Platonic SolidsDate: 01/01/97 at 15:53:30 From: A. Kale Rodabough Subject: "Regular" 3-D shapes A friend gave me a gift this holiday season. It is two cubes with numbers on them which are used to count down the days from the first of December to Christmas (i.e., 25 to 00). I thought it would be neat to have three cubes with numbers on them to count down all the days of the year until Christmas. I realized this was impossible. But, if a seven-sided "cube" exists, it IS possible. The numbers on the first "cube" are: 1,2,3,4,6-(doubles as a 9),8, and 0. The numbers on the second "cube" are: 1,2,3,5,6(9),7, and 0. The numbers on the third "cube" are: 1,2,3,4,5,7, and 8. The difficulty comes in getting the numbers 333, 222, and 111. Also, every two number combination (i.e., 11,22,33,44,55,66(99),77,88,00) is difficult to get. I also need to make sure the digits for each number are available for every day of the year (i.e., placement of digits is important). My only problem is how to "cut out" a regular seven-sided "cube". I got to thinking about the regular polyhedra that I know (tetrahedron, cube, dodecahedron, and the 20-sided die in the game "Scattergories"). How does someone make a "regular" 5-hedron, or 7-hedron, or for that matter N-hedron? I can see that each polyhedron has a "center" where n-rays extend out from. The angles between each consecutive ray are equal. If you "cut off" each ray so that each line-segment has equal length, and "place" infinite planes on each "end-point", the lines where the planes intersect define the corners/edges of the n-hedron. However, I can only visualize the previously mentioned polyhedra. Can you send me a picture, or description, of a regular 7-hedron? Any other info you could send me would be of great interest! Kale Rodabough Date: 01/01/97 at 18:42:13 From: Doctor Pete Subject: Re: Hi, I am very sorry to tell you that there are only 5 "regular" solids. By "regular," I mean having the following properties: 1) They are is convex; i.e., they have no dimples. 2) The faces are all regular n-gons, each of the same type. 3) Each vertex has the same arrangement of faces around it. With this in mind, the names of these solids (called Platonic solids) are: faces edges vertices face type --------------------------------------------------------- tetrahedron 4 6 4 triangle (3) hexahedron(cube) 6 12 8 square (4) octahedron 8 12 6 triangle (3) dodecahedron 12 30 20 pentagon (5) icosahedron 20 30 12 triangle (3) If we eliminate requirement (3), then we obtain many more, in particular the convex deltahedra. The tetrahedron, octahedron, and icosahedron, having triangular faces, are a subset of the convex deltahedra. These have 4, 6, 8, 10, 12, 14, 16, and 20 sides. In any case, a 7-hedron cannot be "regular." For more information, see: http://www.ugcs.caltech.edu/~peterw/studies/platonic/ -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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