Choice of Icosahedron by Fuller for geodesic dome

The Icosahedron is not a superior choice compared to the
Icosidodecahedron as it appears from several considerations of
advantage.
In the latter case it is possible to build a dome using continuous
interwoven relatively flexible strips that can be rigidized by adding
similar strips to make curved I-beams. A continuous reinforcement is
more favourable for force distribution.Construction of higher frequency
triangular elements is possible here also, there is material economy
and stress distribution is improved without concentrations at so many
mechanical joint interfaces.
The more symmetric Platonic solid shell using icosahedron faces and
sub-divisions of higher frequency is not necessarily a better choice
than that of an Archimedean solid shell icosidodecahedron.
What continuous, unbroken great circle arrangements are there on a
sphere of associated solid polyhedra with full spherical symmetry?
To this question only three cases are seen as an answer from Wolfram
site for Archimedean and Platonic solids.
Among Platonic solids we have only one case corresponding to an
octahedron.Here three great circles in mutually perpendicular planes
(E.g.,on a globe containing 3 great circles: Equator /
0 deg Greenwich /
-90 deg US Central Time meridians). Among other
cases using a cube or tetrahedron, 3 lines come together at a vertex,
so a continuous projected great circle on the sphere cannot be drawn
from them, they are not suitable choices.
Among Archimedean solids there are only two cases, those projected from
the icosidodecahedron and the cuboctahedron onto the sphere.
The former has particular appeal, more than any other
arrangement_ perhaps because of its association with the GoldenRatio
(phi ~ 1.618 ) in several ways. On the surface of sphere, tan (half
interior angle of pentagon) is phi and tan of (pentagon center to
corner angle seen at sphere center) equals 1/phi.The side of
Icosidodecahedron solid/Sphere radius ratio is anyhow golden, from
geometry of a plane decagon central section. There are only six
cutting planes/great circles producing 32 faces (20 spherical
triangles and 12 spherical pentagons). Cutting plane is at arctan(0.5)
to axis of symmetry through center of pentagons. It is a strange but
pretty combination of dodecahedron and icosahedron.A RealTime3D view
with great circles looks even better than a solid model. It is rather
easy to make a model, I made one of 500 mm dia from 6 plastic strips
each divided into 10 equal parts for great circle intersection joints.
Three layers are carefully interwoven/locked together. Practically
chose 11equal parts, one overlapped for assembly convenience.
Comments on any aspect are welcome.
Narasimham
(A part of this was posted earlier on de.sci.mathematik 23 Oct 2004.
Favourite Icosododecahedron based Great Circles)
Reply to
Narasimham
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What - with the entire I-beams being woven - or with there being multiple woven layers inside one another? Neither sound terribly promising to me.
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Illustrates the general pattern.
The construction is not suitable for "conventional" strut-and-hub domes, though. In that context, the shape isn't even stable.
It works reaonably well as a frame tent, though:
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Use on a larger scale seems likely to run into material science-related problems.
Of course Fuller was well aware of the pattern - as illustrated by domes such as this one:
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Reply to
Tim Tyler
No, I was may be too brief. The flanges/web appear as separate structural elements only during initial assembly. But later on, the I-beam gets rigid when its flexible flanges are bolted/fixed using an in-between spacer radial web. Two spheres of slightly different diameter are separately interwoven, placed so that one is enclosed inside the other and is connected at 10 points (or multiple frequencies) by means of radial normal shear load resisting webs. Then we have an extraordinary stiffness of the curved I-beam along all great circles for the entire shell.
Yes, the triaxial weaves imparts shear rigidity eliminating in-plane mechanism like movements. When distance between two inter-connected woven layers is large compared to woven layer thickness, we refer to it as "Isogrid" construction also. Here in-plane stiffness is not orthogonally, but isotropically distributed... to be sufficiently near to a sandwich construction with its inherent flexural rigidity advantage.
Right, but it can be easily built up in the manner suggested, without anyway giving up the overall convenience of fabrication. The flanges can be designed proper length or if too long, can even be coiled up into smaller circles prior to assembly.
In the context mentioned above an integrated flange/web action gives high stability and rigidity from the continuous I-beams .
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A thoroughly enjoyable set of small icosi-dodeca tent constructions.
On the contrary, for deeper spherical domes (longitudinal stress compressive and circumferential stress tensile) the triaxial is quite efficient. This can be also checked out on FEA. We know how Fuller pooh-poohed the warnings regarding deep shell tensile circumferential stresses concerns and how his isotropic triaxial beam orientations of the standard geodesic dome triangles efficiently met the loads in tension and compression.
Else, what are the other material science problems?
as this one:
Care to comment about its critical design features and advantages vis-a-vis the multi frequency geodesic dome?.
If we take a figure of merit or structural performance factor such as:
(Static loads on dome + dome weight) * Enclosed Volume / (Plan area * structural weight)
the icosi-dodeca may have a better performance. In any case it appears that this construction would be more efficient than the conventional hub-strut type of geodesic dome construction.
Thanks again for any comments.
Narasimham
Reply to
Narasimham
My models do illustrate the same geometry as Fuller's dome in Japan - albeit at a lower frequency.
None of the woven domes use "genuine" great circles.
They all tend to deviate from being precise great circles.
This image is quite a good one to see the distortions around the pentagons on:
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Such distortions are part of the pattern.
The "octahedron" in the Inagamachi Golf Club Field House dome refers to the shape of the unit cell. Each hexagonal cell was composed of six octahedral blocks - each of which was constructed out of right-angled angle iron.
Each of the black triangles visible on the surface of the dome is the top face of one of these octahedra.
The actual dome itself is of a hex-pent-based geometry - i.e. the vent on top of the dome is a pentagon - rather than a square.
Reply to
Tim Tyler
The three-dimensional trusses I can imagine based on the pattern all seem relatively ugly to me.
Failure to form a proper three-dimensional truss would limit the use of the pattern on large scales - IMO.
If you are bending material, that becomes physically challenging when your materials are steel or are wood much thicker than a floorboard.
Material that is weak enough to bend is probably not going to be approved of by the building inspectors as a potentially load-bearing structural roofing material.
Variations on the construction that use straight members - e.g. like the "Nailbanger's Nightmare":
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...and my own "Leonardo Fuller" domes:
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...still have the problem that forces are applied in the middle of struts. While having some virtues in terms of simplicity, the result is substantially weaker than a hub- and-strut affair made of the same material as a result of this problem.
Generally speaking the pattern is good if you want small and cheap - but I don't think it scales up terribly well.
Reply to
Tim Tyler

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