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Wish upon a Stellated Dodecahedron

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Presentation on theme: "Wish upon a Stellated Dodecahedron"— Presentation transcript:

1 Wish upon a Stellated Dodecahedron
Ethan Bolker Family Friday National Museum of Mathematics July 12, 2019

2 quintessence air fire earth water Plato

3 Paolo Ucello? Venice Basilica of St. Mark ~1430

4 Johannes Kepler Harmonice Mundi 1619

5 Kepler’s Echinus

6 Echinus = hedgehog http://bestiary.ca/beasts/beastgallery217.htm#
Bestiarius - Bestiary of Anne Walsh Kongelige Bibliotek (National Library of Denmark)

7 Edible sea urchin Echinus esculentus 20cm

8 Mathematical Snapshots
Hugo Steinhaus Mathematical Snapshots Stechert 1937 Oxford 1950 Dover 1999

9 Build a stellated dodecahedron?

10 Cundy and Rollett, with construction tips
Net Cundy and Rollett, with construction tips

11

12 {Picture of the new pentagrams}
Use 12 of these {Picture of the new pentagrams}

13 Voila!

14

15 great stellated dodecahedron
Kepler’s Ostrea (Jamnitzer 1568) great stellated dodecahedron

16 Why oyster? Search google (books) for Ostrea Kepler, find
Joannis Kepleri astronomi opera omnia 

17 Twist a pyramid

18 Twist a star

19 Rubik’s stellated dodecahedron
Scramble, try to unscramble Harder than Rubik’s cube? What configurations are possible? Swap stars and pyramids?

20 Gadget vs. cube 6 colors 60 facets 6 axes 24 degree 5 moves
Swap any pair of facets Pyramids (stars) commute, usually with each other too 6 colors 54 facets 3 axes 18 degree 4 moves Swap any pair of centers, edges, corners Only rotations about the same axis commute

21 Gadget vs. cube 6 colors 60 facets 6 axes 24 degree 5 moves
Swap any pair of facets Pyramids (stars) commute, usually with each other too 6 colors 54 facets 3 axes 18 degree 4 moves Swap any pair of centers, edges, corners Only rotations about the same axis commute

22 The gadget group Acts as a group of permutations of the 60 facets
Kernel has order 120 ×10!6 = 2.7 × 1041 Configuration space 60!/|kernel| ~ 3 × 1040 Study action: orbits, macros David Joyner: Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys


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