Assorted Pictures and Movies

12_around_1 (JPEG 26k): Raytrace of 12 transparent spheres closepacked around 1, the template for Fuller's VE (Vector Equilibrium). Interconnecting sphere centers produces VE.

12_around_1.MOV (Quicktime Movie 109k): raytraced animation of the closepacked sphere cluster spinning.

icogeo shows how geodesic sphere is derived from icosa.

icosasphere 9 frequency geodesic sphere.

spinning icosasphere 9 frequency geodesic sphere movie (99K).

27 F icosa geodesic sphere 27F icosa geodesic dome buckeyball.JPEG (JPEG 30k): truncated icosahedron
Pentagonal faces result from truncating the 12 vertices of the icosa. Triangular faces of icosa become hexagonal as result of same truncation.
A geodesic sphere is simply the buckeyball with the pentagons and hexagons triangulated so that all vertices lie on a circumscribing sphere.

gold_icosa.JPEG (JPEG 97k): The icosahedron, one of only 3 regular triangular polyhedra, has 12 vertices and 20 equilateral triangular faces.

prime_structures.JPEG (JPEG 118k): RBF referred to the tetrahedron, octahedron, and icosahedron as "prime structures of Universe".
Since the triangle is the only (self-stabilizing)polygonal structure, only regular polyhedra formed from triangles are stable structures. It boggles the mind to think that there are only 3 extant.

3_FREQ_GEO.JPG (JPEG 72k): 3 frequency geodesic sphere
Frequency refers to the number of triangular subdivisions of the 20 original icosahedral faces

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The following are more geodesic sphere images:

Geomadonna.JPG (JPEG 135k)

geodesic_sphere.JPG (JPEG 78k)

geotriacon.JPG (JPEG 117k)

geotriacon2.JPEG (JPEG 57k)

The following images and animations relate icosahedron and VE (vector equilibrium):

rt2_Phi_icosa.JPEG (JPEG 41k): Shows the icosa sized to fit within the domain of the VE. Edge length of icosa is rt2/Phi (1.414/1.618) when edge length of VE is 1.

icosamatrix.GIF (GIF 18k): matrix of "boxed" icosa. Icosahedra oriented so that they fit within cells of cubic lattice.

icospin.MOV (Quicktime Movie 74k): shows above matrix spinning on vertical axis.

modicosa.GIF (GIF 18k): VE and icosa of unit edge length

modicosa.MOV (Quicktime Movie 302k): Transformation of VE to icosaphase of jitterbug. 8 triangular faces of VE rotate on 4-way axes inward to form icosa.

icophase.MOV (Quicktime Movie 360k): VE to octahedron jitterbug transformation with icosaphase highlighted.

CLOSEPACKED_SPHERE.JPG (JPEG 140k): Shows VE formed by interconnecting centers of 12 spheres closepacked around 1.

moVE.MOV (Quicktime Movie 275k): Animation showing VE with emphasis on 4 hexagonal planes.

The following show the IVM (isotropic vector matrix), RBF's division of space into octahedra and tetrahedra (or VE and octahedra):

ivm.jpg (JPEG 50k)

ivm2.jpg (JPEG 57k)

Kings Chamber (GIF 18k): geometrical highlights of Kings Chamber in the Great Pyramid of Cheops:
double square R.A.P. (right-angled parallelepiped) is kings chamber (yellow) long diagonal (not shown) of which is axis of circumscribing sphere (inner blue) which circumscribes icosa (red).
Phi rectangle of icosa(pink) has square as gnomon (purple) which is base of pyramid (half octa)with apex and 4 base corners on circumscribing sphere.

gnomon_GIF (GIF 11k): shows square as gnomon of Phi rectangle.

gnomonic_cube.GIF (GIF 12k): series of concentric cubes derived from Phi rectangle.

Octahedron

It seems that no one on the list has focused much on the octahedron lately, including me.

I have decided to go back and study it more.

As the mediato between the tetra and the icosa in our metaphycsical model, the octa may help us to better understand the other two.

6octa_1octa.JPEG shows that 6 little octa in the IVM form 1 big octa (of course there are interstitial tetra in the big octa but all of the big octa edge lengths are present from the little octa).

You can also see the VE by mentally truncating the big octa. The equilateral-triangulated hexagonal planes are also present in the big octa.

In the movie 6octa_1octa.MOV, I have positioned the big octa with 2 triangular faces perpendicular to the vertical.

Watching this, you can see that there are 3 "hexplanes" at 120 degree intervals inclined to a common central vertical axis (a fourth hexplane is perpendicular to the axis because we are really just looking at a VE from a different perspective).

Richard Hawkins