029890.2007.11920458, JSTOR 27642302, MR 2354437, S2CID 7213930.
Conway, J. H.; Radin, C.; Sadun, L. (1999), "On angles whose squared trigonometric functions are rational", Discrete and Computational Geometry, 22 (3): 321–332, arXiv:math-ph/9812019, doi:10.1007/PL00009463, MR 1706614, S2CID 563915, Table 3, p. 331.
Dehn, Max (1901), "Ueber den Rauminhalt", Mathematische Annalen (in German), 55 (3): 465–478, doi:10.1007/BF01448001, S2CID 120068465
Sydler, J.-P. (1965), "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions", Commentarii Mathematici Helvetici (in French), 40: 43–80, doi:10.1007/bf02564364, MR 0192407, S2CID 123317371
Debrunner, Hans E. (1980), "Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln", Archiv der Mathematik (in German), 35 (6): 583–587, doi:10.1007/BF01235384, MR 0604258, S2CID 121301319.
Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry, 99 (1–2): 1–13, arXiv:0901.2989, doi:10.1007/s00022-011-0061-7, MR 2823098, S2CID 17515249.
Coulson, David; Goodman, Oliver A.; Hodgson, Craig D.; Neumann, Walter D. (2000), "Computing arithmetic invariants of 3-manifolds", Experimental Mathematics, 9 (1): 127–152, doi:10.1080/10586458.2000.10504641, MR 1758805, S2CID 1313215
Alexandrov, Victor; Connelly, Robert (2011), "Flexible suspensions with a hexagonal equator", Illinois Journal of Mathematics, 55 (1): 127–155, arXiv:0905.3683, doi:10.1215/ijm/1355927031, MR 3006683, S2CID 12302514.

unhyperbolic.org

Görner, Matthias, "α-polyhedra", Unhyperbolic, retrieved 2024-06-26

web.archive.org

Dupont, Johan L. (2001), Scissors congruences, group homology and characteristic classes, Nankai Tracts in Mathematics, vol. 1, River Edge, New Jersey: World Scientific, p. 4, doi:10.1142/9789812810335, ISBN 981-02-4507-6, MR 1832859, archived from the original on 2016-04-29.

zenodo.org

Dehn, Max (1901), "Ueber den Rauminhalt", Mathematische Annalen (in German), 55 (3): 465–478, doi:10.1007/BF01448001, S2CID 120068465