Matematica De Domos

Geodesic Math
All Artwork, Graphics and Illustrations were created or made by: Jay Salsburg, Design Scientist, excerpt of an article by Joe Clinton

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Producing geodesics from the icosahedron.
The following article is an excerpt of an article by Joe Clinton on the different methods of producing geodesics from the icosahedron. Formatted by Jay Salsburg, Design ScientistUsing analytical geometry (Fuller used spherical trigonometry), calculated on a computer: General procedure 1. find the 3-dimensional coordinates of the vertices of the grid on the spherical surface 2. find geometry using the different Methods 3. calculate the chord lengths, angles etc. with these coordinates and analytical formulas. Joe worked with Fuller on his programs and was funded by NASA on aproject called “Structural Design Concepts for Future Space Missions.” The specific motivation for developing these methods was to have a variety of forms to combine in large space frame domes. For example the Expo dome in Montreal is a combination of a: 32-frequency regular triacon (Class II, method 3) and a 16-frequency truncatable alternate (Class I, method 3) With known parameters andsophisticated analysis, large structures can be optimized by different combinations and different methods; however, for small structures (up to 40') they are not generally relevant. What was called “alternate” breakdown, Joe classifies as “Class I’’; what was called ‘’triacon’’ he classifies as ‘’Class II’’. Joe wrote this section mainly with the intent of communicating the state of development of geodesicgeometries and the hope that it would be an aid to those interested in exploring and expanding this field.

Geodesic Math
DEFINITIONS Axial angle (omega Ω ) = an angle formed by an element and a radius from the center of the polyhedron meeting in a common point. The vertex of the axial angle is chosen as that point common to the polyhedron element and radius. The axial angle Ω may be found ifthe central angle δ is known by the following equation:
element axial Ω Radius (0, 0, 0) center of polyhedron

Ω =
vertex

180 - δ 2

Central angle (delta , δ ) = an angle formed by two radii of the polyhedron passing through the end points of an element of the polyhedron. The vertex of the central angle is chosen as that point common to both radii (the center of the polyhedron).
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The central angle δ may be found by knowing the axial angles Ω1 ,& Ω 2 at each end of an element.
iu ad s

Ω Ω

1

δ = 180 - (Ω1 + Ω 2 )
2

R

(0, 0, 0)

Radius

Chord factor (cf) = the element lengths calculated based on a radius of a non-dimensional unit of one for the spherical form with the end points of the elements coincident with the surface of the sphere.

Ifthe central angle

δ

is known, the chord factor may be calculated as follows:

cf = 2 sin

δ 2

The length of any element for larger structures may be found by the equation: I = cf x r where: r = the radius of the desired structural form I = the length of the new element Dihedral angle (beta β ) = an angle formed by two planes meeting in a common line. The two planes themselves arefaces of the dihedral angle, and the element is the common line. To measure the dihedral angle measure the angle whose vertex is on the element of the dihedral angle and whose sides are perpendicular to the element and lie one in each face of the dihedral angle. Face angle ( alpha α ) = an angle formed by two elements meeting in a common point and lying in a plane that is one of the faces of thepolyhedron.
face of the dihedral element of dihedral

face of the dihedral

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Face = any of the plane polygons making up the surface of the structural form.

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Principle polyhedral triangle (PPT) = any one of the plane equilateral triangles which form the faces of the regular polyhedron.

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Principle side (PS) = any one of the sides of...