| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 523.1 | Rigor gives way to brevity | MODEL::YARBROUGH | | Thu Jun 26 1986 10:04 | 6 |
| Since at each vertex at least three 2d figures meet, the interior
angles of the 2d figures must be < 120 deg., which limits the 2d
figures to 3, 4, or 5 sides. The 3d vertices may thus consist of
3,4, or 5 triangles; 3 squares; or 3 pentagons. The shape of each
vertex determines the whole figure, so there are exactly five such
figures.
|
| 523.2 | Semi-perfect solids? | JON::MORONEY | Madman | Thu Jun 26 1986 14:06 | 9 |
| Now, can anyone describe all of what I call "semi-perfect" 3-d polyhedra?
By these I mean all 3-d convex polyhedra, whose faces are all identical,
but they need not be perfect 2-d polygons. The 5 polyhedra mentioned
in .0 are five of them, but what others are there? Ignore distortions where
the object could be lengthened/shortened along an axis and still meet these
requirements (for example, an octahedron may be compressed/stretched along an
axis and still have 8 identical faces)
-Mike
|
| 523.3 | Reminds me of the circus... | MODEL::YARBROUGH | | Thu Jun 26 1986 17:30 | 15 |
| A simple way of generating an infinite set of such polyhedra is:
Start with one of the 5 polyhedra in .0, then at the center of each
face erect a very short perpindicular to the face. From the top
of the perp. connect to each of the surrounding edges. This generates
a new figure with identical isoceles triangles as faces. One can
also connect the perps. to the center of each surrounding edge to
form a figure made of identical right triangles. Having done that,
one can add perpindiculars at the centers of each edge, which makes
the figure somewhat rounder and may make the triangles scalene.
It's all somewhat like raising a big tent from inside.
In each case the height of the perps. determines the angles in the
triangles.
|
| 523.4 | Not quite... | JON::MORONEY | Madman | Fri Jun 27 1986 11:34 | 5 |
| re .3: That'll only work once, only when starting with a "perfect" solid,
since if you repeat the process on the resulting figure, the faces will not be
identical.
-Mike
|
| 523.5 | Once is sufficient | MODEL::YARBROUGH | | Mon Jun 30 1986 09:24 | 4 |
| I didn't mean to imply that the process was recursive, only that
the heights of the perpindiculars are continuously variable. Each
height produces a slightly different figure with the same topology
but different ratios among the sides.
|