www.ConcurrentInverse.com (c) copyright 2005 Wm.C.Corwin APPENDIX A [x] denotes x**0.5 trig half angle cosine cos(A/2) = [1/2 + (1/2)cos(A)] cos(2pi/5) = ([5] - 1)/4 sin(2pi/5) = [([5] + 5)/8] cos(pi/5) = ([5] + 1)/4 sin(pi/5) = [(5 - [5])/8] equilateral triangle height 3 [3]/2 center to one side 1 center to vertex 2 side 2[3] 1 side/height 2/[3] area 3[3] [3]/4 pentagon each side subtends 4 pi/10 angle at vertex for each triangle 3 pi/10 right triangle in each triangle central angle pi / 5 vertex angle 3 pi/10 sin(pi/5) [(5-[5])/8] cos(pi/5) ([5] + 1)/4 inscribed radius (1+[5])/4 1 [5+2[5]]/(2[5] = 0.68819 superscribed radius 1 [5] - 1 [5+[5]]/[10] = 0.85065 height (5+[5])/4 [5] [5+2[5]]/2 = 1.53884 side [(5-[5])/2] 2*[5-2[5]] 1 = 1 break (3+[5])/4 [5] - 2 height/side [5 + 2[5]]/2 = 1.53884 area/5 [5+[5]]/4[2] [5-2[5]] [5+2[5]]/(4[5]) area area in square sides ([5][5+2[5]])/4 note: The derivation of the expressions have been posted at http://www.issi1.com/corwin/calculator/proof.txt and http://www.issi1.com/corwin/calculator/pentagon.txt, and have shown to be consistent at http://www.issi1.com/corwin/calculator/pentagon.html, http://www.issi1.com/corwin/calculator/dodec.html, and http://www.issi1.com/corwin/calculator/ahedron.html . If inconsistencies were found in corroberating derivations the conflicts were resolved by graphical constructions or consistency checks using the concurrent inverse feature in the javascript pages. APPENDIX B SUMMMARY dodecahedron face normal angle arccos(1/[5]) 63.4349488 deg angle,top row vertices arccos([5]/3) angle, middle row vrtc edge length [50 - 22*5**0.5] = 0.89805595 1 2[3-[5]]/[6] = inscribed radius 1 [25+11[5]]/[5*8]= [5+2[5]]/[15] = center of edge radius [(5 -[5])/2] = 1.17557 (3+[5])/4 = [3+[5]]/[6] = superscribed radius [3][5 - 2[5]] = 1.25840857 [3/2][3+[5]]/2 1 pentagon height [5/2][ 3 - [5]] = 1.381966 [5+2[5]/2 = [5+[5]]/[6] = pentagon area ([5][5+2[5]])/4 surface area 30[2][65 - 29[5]] 3[5][5+2[5] volume 10[2][65 - 29[5]] = 5.55029 5[47+21[5]]/[2*7*8*9] 4pi/3 = 4.1887 icosahedron face normal angle arccos(5**0.5/3) = 41.8103 face normal from top middle row from top edge length [6][7 - 3[5]] = 1.32317 1 [2][5-[5]]/[5] inscribed radius 1 [7+3[5]]/(2[6]) [5+2[5]]/[15] center of edge radius [3/2][3 - [5]] [3+[5]]/(2[2]) [5+[5]][10] superscribed radius [3][5 - 2[5]] [5+[5]]/(2[2]) 1 triangle heigth (3/[2])[7 - 3[5]] = 1.145898 [3]/2 [3/10][5-[5]] surface area 30[3](7 - 3[5]) 5[3] 10[3](5-[5])/5 volume 10[3](7 - 3[5]) = 5.05406 = 5.05405 APPENDIX C IDENTITIES simplification by removing radicals in the denominator Phi = (1+[5])/2 = 2/(1-[5]) = 1/phi = phi + 1 (a+b*5**0.5)**0.5/(c+d*5**0.5)**0.5 = ((ac-bd*5) +(bc-ad)*5**0.5))**0.5 / (c**2 - 5*d**2)**0.5 [a+b[5]]/[c+d[5]] = ((ac-bd[5] +[bc-ad)[5]] / [c**2 - 5d**2] examples; let % = 5**0.5 and [x] = x**0.5 [a+b%] / [c+d%] = [ac-5bd + (bc-ad)%]/ [c**2 - 5d%] [5-%]/[3-%] = [10+2%] / 2 = [5+%] / [2] [3-%]/[5+%] = [20-8%] /[20] = [5-2%]/[5] [5-%]/[5+%] = [30-10%] / [20] = [3-%] / [2] ; [15-5%]/[10] = [3-%]/[2] [5+%]/[3-%] = [20+8%] / 2 = [5+2%] [3-%]/[7+3%] = [36-16%]/[4] = [9-4%] [5+%]/[7+3%] = [20-8%]/2 = [5-2%] [3-%]/[3+%] = [14-6%]/[4] = [7-3%]/[2] [3+%]/[5+%] = [10+2%]/[20] = [5+%]/[10] [3-%]/[5+%] = [5-2%]/[5] ; [20-8%]/[20] = [5-2%]/[5] [7-3%]/(5+%) = [7-3%][6-2%]/4[5] = [72-32%]/4[5] = [9-4%]/[10] (5+%)/(3-5%) = (40+28%) / -116 = (10+7%)/-29 = [345+140%]/-29 = [5][69+28%]/-29 [7-3%]/(3-5%) = [7-3%][134+30%] / -116 = [488-192%]/-116 = [61-24%] / -29[2] ; [61-24%] / -29[2] [3+%]/[5+%] = [10+2%]/[20] = [5+%]/[10] [5+%]/(1+%) = [5+%][6-2%]/-4 = [20-4%]/-4 = [5-%]/-2 [3+%]/(1+%) = [3+%](1-%) / -4 = -[3+%][6-2%] / 4 = -[8]/4 = -1/[2] ; -1/[2] (5+%) = [10][3+1%] (3+%) = [2][7+3%] (7+3%) = [94+42%] [7+3%]/(5+%) = [7+3%][30-10%]/20 = [7+3%][3-1%]/2[10] = [3+1%]/[20] (3+%)/[7+3%] = [14+6%][7-3%] /[4] = [49-45]/[2] =[2] (3+%)/(5+%) = (5+%)/10 = [3+%]/[10] (3-%)/[5+%] = [14-6%][5-%]/[20]=[7-3%][5-%]/[10]=[50-22%]/[10]=[25-11%]/[5] (5-%)/(3-%) = (10+2%)/4 = (5+%)/2 [30+10%]/2 (5-%)/[5+%] = [30-10%][5-%] /[20] = [20-8%]/[2] = [10-4%] ; [200-80%]/2[5] = [10-4%] [3+%]/[5-%] = [20+8%]/[20] = [5+2%]/[5] (5+%)/[3+%] = [10] (5+%)/[5-%] = [3+%][5+%] /[2] = [20+8%]/[2] = [10+4%] One way to check these would be to find two with the same numerator or denominator, make an identity with the denominator or numerator cancelling that of the first and replacing it with that for the last; then multiplying the first with the new yields the last result. 1/[3+%] = [3-%] / 2 1/[3-%] = [3+%] / 2 1/[5-2%] = [5+2%] / [5] 1/[5+2%] = [5-2%] / [5] 1/([5+2%] + [5-2%]) = ([5+2%] - [5-2%]) / 4% Above the three things taken two at a time give six relationships; I used three in order to check the work. There may be sets closed under multiplication and division. These resemble complex numbers; maybe there are similar relationships. If the simplification did not exist the expressions would easily become unmanageably unweildy. ----------------------------------------------------------------------------- My discussion of possible and impossible things is at http://www.issi1.com/corwin/calculator/possible.txt Ray tracing is at http://www.issi1.com/corwin/calculator/icosahedron.jpg Other details are at http://www.issi1.com/corwin/calculator/platonic.txt http://www.issi1.com/corwin/calculator/unit_v.txt This page is on line at http://www.issi1.com/corwin/calculator/proof.txt Further theory is at http://www.issi1.com/corwin/calculator/addition.txt After doing these computations I told whoever I thought may be interested and George Hart replied to me that it was well known and in the books: Regular Polytopes by H.S.M.Coxeter Prof. Mathematics University of Toronto isbn 0486614808 Dover Zome Geometry by George Hart and Henri Picciotto KeyCurriculumPress 2001 13 other references that look like they may be of particular use for computations are among the 187 listed at: ref: http://www.georgehart.com/virtual-polyhedra/references.html mathematical text focusing on combinatorial issues Branko Grunbaum, Convex Polytropes, Interscience, 1967 Branko Grunmaum. Regular Polyhedra---Old and New" Aequationes Mathematicae Vol 15 pp 118-120 more regular polyhedra five fold symmetry Istvan Hargittai editor, Fivefold Symmetry, World Scientific, 1992 computational method for locating vertex coordinates, with exact formulas for angles Andrew Hume, Exact Descriptions of Regular and Semi-Regular Polyhedra and their Duals Computing Science Technical Report #130, AT&T Bell Laboratories, Murray Hill, 1986 non-rigorous computations of strut lengths in tensegrity structures and geodesic domes Hugh Kenner, Geodesic Math and How to Use It, Univ.Cal. Pr. 1976 symmetry groups of polyhedra ... translation of 1884 German Felix Klein, The Icosahedron and the Solution of Equations of the Fifth Degree Dover 1956 exact formulas for constructing all 77 kinds of uniform polyhedra Peter W. Messer, Closed Form Expressions for Uniform Polyhedra and Their Duals, Discrete and Computational Geometry 27 pp 353-375, 2002 reassemblies David Peterson, Two Dissections in 3-D, Journal of Recreational Mathematics, Vol 20 pp 257-270, 1988 Coexeter enumeration complete J. Skilling, The Complete Set of Uniform Polyhedra, Philosophical Transactions of the Royal Society Ser A, 278 pp 111-135, 1975 tables Eric W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1998 references David Wells, The Penguin Dictionary of Curious and Interesting Geometry, Penguin, 1991 design principles Robert Williams, Natural Structure: Toward a Form Language, Eudaemon Pr., 1972 The Geometrical Foundations of Natural Structure: A Source Book of Design, Dover, 1979 another reference that has some dimensional details Shelter upc 6 76553 01995 2 70110 isbn 0-936070-11-0 LibraryofCongress 90-60125 Shelter Publications Bolinus CA www.shelterpub.com Lloyd Kahn shelter@shelterpub.com Box 279 800 307 0131 Domebook 3 page 110 The Wonder of Jena 1922 Carl Zeiss optical works in Jena, Germany Dr. Walter Bauersfield Chord Factors page 126 Scientific American September 1963(65) David R. Kruschke 2135 West Juneau Ave Milwaukee WI 53233 $1.50 Hugh Kenner Connector Kits Dyna Domes Bill Woods Bindu Dome page 129 onion dome Dhyana Mandiram, Guru Swami Rama Lama Domes, Col.Beard; Namaste Dennis R. Holloway Hindu Meditation Temple Minneapolis MN Zomeworks Box 712 Albuquerque NM Cadco of NY State, Inc plywood kit Box 874 Plattsburgh NY 412901 Cansdome tent dome custom made skins 7651 Ave. de la Seine Montreal 434 Quebec Canada Dome East model kits 325 Duffy Ave large tent domes Hicksville NY computer calculations Domebuilders Dome kits, hubs Box 4811 Santa Barbara CA Dyna Domes dome kits, hubs 2226 N 23rd Ave Phoenix AZ 85027 Intergalactic Tool Co. portable tent domes 1601 Haight St San Francisco CA 94117 Geodesic Structures plywood dome kit Dept 15 Box 176 Hightstown NY 08520 Redwood Domes dome kits Aptos CA 95003 Zomeworks Corp Box 712 Albuquerque NM 87103 Synapse custom made prefab domes Box 554 Sander WY 82520 Triadome Ins icosaplydome Box 548 Boulder CO 80302 Fuller Patents 2 682 235 3 203 144 3 354 591 3 197 927 2 881 717 2 914 074 3 114 176 3 063 521 The Dome Builder's Handbook, J.Prenis RunningPress 1973 http://www.runningpress.com Philadelphia contracts@perseusbooks.com http://www.math.uga.edu/~clint/2005/5210/texts.htm clint@math.uga.edu Geometry: Transformations and Symmetry natural occurrances, and many examples in different environments http://ascension2000.com/DivineCosmos/03.htm Fig 3.5L Aluminum-Copper-Iron An Pang Tsai NRIM Tsukuba,Japan Sacred Geometry, Philosophy and Practice Robert Lawlor 1982 Thomas & Hudson Ltd, London LoC 88-51328 ISBN-13 978-0-500-81030 ISBN-10 0-500-81030-3 p51,52,53 goldensection&pentagon 5.3a 5.3b 5.4a 5.4b p85 harmonic mean p98 Workbook 9 The Platonic Solids p107 interesting identity !!! space filling http://www.cut-the-knot.org/htdocs/dcforum/DCForumID4/696.shtml commercial toys dog toys Nobbly Wobbley Multipet.com International Moonachie NJ hol-ee roller JW Pet Co. Teterboro NJ patents 6651590 6622659 D477441 woven dog toy http://en.wikipedia.org/wiki/Image:Soccerball.svg