T.R | Title | User | Personal Name | Date | Lines |
---|
1931.1 | | EVMS::HALLYB | Fish have no concept of fire | Tue Jan 24 1995 19:33 | 9 |
| Do you have any requirements on this "shape"?
There are 5 regular polyhedra, i.e., solid shapes that you can pack
densely in 3 dimensions.
Or would you be satisfied with something more irregular, but which
fit your sphere better? How about something curvy?
John
|
1931.2 | regular polyhedra ok | RANGER::CACCAVALE | | Tue Jan 24 1995 19:59 | 4 |
| regular polyhedra would be fine. Could you please list for me ?
Thanks,
Frank
|
1931.3 | | WIBBIN::NOYCE | Alpha's faster: over 4.2 billion times (per minute) | Tue Jan 24 1995 20:50 | 9 |
| The 5 regular polyhedra are
tetrahedron (triangular pyramid)
cube
octahedron (2 square pyramids glued base-to-base)
dodecahedron (12 pentagons)
icosahedron (20 triangles)
The more sides, the closer fit around the sphere (I assume).
Sorry, I can only derive formulas for the first 3.
|
1931.4 | Formulas for the regular polyhedra | BALZAC::QUENIVET | Margins are often too small. | Tue Jan 31 1995 14:13 | 90 |
1931.5 | | FORTY2::PALKA | | Tue Jan 31 1995 15:18 | 29 |
| re .1
>>> There are 5 regular polyhedra, i.e., solid shapes that you can pack
>>> densely in 3 dimensions.
That's not the usual definition of the regular polyhedra. Usually they
are described as having a number of identical faces and vertices, each
face being a regular polyhedron.
If you want a better shape then you can modify an icosohedron. Imagine
cutting a tiny bit off each of the corners. You modify each triangular
side into a hexagon, with 3 large sides and 3 small sides. Each corner
is now a small pentagon. Now take more off the corners, increasing the
size of the pentagons, shrinking the large sides of the hexagons, and
increasing the small sides of the hexagons. When the small sides of the
hexagons are the same size as the large sides stop. You now have a
shape with 20 hexagons and 12 pentagons. This shape fits a sphere
better than the original icosahedron. It should be easy enough to
calculate the volume of this shape - it is the same as the icosahedron
less the pentagonal pyramids taken from each corner. I expect the same
sphere will fit inside this truncated icosahedron.
You can improve on this shape further, by gluing another pyramid on to
each hexagon and pentagon. These pyramids are very flat, but allow a
slightly larger sphere to be contained in the shape. I'll leave it to
someone else to determine how high these pyramids ought to be, and the
size of the sphere that can be contained.
Andrew
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1931.6 | models available at toy & sports stores | RANGER::BRADLEY | Chuck Bradley | Tue Jan 31 1995 16:32 | 5 |
|
in the u.s., the technical name for the construction in .5 is soccer ball.
in the rest of the world it is a football.
close enough to a sphere for recreational work.
|
1931.7 | | AUSSIE::GARSON | achtentachtig kacheltjes | Tue Jan 31 1995 23:41 | 8 |
| re .4
The English word that you are looking for is "circumscribed".
re .6
Don't count Australia in with the rest of the world on that one. Here a
real football is approximately ellipsoidal. ;-)
|
1931.8 | 60 faces? | HDLITE::GRIES | | Thu Mar 09 1995 18:51 | 4 |
| Is there not a regular polhedra with 60 faces, 90 edges, and 32
vertices? the buckey ball?
|
1931.9 | | RUSURE::EDP | Always mount a scratch monkey. | Thu Mar 09 1995 19:12 | 10 |
| Re .8:
It's not regular.
-- edp
Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.
|
1931.10 | Two different buckyballs | WIBBIN::NOYCE | Alpha's faster: over 4.2 billion times (per minute) | Fri Mar 10 1995 16:49 | 20 |
| To be precise, some of the faces (12?) are regular pentagons, and the
rest are regular hexagons.
Re .5
The other way to modify the icosahedron is to connect the midpoints of
adjacent edges, so that each triangular face is replaced with four
triangles:
/\
/ \
/----\
/ \ / \
/___\/___\
The center triangle is still tangent to your sphere, but the three original
vertices can be brought closer to the sphere's center, until the three outer
triangles are also tangent to the sphere.
You can repeat this recursively, to get a volume that's arbitrarily close to
the sphere.
|