Archimedean synthesis and magic numbers: 'Sizing' giant molybdenum- oxide-based molecular spheres of the keplerate type

Angewandte Chemie - International Edition

Volumn 38, Issue 21, 1999, Pages 3238-3241

  Müller, Achim ^a   Sarkar, Sabyasachi ^a,c   Shah, Syed Qaiser Nazir ^a   Bögge, Hartmut ^a   Schmidtmann, Marc ^a   Sarkar, Shatarupa ^a   Kögerler, Paul ^a   Hauptfleisch, Björn ^a   Trautwein, Alfred X ^b   Schünemann, Volker ^b  

^a BIELEFELD UNIVERSITY   (Germany)

^b UNIVERSITY OF LÜBECK   (Germany)

^c INDIAN INSTITUTE OF TECHNOLOGY KANPUR   (India)

Author keywords

Cage compounds; Cluster compounds; Polyoxometalates; Supramolecular chemistry; Topology

Indexed keywords

MOLYBDENUM; OXIDE;

ARTICLE; CHEMICAL REACTION; CHEMICAL STRUCTURE; CRYSTALLIZATION; MOLECULAR MODEL; RAMAN SPECTROMETRY; SYNTHESIS;

EID: 0038040165     PISSN: 14337851     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1521-3773(19991102)38:21<3238::AID-ANIE3238>3.0.CO;2-6     Document Type: Article

References (36)

1. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
2. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
3. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
4. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
5. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
6. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
7. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
8. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
9. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
10. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
11. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
12. Pentagons are of particular interest in numerous scientific areas: for the mathematician (to construct one of the five regular solids given in Euclid's famous work "Elements (Book XIII)", to which Dürer also refers), for the architect and the art theorist or felix aestheticus (in the sense of Fuller's geodesic domes and with respect to the "Golden Section"), for the solid-state physicist (with regard to the fascinating quasicrystals), for the virologist (because of the structure of spherical viruses), for the classical morphologist amongst the biologists (in particular because of the numerous plant organisms (blossoms) with fivefold symmetry), more recently for the cosmologist (because of a planetary nebula with a fullerene structure), and naturally for the chemist, who can use them to design harmonic molecular constructions. Pythagoras even associated physical well-being with the five-pointed star, which was the symbolic sign of his school, and he wanted - obviously with not very altruistic intentions - to keep secret the existence of the dodecahedron derived from it. a) A. L. Mackay, Crystals and Fivefold Symmetry in Quasicrystals, Networks, and Molecules of Fivefold Symmetry (Ed.: J. Hargittai), VCH, Weinheim, 1990, pp. 1-18; b) H. W. Kroto, K. Prassides, A. J. Stace, R. Taylor, D. R. M. Walton in Buckminsterfullerenes (Eds.: W. E. Billups, M. A. Ciufolini), VCH, Weinheim, 1993, pp. 21-57; c) K. Miyazaki, Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg. Braunschweig, 1987, pp. 69-75 (chap.: Fünfecke und Fünfsterne); d) R. Penrose in Hermann Weyl, 1885-1985: Centenary Lectures (Ed.: K. Chandrasekharan), Springer, Berlin, 1986, pp. 23-52; e) H. S. M. Coxeter, Regular Polytopes, Macmillan, New York, 1963 (in particular the various chapters with "historical remarks"); f) H. Weyl, Symmetry, Princeton University Press, Princeton, NJ, 1952, pp. 41-80 (chap.: Translatory, rotational, and related symmetries); g) E. Schröder, Dürer, Kunst und Geometrie: Dürers künstlerisches Schaffen aus der Sicht seiner "Underweysung", Birkhäuser, Basel, 1980; h) L. Liljas, Prog. Biophys. Mol. Biol. 1986, 48, 1-36; i) V. I. Arnol'd, Huygens and Barrow, Newton and Hooke: Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser, Basel, 1990, pp. 62-66 (chap. 3, paragraph 16: The icosahedron and quasicrystals); j) S. Hildebrandt, A. Tromba, Kugel, Kreis und Seifenblasen: Optimale Formen in Geometrie und Natur, Birkhäuser, Basel, 1996; k) A. Dress, D. Huson, A. Müller in Muster des Lebendingen: Faszination ihrer Entstehung und Simulation (Ed.: A. Deutsch), Vieweg, Wiesbaden, 1994, pp. 183-202; l) A. Stückelberger, Einführung in die antiken Naturwissenschaften, Wiss. Buchgesellschaft, Darmstadt, 1988, p. 12.
13. [1a]). Even Archimedes had recognized the importance of pentagonal patterns for the construction of spherical structures (cf. book 5 of the "Collectiones" of Pappos of Alexandria (around 320 AD), the last famous mathematician of that epoch).
14. [5] and in the present Keplerate 102 metal atoms, and in both cases 12 of them span an icosahedron in an arrangement corresponding to Kepler's early speculative model of the cosmos.
15. 2- type, are particularly suitable for the linkage of nucleophilic molybdate fragments (see: A. Müller, W. Plass, E. Krickemeyer, S. Dillinger, H. Bögge, A. Armatage, A. Proust, C. Beugholt, U. Bergmann, Angew. Chem. 1994, 106, 897-899; Angew. Chem. Int. Ed. Engl. 1994, 33, 849-851; A. Müller, J. Meyer, E. Krickemeyer, C. Beugholt, H. Bögge, F. Peters, M. Schmidtmann, P. Kögerler, M. J. Koop, Chem. Eur. J. 1998, 4, 1000-1006; A. Müller. W. Plass, E. Krickemeyer, R. Sessoli, D. Gatteschi, J. Meyer, H. Bögge, M. Kröckel, A. X. Trautwein, Inorg. Chem. Acta 1998, 271, 9-12).
16. 2- type, are particularly suitable for the linkage of nucleophilic molybdate fragments (see: A. Müller, W. Plass, E. Krickemeyer, S. Dillinger, H. Bögge, A. Armatage, A. Proust, C. Beugholt, U. Bergmann, Angew. Chem. 1994, 106, 897-899; Angew. Chem. Int. Ed. Engl. 1994, 33, 849-851; A. Müller, J. Meyer, E. Krickemeyer, C. Beugholt, H. Bögge, F. Peters, M. Schmidtmann, P. Kögerler, M. J. Koop, Chem. Eur. J. 1998, 4, 1000-1006; A. Müller. W. Plass, E. Krickemeyer, R. Sessoli, D. Gatteschi, J. Meyer, H. Bögge, M. Kröckel, A. X. Trautwein, Inorg. Chem. Acta 1998, 271, 9-12).
17. 2- type, are particularly suitable for the linkage of nucleophilic molybdate fragments (see: A. Müller, W. Plass, E. Krickemeyer, S. Dillinger, H. Bögge, A. Armatage, A. Proust, C. Beugholt, U. Bergmann, Angew. Chem. 1994, 106, 897-899; Angew. Chem. Int. Ed. Engl. 1994, 33, 849-851; A. Müller, J. Meyer, E. Krickemeyer, C. Beugholt, H. Bögge, F. Peters, M. Schmidtmann, P. Kögerler, M. J. Koop, Chem. Eur. J. 1998, 4, 1000-1006; A. Müller. W. Plass, E. Krickemeyer, R. Sessoli, D. Gatteschi, J. Meyer, H. Bögge, M. Kröckel, A. X. Trautwein, Inorg. Chem. Acta 1998, 271, 9-12).
18. 2- type, are particularly suitable for the linkage of nucleophilic molybdate fragments (see: A. Müller, W. Plass, E. Krickemeyer, S. Dillinger, H. Bögge, A. Armatage, A. Proust, C. Beugholt, U. Bergmann, Angew. Chem. 1994, 106, 897-899; Angew. Chem. Int. Ed. Engl. 1994, 33, 849-851; A. Müller, J. Meyer, E. Krickemeyer, C. Beugholt, H. Bögge, F. Peters, M. Schmidtmann, P. Kögerler, M. J. Koop, Chem. Eur. J. 1998, 4, 1000-1006; A. Müller. W. Plass, E. Krickemeyer, R. Sessoli, D. Gatteschi, J. Meyer, H. Bögge, M. Kröckel, A. X. Trautwein, Inorg. Chem. Acta 1998, 271, 9-12).
19. A. Müller, E. Krickemeyer, H. Bögge, M. Schmidtmann, F. Peters, Angew. Chem. 1998, 110, 3567-3571; Angew. Chem. Int. Ed. 1998, 37, 3360-3363.
20. A. Müller, E. Krickemeyer, H. Bögge, M. Schmidtmann, F. Peters, Angew. Chem. 1998, 110, 3567-3571; Angew. Chem. Int. Ed. 1998, 37, 3360-3363.
21. See, for example: A. Müller, S. Q. N. Shah, H. Bögge, M. Schmidtmann, Nature 1999, 397, 48-50; A. Müller, F. Peters, M. T. Pope, D. Gatteschi, Chem. Rev. 1998, 98, 239-271; A. Müller, H. Reuter, S. Dillinger, Angew. Chem. 1995, 107, 2505-2539; Angew. Chem. Int. Ed. Engl. 1995, 34, 2328-2361, and references therein.
22. See, for example: A. Müller, S. Q. N. Shah, H. Bögge, M. Schmidtmann, Nature 1999, 397, 48-50; A. Müller, F. Peters, M. T. Pope, D. Gatteschi, Chem. Rev. 1998, 98, 239-271; A. Müller, H. Reuter, S. Dillinger, Angew. Chem. 1995, 107, 2505-2539; Angew. Chem. Int. Ed. Engl. 1995, 34, 2328-2361, and references therein.
23. See, for example: A. Müller, S. Q. N. Shah, H. Bögge, M. Schmidtmann, Nature 1999, 397, 48-50; A. Müller, F. Peters, M. T. Pope, D. Gatteschi, Chem. Rev. 1998, 98, 239-271; A. Müller, H. Reuter, S. Dillinger, Angew. Chem. 1995, 107, 2505-2539; Angew. Chem. Int. Ed. Engl. 1995, 34, 2328-2361, and references therein.
24. See, for example: A. Müller, S. Q. N. Shah, H. Bögge, M. Schmidtmann, Nature 1999, 397, 48-50; A. Müller, F. Peters, M. T. Pope, D. Gatteschi, Chem. Rev. 1998, 98, 239-271; A. Müller, H. Reuter, S. Dillinger, Angew. Chem. 1995, 107, 2505-2539; Angew. Chem. Int. Ed. Engl. 1995, 34, 2328-2361, and references therein.
25. [8]
26. 3COO ligands. Correspondingly, in the above formula all three types of ligands are denoted as L.
27. 8/9 fragments bound to the inside of the spherical shell could be localized. Crystallographic data (excluding structure factors) for the structure reported in this paper have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC-132027. Copies of the data can be obtained free of charge on application to CCDC. 12 Union Road, Cambridge CB21EZ, UK (fax: (+44) 1223-336-033; e-mail: deposit@ccdc.cam.ac.uk).
28. Some basic comments on the synthesis of giant clusters based on polyoxomolyhdates: in contrast to the synthesis of the Keplerate clusters, which can be readily obtained in high yield, the synthesis of the extremely soluble molybdenum blue compounds based on giant ring-type structures caused considerable problems in the past. However, these are now also readily accessible in the presence of a high electrolyte concentration which destroys the hydration shell that stabilizes the structures in solution (A. Müller. M. Koop, H. Bögge, M. Schmidtmann, C. Beugholt, Chem. Commun. 1998, 1501; A. Müller, S. K. Das, V. P. Fedin, E. Krickemeyer, C. Beugholt, H. Bögge, M. Schmidtmann, B. Hauptfleisch, Z. Anorg. Allg. Chem. 1999, 625, 1187-1192; A. Müller, S. K. Das, H. Bögge, C. Beugholt, M. Schmidtmann, Chem. Commun. 1999, 1035-1036; A. Müller, C. Beugholt, M. Koop, S. K. Das, M. Schmidtmann, H. Bögge, Z. Anorg. Allg. Chem. 1999, 625, in press).
29. Some basic comments on the synthesis of giant clusters based on polyoxomolyhdates: in contrast to the synthesis of the Keplerate clusters, which can be readily obtained in high yield, the synthesis of the extremely soluble molybdenum blue compounds based on giant ring-type structures caused considerable problems in the past. However, these are now also readily accessible in the presence of a high electrolyte concentration which destroys the hydration shell that stabilizes the structures in solution (A. Müller. M. Koop, H. Bögge, M. Schmidtmann, C. Beugholt, Chem. Commun. 1998, 1501; A. Müller, S. K. Das, V. P. Fedin, E. Krickemeyer, C. Beugholt, H. Bögge, M. Schmidtmann, B. Hauptfleisch, Z. Anorg. Allg. Chem. 1999, 625, 1187-1192; A. Müller, S. K. Das, H. Bögge, C. Beugholt, M. Schmidtmann, Chem. Commun. 1999, 1035-1036; A. Müller, C. Beugholt, M. Koop, S. K. Das, M. Schmidtmann, H. Bögge, Z. Anorg. Allg. Chem. 1999, 625, in press).
30. Some basic comments on the synthesis of giant clusters based on polyoxomolyhdates: in contrast to the synthesis of the Keplerate clusters, which can be readily obtained in high yield, the synthesis of the extremely soluble molybdenum blue compounds based on giant ring-type structures caused considerable problems in the past. However, these are now also readily accessible in the presence of a high electrolyte concentration which destroys the hydration shell that stabilizes the structures in solution (A. Müller. M. Koop, H. Bögge, M. Schmidtmann, C. Beugholt, Chem. Commun. 1998, 1501; A. Müller, S. K. Das, V. P. Fedin, E. Krickemeyer, C. Beugholt, H. Bögge, M. Schmidtmann, B. Hauptfleisch, Z. Anorg. Allg. Chem. 1999, 625, 1187-1192; A. Müller, S. K. Das, H. Bögge, C. Beugholt, M. Schmidtmann, Chem. Commun. 1999, 1035-1036; A. Müller, C. Beugholt, M. Koop, S. K. Das, M. Schmidtmann, H. Bögge, Z. Anorg. Allg. Chem. 1999, 625, in press).
31. Some basic comments on the synthesis of giant clusters based on polyoxomolyhdates: in contrast to the synthesis of the Keplerate clusters, which can be readily obtained in high yield, the synthesis of the extremely soluble molybdenum blue compounds based on giant ring-type structures caused considerable problems in the past. However, these are now also readily accessible in the presence of a high electrolyte concentration which destroys the hydration shell that stabilizes the structures in solution (A. Müller. M. Koop, H. Bögge, M. Schmidtmann, C. Beugholt, Chem. Commun. 1998, 1501; A. Müller, S. K. Das, V. P. Fedin, E. Krickemeyer, C. Beugholt, H. Bögge, M. Schmidtmann, B. Hauptfleisch, Z. Anorg. Allg. Chem. 1999, 625, 1187-1192; A. Müller, S. K. Das, H. Bögge, C. Beugholt, M. Schmidtmann, Chem. Commun. 1999, 1035-1036; A. Müller, C. Beugholt, M. Koop, S. K. Das, M. Schmidtmann, H. Bögge, Z. Anorg. Allg. Chem. 1999, 625, in press).
32. H. S. M. Coxeter in A Spectrum of Mathematics: Essays Presented to H. G. Forder (Ed.: J. Butcher). Oxford University Press, Oxford, 1967, pp. 98-107; I. Stewart, Spiel, Satz und Sieg für die Mathematik, Birkhäuser, Basel, 1990, pp. 95-114.
33. H. S. M. Coxeter in A Spectrum of Mathematics: Essays Presented to H. G. Forder (Ed.: J. Butcher). Oxford University Press, Oxford, 1967, pp. 98-107; I. Stewart, Spiel, Satz und Sieg für die Mathematik, Birkhäuser, Basel, 1990, pp. 95-114.
34. Within the context of the Pythagorean knowledge of geometry (from about 500 BC) and the knowledge given in the first book written on natural philosophy, the epochal Timaios, it is demonstrated very impressively that the elementary building blocks of nature, such as these polygons, have to "find" each other, metaphorically speaking, in Euclidean space for the joint construction of complex, and in particular, unusual and harmonic structures (see, for example, ref. [1k]).
35. Further information on Archimedean solids in chemistry can be found in a recent review: L. R. MacGillivray, J. L. Atwood, Angew. Chem. 1999, 111, 1080-1096; Angew. Chem. Int. Ed. 1999, 38, 1018-1033.
36. Further information on Archimedean solids in chemistry can be found in a recent review: L. R. MacGillivray, J. L. Atwood, Angew. Chem. 1999, 111, 1080-1096; Angew. Chem. Int. Ed. 1999, 38, 1018-1033.