PLATONIC SOLIDS By: Eric Aucoin & Kelly Nelson
The man behind the math • Plato was a Greek born around 428 B.C. in the city of Athens. It is believed he was born on this year based on research done by scholars tracing the events of his life. His birth name was Aristocles and gained the nickname Platon which means broad, because he had a broad build. • He was a mathematician and a student and teacher of the Greek philosophers, Socrates and Aristotle respectively.
He founded the Academy in Athens at around 385 B.C which was essentially the very first university in the Western World. Its curriculum included a variety of subjects including astronomy, mathematics, biology and philosophy.
From 387 to 361 B.C. it is believed that he wrote many dialogues such as Meno, Euthydemus, Menexenus, Cratylus, Repuglic, Phaedrus, Syposium and Phaedo. His most influential work at that time being the Republic which discusses the virtues of courage, justice, wisdom and moderation within the individual and within society.
In 367 B.C he was invited to be the personal tutor of Dionysus II who was the new ruler of the city of Syracuse. He accepted the invitation but left for Athens in 365 B.C. when Syracuse went to war. Around this time, Aristotle began to study at his Academy. In 361 B.C, he received a letter asking him to return to Syracuse but left immediately as the situation was worse than the last time he had been there.
Back at the Academy, he spent the rest of his life writing and philosophising and wrote more dialogues such as Parmenides, Theatetus,Sophist, Statesmas, Timaeus, Critias, Philebus, and Laws. It is in the Timaeus dialogue that he makes mention of the Platonic Solids. He identified these solids as being the elements that make up the universe.
In that era, people believed that the universe was made up of five different elements: Earth, Air, Fire, Water and quintessence, the element of the heavens. Plato identified Fire to be the tetrahedron, Earth to be the cube, Air to be the octahedron, Water to be the icosahedron and quintessence to be dodecahedron. He died in Athens in approximately 348 B.C.
What are PLATONIC SOLIDS? Platonic solids are 3-dimensional solids with equivalent faces of regular polygons Five different platonic solids: • Tetrahedron (triangles), • Hexahedron (squares), • Octahedron (triangles), • Dodecahedron (pentagons), • Icosahedron (triangles)
The following are true about Platonic Solids: • The vertices of P all lie on a sphere • All dihedral angles (angles between two planes) are equal • All faces and vertex figures are regular polygons • All solid angles are equivalent (if sphere drawn around object, angles from sphere to solid would all be the same) • All vertices are surrounded by the same number of faces
The following variables will be used: F – faces V – vertices E – edges p – the number of edges of each face q – the number of faces meeting at each vertex
How to find Vertices, Faces & Edges? V = 4p . 4 - (p-2)(q-2) E = 2pq . 4 - (p-2)(q-2) F = 4q . 4 - (p-2)(q-2)
General relationships between the features of each 3D shape… pF = 2E = qV V – E + F = 2 Let’s do an example of a tetrahedron..
Symmetries of 3D Digraphs • Tetrahedron: • Transpositions, reflections • Order of the group = 4! = 24 • Cube and octahedron are dual to each other • Order of group = 48 • Subgroup of direct symmetries and these solids are isomorphic to S4 • Dodecahedron and icosahedron are dual to each other • Order of group = 120 • Isomorphic to the alternating group A5
Practical Applications • Dice & puzzles • Kepler’s platonic solids (model of the solar system) • Molecule structures
Dice and Puzzles • The platonic solids are commonly used to make dice of these shapes are fair. The 6-sided die is most commonly used in board games as Monopoly, but the other dice are used in role-playing games such as Dungeons and Dragons. • Rubik’s cube. There is also the Megaminx and the Gigaminx which each have 12 sides, but 3 and 5 layers each respectively.
Dices and Puzzles continued • -The world record for a single solve of a Megaminx is currently 42.28 seconds. • -To put into perspective how difficult the Megaminx is to solve. A Rubik’s Cube with 3 layers has 43 quintillion permutations, which is in the order of 1018. A Megaminx has approximately 101 unvingtillion permutations, which is in the order of 1066.
Kepler’s Platonic Solids • German Astronomer Johannes Kepler • Attempted to relate the five extraterrestrial planets (at the time) to the five platonic solids • Based on the distance between the planets • Mercury, Venus, Earth, Mars, Jupiter • Saturn represented the sphere enclosing the solids • The order was octahedron, icosahedron, dodecahedron, tetrahedron and hexahedron • This idea had to be abandoned – but it led to the discovery of the three laws of orbital dynamics
Exam Question • Show that the symmetry group of the cube is isomorphic to S4. Hint: Label the vertices of the cube as numbers 1 to 8. • ** Solution will be posted online.
References http://sinearch.com/theory-science/keplers-platonic-solid-model-of-the-solar-system/ http://www.math.utah.edu/~pa/math/polyhedra/polyhedra.html http://www.math.utah.edu/~pa/math/polyhedra/polyhedra.html http://milan.milanovic.org/math/english/solids/solids.html http://www.egs.edu/library/plato/biography/ http://www.biography.com/people/plato-9442588 http://www.geom.uiuc.edu/~sudzi/polyhedra/platonic.html http://abys http://abyss.uoregon.edu/~js/ast121/lectures/lec02.html http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L10.html www.math.wisc.edu/~shamgar/RT-P1-Solutions.pdf http://milan.milanovic.org/math/english/solids/solids.html http://meandering-through-mathematics.blogspot.ca/2011/11/why-are-there-exactly-five-platonic.html http://arxiv.org/pdf/1204.1875v1.pdf http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L10.html http://sierra.nmsu.edu/morandi/notes/platonicsolids.pdf http://www.worldcubeassociation.org/results/regions.php?regionId=&eventId=minx&years=&mixed=Mixed http://www.jaapsch.net/puzzles/megaminx.htm http://www.boxytech.com/mf8-gigaminx-puzzle-cube http://www.examiner.com/article/the-ancient-history-of-the-twenty-sided-die-role-playing-games