Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!sdd.hp.com!decwrl!purdue!mentor.cc.purdue.edu!ahg From: ahg@mentor.cc.purdue.edu (Allen Braunsdorf) Newsgroups: comp.graphics Subject: Re: Why are there only 5 regular convex polyhedra? Summary: Look at a vertex and sum the angles. Message-ID: <13811@mentor.cc.purdue.edu> Date: 10 Sep 90 22:11:45 GMT References: <1990Sep9.095906.26612@rice.edu> <49433@brunix.UUCP> Reply-To: ahg@mentor.cc.purdue.edu (Allen Braunsdorf) Organization: Purdue UNIX Group Lines: 55 In article <49433@brunix.UUCP> dbc@cs.brown.edu (Brook Conner) writes: >Dwayne, > >The Platonic solid are the only regular solids because they are :) >Seriously, I'm sure this is a result of some result in topology somewhere >(although having only a passing acquaintance with topology stemming from >working under a topologist (John Hughes of Foley, van Dam, Feiner, and Hughes) >I have yet to see this proof for myself, so I can't offer pointers to it) I've had to prove it before and it goes like this: A regular polyhedron is one that has some number of (identical) regular polygons for faces. The 5 Platonic solids are the only regular convex polyhedra. Which regular polygons can we use to build a regular polyhedron? To answer this, we must notice that at each vertex of the polyhedron, some number of polygons greater than two share a common vertex. This combination of polygons can't be flat. In fact, the sum of the (identical) angles of the polygons at the common vertex must be less than 360 degrees. This gives us a table of candidates: Sides Angle Number we can join at one vertex ----- ----- ------ -- --- ---- -- --- ------ 3 60 3, 4, or 5 4 90 3 5 108 3 6 120 can't! (3 would be flat, more won't fit) A regular polygon can't have fewer than 3 sides. If you have anything with more sides than a hexagon, things only get worse, so they don't work either. So the only polygons we can use to build out polyhedron are the triangle, square, and pentagon. Combining those polygons in the ways described in the last table gives us: Polygon Number we join at one vertex Polyhedron ------- ------ -- ---- -- --- ------ ---------- Triangle 3 Tetrahedron Triangle 4 Octahedron Triangle 5 Icosahedron Square 3 Cube Pentagon 3 Dodecahedron A "real" proof would bog down in the details more, of course, but that's how you prove it. If you use F+V=E+2 you can show it to people with less hand-waving and figure-drawing. If you really want more, I can 'splain it better. --- Allen Braunsdorf Purdue University Computing Center ahg@cc.purdue.edu UNIX Systems Programmer