lab@qubix.UUCP (Q-Bick) (10/02/84)
I guess I qualify as an engineer. During my first visit to EPCOT (8 Oct
1982), I studied the ball in an attempt to determine its basic geometric
construction. I don't know if I still have the notes, so I'll pass along
what I can remember.
Side point to return to later: each triangle of the geodesic dome is
covered by a triangular pyramid, thus multiplying the total number of
triangles on the dome by 3. Back to the basic dome.
The ground perspective made it difficult to correlate with the most
obvious similar structure - an icosahedron. As you enter EPCOT, you can
immediate see a couple of super-triangles, 8 units per side, on the
front. Further, there is a trough extend from the top of one to the top
of the dome. At that vertex and at the ends of common edge of the first
two triangles, you can see the intersection of FIVE basic triangles.
But waitaminnit - if the dome was based on an icosahedron, I should not
have been able to see as much as I did above and below those first two
triangles (even accounting for curvature). Hmmm...new theory is called for.
It became apparent that there had to be places where SIX curved
triangular faces met. With planar faces, this would be obvious; the
curved faces could conceal this well. I came up with a 30-face
construction; with 8 units per side, this put 1080 on the basic dome and
3240 on the entire ball, minus a few for the base.
Will check my EPCOT photos (I have some from both sides of the dome) and
report back later. However, I am also notoriously absent-minded - if you
don't hear anything in a week or two, nag me about it. (BTW, although
disk cameras are rotten for capturing facial detail, they did VERY well
with the ridges and troughs on the dome!)
--
The Ice Floe of Larry Bickford
{amd,decwrl,sun,idi,ittvax}!qubix!lab
You can't settle the issue until you've settled how to settle the issue.lab@qubix.UUCP (Q-Bick) (10/13/84)
Took a look at my photographs (and at the cover of the EPCOT guide
provided by Eastern Airlines) and reconstructed it, finding a
dodecahedron a better and simpler basis than an icosahedron.
Simply: start with a dodecahedron. Cover each face with a pentagonal
pyramid, where each non-base face is an equilateral triangle. You now
have 60 triangles. Divide each edge into 8 units and connect two edges
of each triangle with a line parallel to the third edge. You now have 1+
(2+1)+(3+2)+(4+3)+(5+4)+(6+5)+(7+6)+(8+7)=64 subtriangles. Put a
tetrahedron on each subtriangle.
3 exposed faces 64 tetrahedra 60 major triangles
--------------- * ------------- * ---------- = 11520
tetrahedron major triangle dome faces on the dome
For the purist looking for the icosahedron, its vertices are easy to
find - wherever five triangles meet (fairly easy to find on the dome).
The edges are totally hidden, as they pass through the ALTITUDES of the
triangles (== apexes of the tetrahedra). However, the troughs are such a
noticeable feature that you would probably never see the icosahedron,
even if you were looking for it. (For that matter, the curvature of the
sphere hides conceals the transition from one pentagon to the next,
concealing even *their* existence.)
BTW, the 30-face value in my former article was the number of skew
equilateral rhombi (two non-coplanar equilateral triangles with a common
edge) covering the dome. The acute ends of the rhombi are the apexes of
the pentagonal pyramids - where 5 triangles meet on the dome. The major
axes of the rhombi are the edges of the icosahedron; their minor axes,
those of the dodecahedron.
--
The Ice Floe of Larry Bickford
{amd,decwrl,sun,idi,ittvax}!qubix!lab
You can't settle the issue until you've settled how to settle the issue.