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
?
http://en.wikipedia.org/wiki/Exact_trigonometric_constants#Uses_for_constants
(15+7[5])/4
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
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian?PRIAIcos.pdf
commercial toys
dog toys
Nobbly Wobbley
Multipet.com International
Moonachie NJ
hol-ee roller
JW Pet Co.
Teterboro NJ
patents 6651590 6622659 D477441
Nobbly Wobbly six five point rings woven dog toy
Multipet.com International MoorachiaNJ
upc 1 84369 5 1023 2
Pet smart 78436951023
rubic icosahedron
http://www.twisypuzzles.com/cgi-bin/puzzle.cgi?pid=25
Thomas Ball
Brainy Toys
http://en.wikipedia.org/wiki/Image:Soccerball.svg
http://www.math.nus.edu.sg/aslaksen/polyhedra/index.html#links
------------------------------------
quaternions William_Rowan_Hamilton Arthur_Cayley
http://en.wikipedia.org/wiki/Quaternoin
rotations in four dimensional space
Kepler_Poinsot_polyhedra Louis_Poinsot
wiki/Uniform_polyhedron