Polyhedral Solids (perhaps that's redundant, but it gets the point across.

Dedicated to Archimedes.


There are 3 classes of sorta-regular solids (polyhedrons) ...
 1.- all faces are the same regular polygon -- there are only 5 of these, see at ...
       http://mathforum.org/alejandre/applet.polyhedra.html
 2.- all faces are a regular polygon (but not the same one) there are a LOT of these -- like domes and soccer balls
 3.- all faces are the same polygon  (but not necessarily regular ones)
 See more at ...
   http://illuminations.nctm.org/ActivityDetail.aspx?ID=125
 There are some cut outs at ...
   http://isotropic.org/polyhedra/ (the most familiar 18 of Types 1 & 2)
   http://www.senteacher.org/wk/3dshape.php
    (but only 4 of the 5 possible type 1's.)
 There are some very good examples( 30 of types 1, 2, & 3)
    -- which
even can EVEN BE ROTATED in 3-D at ...
   http://jcrystal.com/steffenweber/
   http://jcrystal.com/steffenweber/POLYHEDRA/
   http://jcrystal.com/steffenweber/POLYHEDRA/p_00.html
 See also --
   http://mathforum.org/mathtools/cell/m6,8.12.1,ALL,ALL/
   http://www.fi.uu.nl/toepassingen/00297/toepassing_wisweb.en.html
   http://web.aanet.com.au/robertw/Stella.html
   http://web.aanet.com.au/robertw/PolyNav/PolyNavigator.html
   http://www.google.com/search?hl=en&q=polyhedrons+cutouts
   http://www.crystalinks.com/archimedes.html
   http://www.georgehart.com/pavilion.html
   http://www.georgehart.com/virtual-polyhedra/zometool.html
   http://www.georgehart.com/virtual-polyhedra/synestructics.html
   http://www.georgehart.com/virtual-polyhedra/naming.html
   http://en.wikipedia.org/wiki/Polyhedron

==============================================

Having nearly always been fascinated by regular geometric figures,  I set out to
 classify them somehow and comment on them, but discovered that it had already been done.

So, what follows is a list WebSites which more than takes care of the subject and is
   far above my poor power to add or detract.          (apologies to Abe Lincoln)
http://en.wikipedia.org/wiki/Platonic_solid                     5
http://en.wikipedia.org/wiki/Archimedean_solid          13
http://en.wikipedia.org/wiki/Johnson_solid                      92
http://en.wikipedia.org/wiki/Catalan_solid
http://en.wikipedia.org/wiki/Prism_%28geometry%29       inf
http://en.wikipedia.org/wiki/Antiprism                  inf

http://en.wikipedia.org/wiki/Quasiregular_polyhedron
http://en.wikipedia.org/wiki/Snub_polyhedron
http://en.wikipedia.org/wiki/Truncated_polyhedron             w/progression
http://en.wikipedia.org/wiki/Polyhedral_compound
---
http://en.wikipedia.org/wiki/Kepler-Poinsot_polyhedron   ++
http://en.wikipedia.org/wiki/Regular_polytope
http://en.wikipedia.org/wiki/List_of_regular_polytopes
http://en.wikipedia.org/wiki/Metatron%27s_Cube
http://en.wikipedia.org/wiki/Goldberg_polyhedron
http://en.wikipedia.org/wiki/Conway_polyhedron_notation
http://en.wikipedia.org/wiki/Kepler-Poinsot_polyhedron
http://en.wikipedia.org/wiki/Toroidal_polyhedron
http://en.wikipedia.org/wiki/Zonohedron
http://en.wikipedia.org/wiki/Tesseract


http://mathworld.wolfram.com/letters/       <== alphabetical listing
http://mathworld.wolfram.com/Tetrahedron.html           no images??
http://mathworld.wolfram.com/Octahedron.html
http://mathworld.wolfram.com/Cube.html
http://mathworld.wolfram.com/Dodecahedron.html
http://mathworld.wolfram.com/Icosahedron.html

http://mathworld.wolfram.com/TetrahedralGraph.html
http://mathworld.wolfram.com/CubicGraph.html
http://mathworld.wolfram.com/OctahedralGraph.html http://mathworld.wolfram.com/DodecahedralGraph.html
http://mathworld.wolfram.com/IcosahedralGraph.html


http://mathworld.wolfram.com/Cube5-Compound.html


http://en.wikipedia.org/wiki/Project_Euler


===========================

My interest was recently re-kindled when a classmate sent me an E-Mail with the subject of
      “The Beauty of Mathematics

    which linked to ...
   http://www.youtube.com/watch_popup?v=h60r2HPsiuM
   http://www.youtube.com/watch_popup?v=h60r2HPsiuM&feature=youtube_gdata_player
      which starts w/ 12345x8+5=98765  etc etc and ends with an inspirational message,
      after which it then offers a LOT of miscellaneous random choices one of which leads to
      a linked set of videos under the title of 
The Beauty of Mathematics - You Tube.
    Following all of the links is very interesting, but the address-line/URL never changes,
       so I had NO way to get back to some of the heart of the material.
By trail and error I finally got to ...
           http://www.youtube.com/watch?v=ZOqg5bPZ0HE 
    which is a lecturer with neat wall hangings.
    After which there is another set, one of which is
          http://www.youtube.com/watch?v=f81ZEVO6Btw 

Then, too there are still more interesting WebSites to explore, as at ...
         https://www.google.com/#q=%22beauty+of+mathematics%22
         http://videos.sacredgeometryweb.com/category/shape/platonic-solids/
         http://videos.sacredgeometryweb.com/snub-cuboctahedron/
         https://www.google.com/#q=flower+of+life+masonic+secrets+sacred+geometery

    http://peda.com/poly
    http://mathsci.kaist.ac.kr/~drake/tes.html
    http://michael-hogg.co.uk

------------
http://www.world-mysteries.com/sar_sage1.htm
http://en.wikipedia.org/wiki/Fibonacci_number
http://oeis.org/A001622
http://video.mit.edu/watch/metamorphosis-of-the-cube-12077/
http://mit.tv/NN5mLe

https://www.google.com/#psj=1&q=vertices+edges+faces
http://www.ezschool.com/Math/Shapes/ws12.html

Other terms and concepts ---
# of  vertices, edges, faces   FVE   EFV FEV VEF
dual  duality
prism  anti-prism
prism or antiprism.
truncated   snub
star stellated
================================

N.B.::
1. The majority of these solids can be morphed into related but different geometries by
...
                  [ see at   http://en.wikipedia.org/wiki/Truncated_polyhedron ]
    a1 .-  filing off the vertices to increase the number of faces by the number of vertices; 
               then the result is said to be a TRUNCATED solid.
    a2 .-  when the new lengthening/emerging edges equal the shortening/disappearing edges;
               then the result is said to be a UNIFORM TRUNCATED solid.
    a3 .-  continued filing will eventually result in the new faces meeting each other
               as some of the shortened edges reach a length of zero;
               then the result is said to be a RECTIFIED TRUNCATED solid.
  
    b1 .-  building pyramids on the faces to increase the number of faces
               by the twice the number of edges,
while original faces disappear
    b2 .-  continued heightening the pyramids will increase the length of the new edges;
                for n<6 sides/face of the original, the edge length will get to be equal to the orig edges.
                for n>6 sides/face of the original, the edge length will always be greater than the orig edges.
    b3 .-  continued heightening the pyramids will eventually result in the new faces
                becoming co-planar, which will cut the total number of faces in half
                and make all of the orig edges disappear.
    b4 .-  in some cases b2 & b3 may occur at the same height.
    b5 .-  continuing beyond b3 will  violate the requirement of being convex.

2. There are 43380 distinct nets for the icosahedron, the same number as for the dodecahedron .
          [  as see at  https://www.google.com/#q=43380+polyhedron  ]