Day 21: Heart of Mathematics

• I graded #9 and 19; pretty good job, overall. There was a missing shape in #19, however....
• Someone asked about #5

The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia!
Tetrahedron	Hexahedron or Cube	Octahedron	Dodecahedron	Icosahedron
(Animation)	(Animation)	(Animation)	(Animation)	(Animation)

• Regular polyhedra start with regular polygons -- which ones play a role in the regular polyhedra?

• Application of the septagon

• Application of the dodecahedron

• We made them in 3-d: We used "geomags" to construct a set of Platonic solids (Problem I.1), and used them to fill in the table on page 275:

  	# of Vertices	Edges	Faces	faces at each vertex	sides at each face
  Tetrahedron	4	6	4	3	3
  Cube	8	12	6	3	4
  Octahedron	6	12	8	4	3
  Dodecahedron	20	30	12	3	5
  Icosahedron	12	30	20	5	3

  What conclusions can we draw from this data? Is there a pattern?

• What kinds of symmetry do these objects possess?

• The sphere is the "missing link" in the Platonic family: it is the ideal to which the Platonic solids aspire.

• Two dimensional representations of Platonic Solids
  • Put your eye to the face of an object and draw it.
  • Is it possible to draw each objects without any edges crossing?

• The concept of Duality
  • Who's dual with whom?
  • Who's dual alone?
  • Pictures of Duality

• "Is it possible to link three rings together in such a manner that they are indeed linked, yet if we remove any one of the rings the other two become unlinked?"
• Yes!

  Standard diagram of the Borromean rings

  A realization of the Borromean rings as ellipses

  Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration

  
  (Thanks Wikipedia!)

• icosahedra, golden rectangles, and interlocking rings! (see page 381)