geodesic triacontahedron
sipspert
Myself and a design partner have developed a flexible building design
solution called Architecture Resistant to Climatic eXtremes or (ARCX)
it uses Structural insulated Panels (SIPs) as the material of choice
for our buildings and takes advantage of the automated cutting
capabilities of CNC equipment to cut the perfect triangles needed for
the geodesic part of the buildings.
This system is based on the polyhedral mathematics of a dual figures
of the icosahedron and the dodecahedron. This figure is called the
triacontahedron. This is all well known. When this figure is
analyzed the vertices exist in three different spherical planes, one
for the icosahedron, one for the dodecahedron and one for the vertices
formed by the intersection where the two figures cross.
We wanted to take the above mentioned figure and make all the
vertices reside on the same spherical plane and make it a true
geodesic structure. To do this we hired a mathematics professor and
had her work out the angles of the 60 triangles as well as the
dihedral angles for three dimensional shapes. She labelled this shape
a Geodesic Triacontahedron. We used this information in a patent
application filed and accepted in 2007. This math was one part of an
extensive application process.
We have been told several times by a Gerry Toomey that this figure is
not new and has been around for quite some time. When I entered
Geodesic Triacontahedron in this help group nothing comes up. If this
figure pre-exists our calculations then the actual math must exist, ie
the three angles of the triangles and the dihedral angles of each
side. Maybe the answer to this dilemma is for someone to show me the
math because I never could find it.
We certainly want a clean patent application and do not want to take
credit for something we did not invent. So far Mr Toomey has failed
to show me that the actual calculations of the math exists.
This is a very interesting group and I am glad to have been referred
to it. We are working hard to advance our system and would appreciate
any comment and/feedback on our building designs that can be found on
our website shown below. More detail about the design can be found on
the Buckminster Fuller Institute website in the ideas section for the
competitions under ARCX. It offers a more thorough explanation of the
designs.
Thank you for your interest
Michael Morley
SIPsmart Building Systems
700 Mississippi St.
Lawrence, KS 66044
P 785-843-7007
C 785 218-5061
F 785 380-4029
mor...@sunflower.com
www.sipsmart.com
Hugh Simpson
who calls it a dodecahedron. Rob met Bucky and became fascinated with
him. He has built over a dozen dome home kits in the past yrs. He says
that the dodecahedron originated he thought at MIT. Someone from this
group said it was also a tricondehedron and later saw the entire
dimensions for building the tri in the classic Dome 2 book. I would be
very interested in learning more about dome. We are launching
PlanetGreenSolutiins in the next week where we will offer domes,
yurts, earth bag, cargo containers, etc. I studied with Bucky's right
hand man Marshall Thurber. Thanks Hugh Simpson God's Steward
Adrian Rossiter
On Mon, 11 Jan 2010, sipspert wrote:
> geodesic structure. To do this we hired a mathematics professor and
> had her work out the angles of the 60 triangles as well as the
> dihedral angles for three dimensional shapes. She labelled this shape
> a Geodesic Triacontahedron. We used this information in a patent
> application filed and accepted in 2007. This math was one part of an
> extensive application process.
>
> We have been told several times by a Gerry Toomey that this figure is
> not new and has been around for quite some time. When I entered
> Geodesic Triacontahedron in this help group nothing comes up. If this
> figure pre-exists our calculations then the actual math must exist, ie
> the three angles of the triangles and the dihedral angles of each
> side. Maybe the answer to this dilemma is for someone to show me the
> math because I never could find it.
>
> We certainly want a clean patent application and do not want to take
> credit for something we did not invent. So far Mr Toomey has failed
> to show me that the actual calculations of the math exists.
The Schwarz triangle tiling that Gerry mentioned is very well
known. It is illustrated in Coxeter's "Introduction to Geometry",
and also appears on the cover of another of his books
The vertex positions of this shape are unit vectors in the
direction of vertices, edge centroids and face centroids of
icosahedron (or dodecahedron) and can be worked out from
these coordinates
http://en.wikipedia.org/wiki/Icosahedron#Cartesian_coordinates
(For example, the unit vector towards a face centre is the sum of
the three face vertex coordinates, then divide through by the
distance of this summed coordinate from from the origen.)
Here are some example techniques that will calculate the dihedral
angles and face angles from the coordinates of the shape
http://en.wikipedia.org/wiki/Plane_%28geometry%29#Method_1
http://en.wikipedia.org/wiki/Plane_%28geometry%29#Dihedral_angle
http://en.wikipedia.org/wiki/Dot_product#Geometric_interpretation
Similar techniques are included in the Antiprism programs. They
will make the shape and find the dihedral angles with this command
pol_recip icosid | geodesic -i - | off_util -S | off_query -I - Ea
The angles it calculates are
178.07586653298 ( 1.92413346702 between normals)
159.87395402650 (20.12604597350 between normals)
148.78300805172 (31.21699194828 between normals)
and for the face angles
pol_recip icosid | geodesic -i - | off_util -S | off_report -C F
34.43652599605169
58.40516129494199
87.15831270900634
Please let me know if these match your dihedral and face angles
as it would be a useful check for my programs.
Adrian.
--
Adrian Rossiter
adr...@antiprism.com
http://antiprism.com/adrian
Gerry in Quebec
Hello Michael,
Here’s what I sent you via Yahoo e-mail on March 22, 2009, with a copy
to your mathematician, Brigitte Servatius. It includes the interior
angles:
“Based on my reading of your U.S. patent application publication (July
1[7], 2007, no. US2007/0163185 A1), the polyhedron presented is simply
a 2-frequency geodesic sphere generated by ‘basic triacon division’ of
the icosahedron (as opposed to the better-known method, ‘full triacon
division’). These methods of triangular subdivision and vertex
projection have been used for decades in the design of geodesic domes.
Furthermore, the breakdown of the surface of a sphere into identical
triangles was, if I'm not mistaken, demonstrated and quantified by the
19th German mathematician K. H. A. Schwarz (1843-1921). The creation
of Schwarz triangles via basic triacon division is described in detail
(pp. 8-14) in ‘Geodesic Domes’, by Borin Van Loon, first published in
1994 by Tarquin Publications, Stadbroke, Diss, Norfolk, IP21 5JP.
[England]
“According to Van Loon, in the last century Buckminster Fuller used
the term ‘alpha particle’ to refer to the icosahedral Schwarz
triangle, the same triangle for which you and co-applicant David Glenn
Henderson [sic] seem to be claiming novelty in your patent application
publication. Although there's a mathematical error (p. 9) in Van
Loon's presentation of the planar version of the icosahedral Schwarz
triangle, it is in fact the same triangle described in your patent
application. This is borne out by the interior angles that Van Loon
gives in radians for the spherical version of the triangle, namely pi/
2, pi/3 and pi/5. These spherical angles are the exact counterparts of
the interior angles of the planar triangle described in your
application: 87.16 degrees, 58.41 degrees and (by subtraction) 34.44
degrees.
“I just wanted to point out to you what I believe to be ‘prior art’ on
this topic, that is, historical information that seems to contradict
any claim of novelty regarding the polyhedron on which your geodesic
dome is based.”
End of quotes from my correspondence with you/SIP-Smart several moons
ago. So, Michael: I did in fact send you the interior angles of your
building-block triangle,. These were expressed in radians for the
spherical version of the triangle, which convert to the degree values
in the planar version of your triangle.
If you need it spelled out in detail, i.e., how the spherical angles
pi/2 radians (90 degrees), pi/3 radians (60 degrees) and pi/5 radians
(36 degrees) translate into the three planar triangles of your
building-block triangle, I can provide that. If you’d also like to see
the calculation of the dihedral angles between faces, I can do that
too. These will all be straightforward trig equations.
I see Adrian has also provided some references. Thanks, Adrian.
Gerry Toomey in Quebec
> P 785-843-7007 785-843-7007
> C 785 218-5061 785 218-5061
> F 785 380-4029
> morl...@sunflower.comwww.sipsmart.com
Richard Fischbeck
Gerry in Quebec
Using the icosahedron as my starting point, I put together a step-by-
step derivation of key dimensions of the 2v icosa sphere, basic
triacon division. This is what you refer to as the geodesic
triacontahedron.
The illustrated Excel spreadsheet I put in the files section a few
minutes ago is called 2v-icosa-basic-triacon-derivation.xls. It
contains all the necessary diagram and equations to illustrate the
derivation of the edge length of the icosahedron; the three chord
factors of the triacon geodesic sphere; the three face angles; and the
three dihedral angles.
Adrian,
Our angle values match. Here's what I get:
Dihedral angles:
159.8739540265
148.783008051718
178.075866532976
Face angles
58.405161294942
34.4365259960517
87.1583127090063
Gerry
On Jan 12, 6:30 am, Adrian Rossiter <adr...@antiprism.com> wrote:
> Hi Michael
>
>
> On Mon, 11 Jan 2010, sipspert wrote:
> > geodesic structure. To do this we hired a mathematics professor and
> > had her work out the angles of the 60 triangles as well as the
> > dihedral angles for three dimensional shapes. She labelled this shape
> > a Geodesic Triacontahedron. We used this information in a patent
> > application filed and accepted in 2007. This math was one part of an
> > extensive application process.
>
> > We have been told several times by a Gerry Toomey that this figure is
> > not new and has been around for quite some time. When I entered
> > Geodesic Triacontahedron in this help group nothing comes up. If this
> > figure pre-exists our calculations then the actual math must exist, ie
> > the three angles of the triangles and the dihedral angles of each
> > side. Maybe the answer to this dilemma is for someone to show me the
> > math because I never could find it.
>
> > We certainly want a clean patent application and do not want to take
> > credit for something we did not invent. So far Mr Toomey has failed
> > to show me that the actual calculations of the math exists.
>
> The Schwarz triangle tiling that Gerry mentioned is very well
> known. It is illustrated in Coxeter's "Introduction to Geometry",
> and also appears on the cover of another of his books
>
> http://www.amazon.com/Mathematical-Recreations-Essays-Dover-Books/dp/...
> adr...@antiprism.comhttp://antiprism.com/adrian- Hide quoted text -
>
> - Show quoted text -
Adrian Rossiter
On Wed, 13 Jan 2010, Gerry in Quebec wrote:
> Adrian,
> Our angle values match. Here's what I get:
Thanks for the feedback. Michael also wrote to say that
his figures matched.
TaffGoch
Application number: 11624470 - Filed Jan 18, 2007
http://www.google.com/patents?id=HaOAAAAAEBAJ&printsec=abstract
Blair Wolfram
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Gerry in Quebec
Incorporated Kauai dome in Hawaii. In at least one pic, both the short
seam of the plywood and the tubular metal struts are visible.
http://losttrails.com/portfolio/dome/
Gerry in Quebec
On Jan 15, 6:35 pm, Blair Wolfram <thedome...@gmail.com> wrote:
> To maximize the strength of a standard 4' x 8' sheet of plywood, I don't
> split the triangle down the middle. Cutting the plywood in a trapezoid to
> cover the lower 3/4 of a geodesic triangle leaves a much shorter seam. Add a
> small 'tip' to the trapezoid to complete the triangle.
>
> Blairhttp://www.hurricanedomes.com
>
>
>
> On Fri, Jan 15, 2010 at 4:53 PM, TaffGoch <taffg...@gmail.com> wrote:
>
> > Anyone can read, for themselves, the patent application at Google Patents:
>
> > Application number: 11624470 - Filed Jan 18, 2007
> >http://www.google.com/patents?id=HaOAAAAAEBAJ&printsec=abstract
> > _______________________
>
> > I like the emphasis on the reduction of waste, when cutting SIPs to cover
> > triangles (approximating right triangles.) Of course, the same principle can
> > be (and is often) applied when building a "typical" geodesic dome. You have
> > to split each triangle down the middle, most likely including an additional
> > support stud at that "join." The method is depicted in "Domebook 2," so,
> > it's been around awhile.
>
> > -Taff
>
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