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Platonic Solids and Rubik's Cubes* Jordan Vosman Melanie Stewart What is a Platonic Solid? A polyhedron that: 1. Is Convex 2. All of its faces are identical regular polygons 3. The same number of faces at each vertex Also, there are only five Platonic Solids Euler’s Formula for Platonic Solids • # Vertices - # Edges + # Faces = 2 • Example: Dodecahdron • 20 Vertices • 30 Edges • 12 Faces 20 – 30 + 12 = 2 Why are there only five Platonic Solids? If each face is a regular triangle then: • There cannot be more than five faces to a vertex, because if there are six or more, the sum of the angles at the vertex would be 360° or higher, resulting in a flat surface or hills and valleys. • This gives us the Tetrahedron (3), Octahedron (4), and Icosahedron (5) If each face is a square: • Four squares meeting at a vertex results in a flat surface, so only three squares meeting at a vertex will work • This gives us the Cube If each face is a regular pentagon: • Similar to the cube, as the maximum number of pentagons meeting at a vertex is three. • This gives us the Dodecahedron For Hexagons: • Only three hexagons can meet at a vertex, but this results in a flat surface. • Thus, there are no Platonic solids with regular n-gonal faces for n ≥ 6. Duality of Platonic Solids Cube: 6 faces and 8 vertices === Octahedron: 8 faces and 6 vertices Dodecahedron: 12 faces and 20 vertices === Icosahedron: 20 faces and 12 vertices Tetrahedron is a dual of itself The Cycle of Platonic Solids Tetrahedron Cube Octahedron Dodecahedron Icosahedron History • Pythagoras knew of the Tetrahedron, Cube, and Dodecahedron (~500 BC). • Later, Plato added the Octahedron and Icosahedron (~400 BC). • Plato speculated that the five shapes were the fundamental components of the physical universe. • "The Theory Of Everything" • Aristotle disproved this Examples of Platonic Solids in Nature Platonic Solids Rubik's Cubes* The Rubik's Cube • 3 x 3 x 3 • Invented in 1974 by a Hungarian architecture professor, Erno Rubik • The highest selling toy ever (>350,000,000) • The world record solve is 4.74 seconds • There are 43,252,003,274,489,856,000 different permutations for the Rubik's Cube. Other Cube Puzzles 2x2x2 or Pocket Cube 3,674,160 permutations Current world record solve is 0.49 seconds 4x4x4 or Rubik's Revenge 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 different permutations or 7.4 Septillion Current world record solve is 21.54 seconds Other Cube Puzzles 5x5x5 or Professor's Cube 282,870,942,277,741,856,536,180,333,107,150,328,293, 127,731,985,672,134,721,536,000,000,000,000,000 permutations or 282.87 Deodecillion World Record Solve is 41.27 seconds 6x6x6 or V-Cube 6 1.57 x 10^116 Permutations or 157 Septentrigintillion World Record solve is 1:32.77 minutes Shape Shifters • Don't hold their solved shape when scrambled • Possible if each piece is not the same size Tetrahedron - Pyraminx Was invented by Uwe Meffert in 1969 (Only) 933,120 permutations Current World Record solve is 1.32 seconds Other Tetrahedral Puzzles Octahedron – Face Turning Octahedron 31,408,133,379,194,880,000,000 possible permutations Dodecahedron Puzzles • Megaminx • 100 669 616 553 523 347 122 516 032 313 645 505 168 688 116 411 019 768 627 200 000 000 000 possible permutations • Or 100.7 unvigintillion • World record solve is 34.40 seconds Gigaminx – Teraminx - Petaminx Zettaminx Yottaminx Icosahedron - Icosaix Newest of the Platonic solid puzzles, first being manufactured in 2015 References • Akiyama, J. and Matsunaga, K., 2015. Treks into Intuitive Geometry: The World of Polygons and Polyhedra. pp. 143-152. • Baragar, A., 2001. A Survey of Classical and Modern Geometries. pp. 97-100. • Megaminx. https://en.wikipedia.org/wiki/Megaminx • Millenium Education. Platonic Solids. http://www.absoluteempowerment.com/platonic-solids.php • Octahedron. http://www.jaapsch.net/puzzles/octaface.htm • Pyraminx. http://rubiks.wikia.com/wiki/Pyraminx • Rubik’s. Cube Facts. https://www.rubiks.com/about/cube-facts/ • The Rubik Zone. Number of Combinations. http://www.therubikzone.com/Number-Of-Combinations.html • World Cubing Association. https://www.worldcubeassociation.org/results/regions.php Exam Question • A) Name the five Platonic solids • B) How many faces does each Platonic solid have? • C) What is the shape of each face for each Platonic solid?.
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