William C. Corwin.info www.ConcurrentInverse.com www.byeless.com 2005 May 27 Title: possibilities: byeless, platonic angles, heptagonal angles, 3 utilities, concurrent inverse, quartic-quintic Alexander Bogomolny alexb@cut-the-knot.com http://www.cut-the-knot.com/cgi-bin/dcforum/ctk.cgi /do_you_know/sgroups.shtml http://www.cut-the-knot.org/htdocs/dcforum/DCForumID3/606.shtml cc: ian.gatland@physics.gatech.edu computational physics http://www.physics.gatech.edu/academics/tutorial/mathematics.htm McNamara@physics.syr.edu aam@ohysics.syr.edu easchiff@syr.edu http://physics.syr.edu/courses/java-suite/crosspro.html george@georgehart.com www.georgehart.com/virtual-polyhedra/platonic-info.html dualspace@hotmail.com members.tripod.com/~Paul_Kirby/appletcross/CrossProduct.html dsantos@ccp.edu faculty.ccp.cc.pa.edu/dept/math/_plotter.html hwalters@contracosta.cc.ca.us contracosta.cc.ca.us/math/mathlinks.htm admin@goldenmuseum.com?subject=goldenmuseum_design goldenmuseum.com feedback@vb-helper.com www.vb-helper.com/tutorial_platonic_solids.html derivation enquiry@ronknott.com http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#trigmore ref: St.Andrews Scotland home.aanet.com.au/robertw/Stella.html This letter is on line at http://www.issi1.com/corwin/possible.txt It is good to establish what is possible and what is not possible so that one will not spend a lot of time in futile efforts. It is widely accepted that to have a robin or tournament schedule with no byes or slots of time that a team has to sit out, that the number of teams must be a power of 2. It is not the case, although it is difficult. You may be interested in byeless.com with regards to your discussion on what is possible and what is not possible. I developed some systematic methods to arrive at solutions which I do not wish to disclose as it would jeopardize my copyright rights. It is very difficult to somewhat randomly rearrange things to guess the answer; I have done it for 18 even; then some patterns may have become apparent. It would be impossible with completely random guesses since (18-1)!*(18-2)!*... is very large. Obviously many people are convinced that it is an impossibility. It would be interesting to determine if there are solutions that have different properties such as the amount of disorder and the number of solutions and to see if there are any isomorphisms with statistical things in nature. My results are at http://www.issi1.com/corwin/byeless.html . The three houses and three utilities problem is equivalent to the four color map problem, which I think has been proven lately; for if each house is one color and each utility is a different color, and the background is a fifth color, and each house is surrounded by the combination of the three utilities, five colors would be required. You may be interested in http://www.issi1.com/corwin/platonic.txt with regards to your platonic solids examples. I have shown that the golden ratio applies to the dihedral angles (cosine of the supplement). I derived the unit vectors to the vertices using a variation method; I took three vectors, the length of one pentagon side, lying in the plane and raised them equally until the ends were the distance of a chord apart. Alternatively each of five equal vectors from the origin equally spaced could be raised equally from a plane until the distance between adjacent ends equals the length of the vector. I only remember a very awkward long equation in an old CRC handbook as any other reference. My unit vectors are at http://www.issi1.com/corwin/unit_v.txt and discussed at http://www.issi1.com/corwin/platonic.txt (heptagonal mentioned) and some relevant ray tracing is at http://www.issi1.com/corwin/icosahedron.jpg . Further insight into what is and is not possible and other things may be gained by studying of the calculus of variations and the method of steepest descent which may be in Morse & Feshbach. I don't have my books here so I cannot reference the proper works properly. Parenthetically, Also, I think that dimensional analysis, PercyWilliamBridgeman isbn 0404147747, needs more attention. Also, http://www.georgehart.com/research/multanal.html . I realized that the concurrent inverse feature was possible after viewing a Russian units conversion page, http://www.ur.ru/~sg/transl transl@sg.ur.ru . I improved on the floating point display and made some self checking calculators at ConcurrentInverse.com . Also, my triangle calculator allows any of the angles or sides to be adjusted on the fly without considering which process applies. Also, the quartic calculator may be relevant to your quintic impossibility discussion. You may reference my things as long as you give credit and don't copy them without giving me a cut of profits.