Day 13 in Mat115

Today:

• Announcements:
  • Exam 1 returned, with discussion of the results and curve
  • Your score is out of 100.
  • Curve: add 10 to your score (some students will have more than 100! Well done....)
  • Some common errors
  • There's a key on-line, made up of your peers' work.
  • I forgot to put up the homework over platonic solids, and your reading assignment -- it's coming (by the end of today, or it's free!).

• More Platonic solids
  • We began our discussion with the convex regular polygons:

    "A regular polygon is a polygon which is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star."

  • We discovered, by looking at some regular convex hexagonal graph paper, that no platonic solid could use hexagons (because folding them up into a solid would force the hexagons to overlap). That left equilateral triangles, squares, and regular pentagons as candidates for the platonic solids.

  • A cube is the most familiar example of a Platonic solid. The special thing about a Platonic solid is that
    • Each face, each edge, and each vertex is exactly equivalent to every other face, edge, or vertex (respectively), and
    • All faces are congruent (identical) convex regular polygons.
    Turns out that there are five distinct platonic solids.

  • Johannes Kepler thought that the orbits of the planets (of which five were known at the time) were somehow related to the various Platonic solids, as shown in this figure.

    "The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids."

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

  • Last time we made a set of Platonic solids in 3-d, using two different methods to construct them:
    • "geomags" and
    • Platonic solids in 2-D (paper template)
    We can use a set of Platonic solids, and symmetry and counting arguments, to fill in the following table:

    	# of Vertices	Edges	Faces	faces at each vertex	sides at each face
    Tetrahedron
    Cube
    Octahedron
    Dodecahedron
    Icosahedron

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

  • The concept of duality:
    • Each platonic solid has a "mate", which we can identify from the table.
    • One of them, however, has a rather odd mate!

  • Links: Borromean Rings:
    • "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!