Question about Strength of Octahedron

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homespun

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Jun 29, 2010, 4:55:11 PM6/29/10
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A square-based pyramid made of just sticks is not very strong unless the square base has members to triangulate it. But two of them together to make up an octahedron is quite strong, even though the square itself has not been triangulated. Any comments on why this works?
Can a numerical comparison be made as to how much stronger a tetrahedron is than an octahedron?
Does the octahedron become even stronger when included as part of an extended OCTA-TETRA structure?

I'm hoping a discussion will develop on list about these questions.
Dan Suttin

TaffGoch

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Jun 29, 2010, 10:35:37 PM6/29/10
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Dan,
 
A triangle is a triangle -- and THAT is all that counts, regarding deformability. It can't be deformed.
 
The only difference between tetrahedrons and octahedrons (or icosahedrons, etc.) will be the lengths of the "sticks." If you use equal lengths for these polyhedral skeletons, there won't be any difference in compression strength of the sticks.
 
However, the difference between tetrahedron and octahedron will be how the sticks are loaded. When loaded from above (i.e.; gravity loading,) the ANGLE of the stick, from vertical, will be different, and will produce different vertical-vs-horizontal components of the load. Since a tetrahedron has "more vertical" sticks, compared to an octahedron (or square-base pyramid,) the vertical load component will be greater for the tetrahedron, and the horizontal component less.
 
Additionally, if you examine the tension produced in the "skirt" sticks, due to the vertical load on the vertically-angled sticks, the tension in a square-based pyramid ground-level stick will be greater than the skirt sticks in a tetrahedron. (Can you visualize that, without a diagram?)
 
This issue is apparent in a geodesic dome, as the horizontal struts, around the "belt" of the dome will be under tension, and NO compression. This is why hub design is so critical. On the top of a dome, the hubs have to be strong in compression, while along the equator, the hubs have to be strong in tension.
 
Relative comparison of compression/tension, octahedron-vs-tetrahedron, can be calculated with simple trigonometry. When calculating horz/vert components of geodesic dome struts can be more complicated, since many struts are angled in more than one plane. You have to include calculations for x,y,z components of the load forces -- not just x & y (tetrahedron & octahedron.)
 
Confused yet?
 
Taff

Hugh Simpson

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Jun 30, 2010, 9:50:04 AM6/30/10
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Dan we market the dodecahedron dome kits as our supplier who is a long
time dome builder says it is the strongest dome structure of all of
them. This is what we will be using on the Gulf Coast with our new
http://www.operationsavethegulf.com project. Thanks Hugh

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Blair Wolfram

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Jun 30, 2010, 4:27:38 PM6/30/10
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Hugh;
 
I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
 
Blair

Hugh Simpson

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Jun 30, 2010, 4:44:43 PM6/30/10
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My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.

Ken G. Brown

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Jun 30, 2010, 4:56:53 PM6/30/10
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Do you have any good links for domes that have actually withstood severe weather or other disasters?

Thx,
Ken


At 4:44 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.
>

>On 6/30/10, Blair Wolfram <<mailto:thedo...@gmail.com>thedo...@gmail.com> wrote:
>
>Hugh;
>
>I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
>
>Blair
>
>
>
>
>On Wed, Jun 30, 2010 at 8:50 AM, Hugh Simpson <<mailto:hughms...@gmail.com>hughms...@gmail.com> wrote:
>
>Dan we market the dodecahedron dome kits as our supplier who is a long
>time dome builder says it is the strongest dome structure of all of
>them. This is what we will be using on the Gulf Coast with our new

><http://www.operationsavethegulf.com/>http://www.operationsavethegulf.com project. Thanks Hugh

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Hugh Simpson

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Jun 30, 2010, 4:59:10 PM6/30/10
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Hi, Ken! I will ask my manufacturer for you.

On 6/30/10, Ken G. Brown <kbr...@mac.com> wrote:
Do you have any good links for domes that have actually withstood severe weather or other disasters?

Thx,
Ken



At 4:44 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.
>

>On 6/30/10, Blair Wolfram <<mailto:thedo...@gmail.com>thedo...@gmail.com> wrote:
>
>Hugh;
>
>I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
>
>Blair
>
>
>
>

>On Wed, Jun 30, 2010 at 8:50 AM, Hugh Simpson <<mailto:hughms...@gmail.com>hughms...@gmail.com> wrote:
>
>Dan we market the dodecahedron dome kits as our supplier who is a long
>time dome builder says it is the strongest dome structure of all of
>them. This is what we will be using on the Gulf Coast with our new



>
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Richard Fischbeck

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Jun 30, 2010, 5:02:32 PM6/30/10
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$40,000 sounds like a ton of money. But what do I know. I bet your heart is in the right place.


On Wed, Jun 30, 2010 at 4:44 PM, Hugh Simpson <hughms...@gmail.com> wrote:
My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. 
 

Ken G. Brown

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Jun 30, 2010, 5:03:52 PM6/30/10
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Anything at all to show how strong the domes are would be good. I just thought of large snow loads too since you mention Colorado.
Of course, any and all dome pics are of interest, especially during construction.

Ken

At 4:59 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>Hi, Ken! I will ask my manufacturer for you.
>

>On 6/30/10, Ken G. Brown <<mailto:kbr...@mac.com>kbr...@mac.com> wrote:
>
>Do you have any good links for domes that have actually withstood severe weather or other disasters?
>
>Thx,
>Ken
>
>
>
>At 4:44 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>>My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.
>>
>
>>On 6/30/10, Blair Wolfram <<mailto:<mailto:thedo...@gmail.com>thedo...@gmail.com><mailto:thedo...@gmail.com>thedo...@gmail.com> wrote:
>>
>>Hugh;
>>
>>I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
>>
>>Blair
>>
>>
>>
>>
>
>>On Wed, Jun 30, 2010 at 8:50 AM, Hugh Simpson <<mailto:<mailto:hughms...@gmail.com>hughms...@gmail.com><mailto:hughms...@gmail.com>hughms...@gmail.com> wrote:
>>
>>Dan we market the dodecahedron dome kits as our supplier who is a long
>>time dome builder says it is the strongest dome structure of all of
>>them. This is what we will be using on the Gulf Coast with our new
>

>><<http://www.operationsavethegulf.com/>http://www.operationsavethegulf.com/><http://www.operationsavethegulf.com>http://www.operationsavethegulf.com project. Thanks Hugh

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Richard Fischbeck

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Jun 30, 2010, 4:58:24 PM6/30/10
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Oh boy. This will be interesting! Be kind, Blair...

On Wed, Jun 30, 2010 at 4:44 PM, Hugh Simpson <hughms...@gmail.com> wrote:

Richard Fischbeck

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Jun 30, 2010, 5:14:44 PM6/30/10
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50,000 psi?!

"At the bottom of the trench, where the plates meet, the water column above exerts a pressure of 108.6 megapascals (15,750 psi),"



On Wed, Jun 30, 2010 at 4:44 PM, Hugh Simpson 

Gerry in Quebec

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Jun 30, 2010, 5:22:36 PM6/30/10
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Blair,
Yes, the 2 non-triangulated platonic solids are pretty wobbly as stick
structures.... I think you meant the cube, not the square.

I recently made a model of one of the Clinton-Goldberg polyhedra, the
"I 2,0", following Clinton's equal central angle conjecture which we
discussed on the list a while back.

The model is in the icosahedral family. The model has 120 struts of
equal length, 80 vertices, and 42 faces (12 pents and 30 irregular,
non-planar hexagons, no triangles). It's similar to the dual of the 2v
icosa class I geodesic sphere. I used 3/16" stove bolts and 8" lengths
of half-inch pex pipe (60 black struts, 60 white). At that scale, the
model was a wee bit wobbly even after I tightened the nuts. I think it
would have been just about right at about 6.5" per strut instead of
7.5. The model is now hanging from the cupola of my workshop and looks
pretty cool. Diameter is about 42 inches. I installed a threaded rod
from "north pole to south pole" to help it retain its shape.

Gerry in Quebec

On Jun 30, 4:27 pm, Blair Wolfram <thedome...@gmail.com> wrote:
> Hugh;
>
> I don't know who would have made this claim, but the dodecahedron is not a
> stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and
> icosahedron are stable, and the square and dodecahedron are unstable.
>
> Blair
>

homespun

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Jun 30, 2010, 6:31:14 PM6/30/10
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Gerry,
Could you please send picture(s) of the model you just wrote in about?
Thanks,
Dan

Blair Wolfram

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Jun 30, 2010, 6:51:38 PM6/30/10
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Gerry;
 
I too have just made two sizes of tessellated dodecahedrons based on Clinton's equal central angle conjecture. The larger is a hex/pent dual of a 6 frequency. After tightning the nuts I could spend an hour getting the hexagons symmetrical, as pinching one into shape deforms it's adjacent hexagon. I started with the pentagons and worked my way out. Without looking it up it must have 360 struts.
 
Blair

 

Blair Wolfram

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Jun 30, 2010, 6:56:48 PM6/30/10
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Hi Dick;
It's good to hear you're out there...
Blair

Richard Fischbeck

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Jun 30, 2010, 7:17:30 PM6/30/10
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I'm out-there, alright! What's life without a dream.

Build any domes lately ? ! 

Gerry in Quebec

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Jun 30, 2010, 7:23:11 PM6/30/10
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Hi Dan and Blair,
Will do.... I'll try to take either a webcam image or else a pic with
a regular digitial camera if I can rope someone into helping out since
we don't have a working digital camera in the house at the moment.
Blair's equal-central-angle sphere sounds more interesting than
mine.... Blair, send a pic or two if possible. Cheers,
Gerry

On Jun 30, 6:31 pm, "homespun" <uncle...@homespun4homeschoolers.com>
wrote:

Gerry in Quebec

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Jul 1, 2010, 8:54:08 AM7/1/10
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I've uploaded a webcam image of the Icosa 2,0 Goldberg-Clinton
polyhedron (rather fuzzy, I'm afraid), together with a line drawing
just to make the arrangement clear. The latter illustration is
actually the dual of the 2v icosa class I geodesic sphere, which is
slightly different from polyhedron I2,0 in that it has two edge
lengths instead of one, and not all vertices are equidistant from the
centre of the polyhedron. Filename: Goldberg-Clinton-I2,0-pex-
tubing.jpg.
Gerry


On Jun 30, 6:31 pm, "homespun" <uncle...@homespun4homeschoolers.com>
wrote:

sergio_cohen

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Jul 1, 2010, 4:00:51 PM7/1/10
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Ken, There I upload 3 pictures of geodesic domes in the Artic and
Antartic. Hope it serves. I didnt know how to create a folder, files
are getting many. See you, Sergio

On 30 jun, 17:56, "Ken G. Brown" <kbr...@mac.com> wrote:
> Do you have any good links for domes that have actually withstood severe weather or other disasters?
>
> Thx,
> Ken
>
> At 4:44 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>
>
>
> >My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.
>
> >On 6/30/10, Blair Wolfram <<mailto:thedome...@gmail.com>thedome...@gmail.com> wrote:
>
> >Hugh;
>
> >I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
>
> >Blair
>
> >On Wed, Jun 30, 2010 at 8:50 AM, Hugh Simpson <<mailto:hughmsimp...@gmail.com>hughmsimp...@gmail.com> wrote:
>
> >Dan we market the dodecahedron dome kits as our supplier who is a long
> >time dome builder says it is the strongest dome structure of all of
> >them. This is what we will be using on the Gulf Coast with our new
> ><http://www.operationsavethegulf.com/>http://www.operationsavethegulf.comproject. Thanks Hugh
> >> <mailto:GeodesicHelp%2Bunsu...@googlegroups.com>GeodesicHelp+unsubscrib e...@googlegroups.com
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sergio_cohen

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Jul 1, 2010, 4:03:01 PM7/1/10
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Sorry I forgot, the names of the files are domopolonorte and
domopolosur or something...Sergio

On 30 jun, 17:56, "Ken G. Brown" <kbr...@mac.com> wrote:
> Do you have any good links for domes that have actually withstood severe weather or other disasters?
>
> Thx,
> Ken
>
> At 4:44 PM -0400 6/30/10, Hugh Simpson apparently wrote:
>
>
>
> >My dome manufacturer has well over 35 yrs experience building his first in Colorado which is still standing. Our dome will withstand up to 200 mph winds and has been able to withstand 50,000 psi which simulates the four worst earthquakes. So I figure that is stable enough.
>
> >On 6/30/10, Blair Wolfram <<mailto:thedome...@gmail.com>thedome...@gmail.com> wrote:
>
> >Hugh;
>
> >I don't know who would have made this claim, but the dodecahedron is not a stable polyhedron. Of the 5 Platonic solids, the tetrahedron, octahedron and icosahedron are stable, and the square and dodecahedron are unstable.
>
> >Blair
>
> >On Wed, Jun 30, 2010 at 8:50 AM, Hugh Simpson <<mailto:hughmsimp...@gmail.com>hughmsimp...@gmail.com> wrote:
>
> >Dan we market the dodecahedron dome kits as our supplier who is a long
> >time dome builder says it is the strongest dome structure of all of
> >them. This is what we will be using on the Gulf Coast with our new
> ><http://www.operationsavethegulf.com/>http://www.operationsavethegulf.comproject. Thanks Hugh
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dick.fi...@gmail.com

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Jul 2, 2010, 12:43:37 PM7/2/10
to Geodesic Help Group
The Eden Project domes show how to stabilized a vertex of a non-
triangulated dome. It is hexagonal, not dodecahedral, but the same
principle applied. Each vertex is connected three ways to a grid of
cables.

http://www.acrosstheplanet.net/wp-content/uploads/2009/03/800px-eden_project2.jpg

Hope this helps.
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