King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts
London: Profile Books, 2006, 2007
ISBN 978-1-84668-007-6
(4) Coxeter’s definition of his discipline, often recited, was this: “Geometry is the study of figures and figures. Figues as in shapes” - triangles, cubes, dodecahedrons - “and figures as in numbers.”
(8) The spearmint molecule and caraway molecule are chiral twins - one molecule is the mirror reflection of the other, and with that minor difference the molecules have considerably different effects on our taste buds.
(10) Analogous figures exist in higher dimensions - the fourth dimension, for example, contains the simplex (the 4-D analog to the tetrahedron), and the hypercube (the 4-D analog to the cube). And in higher dimensions still, polytopes morph into more and more complex cousins of the originals, some continuing to infinity.
(20) Later that evening, relaxing in the hotel lobby, Coxeter met with another fan, Texan Glenn Smith, a self-described “geometry groupie,” who makes a successful living in the sesame business.
(25) Euclid (365-25 BC) proved there are only five Platonic solids. And given the above-mentioned restrictions, only three regular polygons (the equilateral triangle, square, and pentagon) can be used in the construction of the Platonic solids. This is because the sum of polygon angles that meet at a vertex must be less than 360º in order to form a convex solid.
(65) [HT] Flather’s models: The series included more than fifty stellations of the icosahedron. Littlewood accepted Flather’s models as a gift to Trinity and Coxeter agreed to write an accompanying enumeration and description, which became The 59 Icosahedron.
(69) In four dimensions the six regular polytopes include: the simplex or 5-cell, each cell being a tetrahedron, and three tetrahedron meeting any an edge; the 8-cell, or tesseract, made of eight cubes, three cubes meeting at every edge; the 16-cell made of sixteen tetrahedra; the 24-cell made of octahedra; the 120-cell made of dodecahedra; and the 600-cell made of tetrahedra.
Schäfli proved that in higher dimensions regular polytopes become a rarer breed. Only three regular polytopes exist in five or more dimensions, continuing to infinite dimensions: these are the simplex (the generalized tetrahedron), the hypercube or “measure polytope” (the generalized cube), and the orthoplex or cross polytope (the generalized octahedron).
(74) In the 1880s, [Alice] Boole Stott rediscovered the six polytopes in four dimensions and then, using a ruler and compass, cardboard and paint, she produced complete model sets of their central sections.
(92) Icosahedra and dodecahedra do not exist in dimensions higher than four, which suited Coxeter fine. “Four is my favorite dimension,” he once said. “The things that happen in four dimensions are extra special and agreeable.”
(99) For every symmetry in the laws of physics, there must exist a conservation law (if there is symmetry, something is conserved.)
(112) [JL] Synge [nephew of JM Synge] also wrote a fantastical mathematical novel, Kandelman’s Krim. Coxeter loved it and plundered its pages, excerpting twelve passages in his book _Introduction to Geometry_…
(161) “In Italy today, Emma Castelnuovo has popularized and developed a [new approach to Euclidean geometry], he said. “Her book, La Geomatria Intuitiva, describes the teaching of geometry with apparatus resembling Meccano. The book, beautifully illustrated, shows how geometrical shapes are used in the architecture of Italy.”
(163) After his Pittsburgh talk [1967], he traveled to Minneapolis where he was coming to the end of a long-running pet project, working for four years with a group of mathematicians on educational geometry films, "Dihedral Kaleidoscopes" and "Symmetries of the Cube" (two in a series of five films).
(178) But in Coxeter’s eyes, one of Fuller’s downfalls was his use of preexisting material without acknowledgement.
(191) By contrast, Coxeter and Greitzer’s book Geometry Revisited, which has 153 pages of text, has roughly 160 separate diagrams - an average of over one per page!…
(195) He [Claude Shannon] posited that the design of such a communication system was analogous to the sphere-packing problem of the geometer - sphere packing was a strategy for efficiently storing and encoding data to eliminate errors.
(202) Geometer’s Sketchpad, now in its fourth edition, has met with enthusastic response.
(203) He [Walter Whiteley] tries to detemine whether a protein’s regoins will be rigid or flexible, because this is the property that dictates how a protein interacts. Working in the York Math Lab, Whiteley and his students devise computer algorithms that shorten the biochemist’s search, tinkering with the geometric models, adjusting their struts and nodes, trying to discover how many rigid and flexible vertices each sample protein structure might have.
(212) He really felt that mathematics was part of the humanities as well as science.
(216) Crystals, in fact, are classified by seventeen planar symmetry groups (planar meaning 2-D; in 3-D there are 230 crystallographic space groups), the collection of all motions - translations, rotations, reflections, glide-reflections, screw motions, and rotary reflections - that, when they act on the crystal structure, leave the structure invariant.
(222) If mathematics is “the queen of the sciences” what is the king? To which Coxeter responded: “Maybe the King of the Sciences is Ecology."
(223) [English sculptor John] Robinson had sent Coxeter a book of his most recent sculptures - Symbolic Sculpture, The Universe Series. Coxeter appreciated what he saw: exquisite executions in bronze, wood, and wool tapestry, of many geometrical concepts; the golden rule, the Archimedean spirals, golden spirals, cones, knots, pyramids, triangles, ovoids, Möbius bands, circles, and tangents.
(230) … John Conway; Marc Pelletier, a geometric model-maker from Boulder, CO; and geometry lover Glenn Smith from Texas….
(233) Jeff Weeks, a freelance geometer from Canton, New York and the recipient of a 1999 MacArthur fellowship. Weeks is also the author of The Shape of Space, a book exploring the possible shapes of the universe.
(246) …he [Gyorgy Darvas] edits Symmetry: Culture and Science, published by the Symmetry Society…
(251) He [Coxeter] reconsidered his offer [to leave his house to the University of Toronto], however, when he perceived a change in the university’s pedagogical approach - shifting from “learning for its own sake” to “learning for opportunity.”
……There [Stockholm] he spoke on another of his signature subjects, the “Rhombic Triacontehedron,” and he planned to use a new type of model invented by his friend, geometer and geophysicist Michael Longuet-Higgins, at the Scripps Institution of Oceanography, UCSD. Called RHOMBO, the model’s component parts are six-faced solid blocks that click together by a patented system of magnets.
(265) Morley’s Trisector Theorem; The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
(267) NB: 7 triangles around central equilateral triangle. Any triangle will always have an equilateral triangle at the center of the trisection of its vertices.
London: Profile Books, 2006, 2007
ISBN 978-1-84668-007-6
(4) Coxeter’s definition of his discipline, often recited, was this: “Geometry is the study of figures and figures. Figues as in shapes” - triangles, cubes, dodecahedrons - “and figures as in numbers.”
(8) The spearmint molecule and caraway molecule are chiral twins - one molecule is the mirror reflection of the other, and with that minor difference the molecules have considerably different effects on our taste buds.
(10) Analogous figures exist in higher dimensions - the fourth dimension, for example, contains the simplex (the 4-D analog to the tetrahedron), and the hypercube (the 4-D analog to the cube). And in higher dimensions still, polytopes morph into more and more complex cousins of the originals, some continuing to infinity.
(20) Later that evening, relaxing in the hotel lobby, Coxeter met with another fan, Texan Glenn Smith, a self-described “geometry groupie,” who makes a successful living in the sesame business.
(25) Euclid (365-25 BC) proved there are only five Platonic solids. And given the above-mentioned restrictions, only three regular polygons (the equilateral triangle, square, and pentagon) can be used in the construction of the Platonic solids. This is because the sum of polygon angles that meet at a vertex must be less than 360º in order to form a convex solid.
(65) [HT] Flather’s models: The series included more than fifty stellations of the icosahedron. Littlewood accepted Flather’s models as a gift to Trinity and Coxeter agreed to write an accompanying enumeration and description, which became The 59 Icosahedron.
(69) In four dimensions the six regular polytopes include: the simplex or 5-cell, each cell being a tetrahedron, and three tetrahedron meeting any an edge; the 8-cell, or tesseract, made of eight cubes, three cubes meeting at every edge; the 16-cell made of sixteen tetrahedra; the 24-cell made of octahedra; the 120-cell made of dodecahedra; and the 600-cell made of tetrahedra.
Schäfli proved that in higher dimensions regular polytopes become a rarer breed. Only three regular polytopes exist in five or more dimensions, continuing to infinite dimensions: these are the simplex (the generalized tetrahedron), the hypercube or “measure polytope” (the generalized cube), and the orthoplex or cross polytope (the generalized octahedron).
(74) In the 1880s, [Alice] Boole Stott rediscovered the six polytopes in four dimensions and then, using a ruler and compass, cardboard and paint, she produced complete model sets of their central sections.
(92) Icosahedra and dodecahedra do not exist in dimensions higher than four, which suited Coxeter fine. “Four is my favorite dimension,” he once said. “The things that happen in four dimensions are extra special and agreeable.”
(99) For every symmetry in the laws of physics, there must exist a conservation law (if there is symmetry, something is conserved.)
(112) [JL] Synge [nephew of JM Synge] also wrote a fantastical mathematical novel, Kandelman’s Krim. Coxeter loved it and plundered its pages, excerpting twelve passages in his book _Introduction to Geometry_…
(161) “In Italy today, Emma Castelnuovo has popularized and developed a [new approach to Euclidean geometry], he said. “Her book, La Geomatria Intuitiva, describes the teaching of geometry with apparatus resembling Meccano. The book, beautifully illustrated, shows how geometrical shapes are used in the architecture of Italy.”
(163) After his Pittsburgh talk [1967], he traveled to Minneapolis where he was coming to the end of a long-running pet project, working for four years with a group of mathematicians on educational geometry films, "Dihedral Kaleidoscopes" and "Symmetries of the Cube" (two in a series of five films).
(178) But in Coxeter’s eyes, one of Fuller’s downfalls was his use of preexisting material without acknowledgement.
(191) By contrast, Coxeter and Greitzer’s book Geometry Revisited, which has 153 pages of text, has roughly 160 separate diagrams - an average of over one per page!…
(195) He [Claude Shannon] posited that the design of such a communication system was analogous to the sphere-packing problem of the geometer - sphere packing was a strategy for efficiently storing and encoding data to eliminate errors.
(202) Geometer’s Sketchpad, now in its fourth edition, has met with enthusastic response.
(203) He [Walter Whiteley] tries to detemine whether a protein’s regoins will be rigid or flexible, because this is the property that dictates how a protein interacts. Working in the York Math Lab, Whiteley and his students devise computer algorithms that shorten the biochemist’s search, tinkering with the geometric models, adjusting their struts and nodes, trying to discover how many rigid and flexible vertices each sample protein structure might have.
(212) He really felt that mathematics was part of the humanities as well as science.
(216) Crystals, in fact, are classified by seventeen planar symmetry groups (planar meaning 2-D; in 3-D there are 230 crystallographic space groups), the collection of all motions - translations, rotations, reflections, glide-reflections, screw motions, and rotary reflections - that, when they act on the crystal structure, leave the structure invariant.
(222) If mathematics is “the queen of the sciences” what is the king? To which Coxeter responded: “Maybe the King of the Sciences is Ecology."
(223) [English sculptor John] Robinson had sent Coxeter a book of his most recent sculptures - Symbolic Sculpture, The Universe Series. Coxeter appreciated what he saw: exquisite executions in bronze, wood, and wool tapestry, of many geometrical concepts; the golden rule, the Archimedean spirals, golden spirals, cones, knots, pyramids, triangles, ovoids, Möbius bands, circles, and tangents.
(230) … John Conway; Marc Pelletier, a geometric model-maker from Boulder, CO; and geometry lover Glenn Smith from Texas….
(233) Jeff Weeks, a freelance geometer from Canton, New York and the recipient of a 1999 MacArthur fellowship. Weeks is also the author of The Shape of Space, a book exploring the possible shapes of the universe.
(246) …he [Gyorgy Darvas] edits Symmetry: Culture and Science, published by the Symmetry Society…
(251) He [Coxeter] reconsidered his offer [to leave his house to the University of Toronto], however, when he perceived a change in the university’s pedagogical approach - shifting from “learning for its own sake” to “learning for opportunity.”
……There [Stockholm] he spoke on another of his signature subjects, the “Rhombic Triacontehedron,” and he planned to use a new type of model invented by his friend, geometer and geophysicist Michael Longuet-Higgins, at the Scripps Institution of Oceanography, UCSD. Called RHOMBO, the model’s component parts are six-faced solid blocks that click together by a patented system of magnets.
(265) Morley’s Trisector Theorem; The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
(267) NB: 7 triangles around central equilateral triangle. Any triangle will always have an equilateral triangle at the center of the trisection of its vertices.
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