The Art, History and Mathematics of Polyhedra

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• The Art, History and Mathematics of Polyhedra
• Professor Alan F. Beardon
• University of Cambridge
• AIMS 23 September 2004

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Pyramids at GizaEgypt, 4500 years ago
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The Great Pyramid c. 2800 BCNational Geographic
April 1988
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Neolithic Platonic polyhedra2000 BC
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Giants Causeway
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Roman ornamentsdodecahedron and icosahedron
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Alhambra
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Escher 3 geometries
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Polyhedral ArtGold-plated lion ball Forbidden
City, Beijing 1736-1796
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Amethyst
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Zeolite
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Snow in England
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Snowflakes
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Snow
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Inscription over doors of Platos Academy 387
B.C.

• "Let no one destitute of geometry enter my doors."

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Platonic Polyhedra
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Platonic solids from Kepler's Mysterium
Cosmographicum, 1596
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Archimedes 287 212 BC

• Oftimes Archimedes servants got him against
  his will to the baths, to wash and anoint him,
  and yet being there, he would ever be drawing out
  of the geometrical figures, even in the very
  embers of the chimney. And while they were
  anointing of him with oils and sweet savours,
  with his fingers he drew lines upon his naked
  body, so far was he taken from himself, and
  brought into ecstasy or trance, with the delight
  he had in the study of geometry.
• Plutarch
• A Roman soldier, running upon him with a
  drawn sword offered to kill him and that
  Archimedes, looking back, earnestly besought him
  to hold his hand a little while, that he might
  not leave what he was then at work upon
  inconclusive and imperfect but the soldier,
  nothing moved by his entreaty, instantly killed
  him. Plutarch

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Archimedean Polyhedra
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Polyhedra with the same vertex structure but
different symmetries
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Leonardo da Vinci Experiments in perspective
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Experiments in perspective
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Polyhedral Art

• Lorenz Stoer
• Geometria et Perspectiva (1567)

• Title page of Hans Lencker's Perspectiva (1571)

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Trattato dAbacoby Piero della Francesca

• (806400 - ?597196800000)1/6
• ?108 - ?12 ?48
• 6?3 - 2?3 4?3

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Fra Giovanni's Intarsia Verona c.1520
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Kepler 1571 to 1630

• During his wedding celebration he noticed that
  the volumes of wine barrels were estimated by
  means of a rod slipped in diagonally through the
  bung hole, and he began to wonder how that could
  work

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Keplers model of the solar system Mysterium
Cosmographicum, 1596
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Euler 1707 to 1783

• Entered University of Basel at the age of 14
• Masters degree in philosophy at age of 17
• Medical lieutenant in the Russian Navy
• Professor of Physics, St Petersburg Academy
• Professor of Mathematics, St Petersburg Academy
• Director of the St Petersburg Academys Geography
  section
• Supervisor of Observatory and Botanical Gardens,
  Berlin
• Financial Adviser to the Government, Berlin
• Composer of music.
• Euler had 13 children and claimed that he
  made some of his greatest mathematical
  discoveries while holding a baby in his arms and
  with children playing around his feet.
• He became blind in 1766. St Petersburg
  Academy continued to publish Eulers work for
  nearly 50 years after his death.

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• eip -1

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Eulers Relation

• Number of faces F
• Number of edges E
• Number of vertices V
• F E V 2
• For a cube,
• F6, E12, V8
• 6 12 8 2

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Woodworking instructionsAbraham Sharp 1718
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Woodworking instructionsAbraham Sharp 1718
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Szilassi heptahedronEach face meets all the
other faces
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Cauchy 1789 - 1857

• Cauchy is mad and there is nothing that can be
  done about him, although, right now, he is the
  only one who knows how mathematics should be
  done. Abel. 1826
• Cauchy lived until the age of 68 and published
  789 papers.