Wm.C. Corwin.info, Ph.D.,P.E.
www.ConcurrentInverse.com
billc@issi1.com
(c) copyright 2005 Wm.C.Corwin
TITLE: Proof of simple exact formulas for dimensions of the
dodecahedron and icosahedron.
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The proof of the exact formulas in closed form for the dodecahedron
and icosahedron is at http://www.issi1.com/corwin/calculator/proof.txt .
I have not yet read the formulas or their derivation elsewhere but
George Hart has informed me that they have been published. I don't
think the derivation is on the internet so I am putting it there so
references and links to them can be made. Interest in proofs
of exact formulas for pentagonal symmetry was expressed at
http://www.cut-the-knot.org/htdocs/dcforum/DCForumID4/606.shtml .
I have used the numerical results to make ray tracing applications at
http://www.issi1.com/corwin/icosahedron.jpg .
RESULTS:
dodecahedron
face normal angle arccos(1/5**0.5) 63.4349488 deg
edge length (50 - 22*5**0.5)**0.5
= 0.89805595
inscribed radius 1
center of edge radius ((5 - 5**0.5)/2)**0.5
= 1.17557
superscribed radius 3**0.5 * (5 - 2*5**0.5)**0.5
= 1.25840857
pentagon height (5/2)**0.5 * ( 3 - 5**0.5)**0.5
= 1.381966
surface area 30 * 2**0.5 * (65 - 29*5**0.5)**0.5
volume 10 * 2**0.5 * (65 - 29*5**0.5)**0.5
= 5.55029
4pi/3 = 4.1887
icosahedron
face normal angle arccos(5**0.5/3)
edge length 6**0.5 * (7 - 3*5**0.5)**0.5
inscribed radius 1
center of edge radius (3/2)**0.5 * (3 - 5**0.5)**0.5
superscribed radius 3**0.5 * (5 - 2*5**0.5)**0.5
triangle heigth 3/2**0.5 * (7 - 3*5**0.5)**0.5
surface area 30 * 3**0.5 * (7 - 3*5**0.5)
volume 10 * 3**0.5 * (7 - 3*5**0.5)
= 5.05406
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My discussion of possible and impossible things is at
http://www.issi1.com/corwin/calculator/possible.txt
Other relevant things are at
http://www.issi1.com/corwin/platonic.txt
http://www.issi1.com/corwin/unit_v.txt
Ray tracing example
http://www.issi1.com/corwin/icosahedron.jpg
After doing these derivations 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
Dover isbn 0486614808
Zome Geometry by George Hart and Henri Picciotto KeyCurriculumPress 2001
At
http://www.georgehart.com/virtual-polyhedra/references.html
13 of the 187 references may be particularly useful in calculations.