Neat pics

Cf.; to rotate it click here and use your mouse (hold down the left key).


You may wonder what to call
the figure shown in the top row
in various stages of undress.
Well, if you like Latin
you may call it the “stella octangula”;
if you prefer English
try “stellated octahedron”;
and if you like a systematic naming scheme
that gives
lots of numerical information about it
and admits many useful generalizations
(including to arbitrarily high dimension;
it is really a very powerful notation),
try “{4,3}[2{3,3}]{3,4}”.
The last is the version for compounds
of notation due to Herr Schläfli,
and is therefore called a
Schläfli symbol”.
What those numbers signify, roughly, in English is
a cube {4,3} containing (circumscribed around)
two reciprocal (i.e., dual) tetrahedra {3,3} which intersect in
an octahedron {3,4}
(so three of the five Platonic solids appear in it).
Conglomerates of this nature are generally called compounds,
more specifically, polyhedral compounds.

Here is a big brother of the
{4,3}[2{3,3}]{3,4} cube/tetrahedra/octahedron compound:

This consists of
an outside dodecahedron {5,3}
whose 20 vertices are also the vertices of
five tetrahedra {3,3}s which intersect in an
icosahedron {3,5}.
Its Schläfli symbol is, fairly naturally,
In this figure, each of the five tetrahedra has its own color,
red, blue, light blue, green, or brown.
The edges of each tetrahedron are in that color,
as well as
the faces of the icosahedron which arise from that tetrahedron.

Here is what the five tetrahedra look like,
without the surrounding (circumscribed) dodecahedron
(that is, the compound of five tetrahedra),
first as (colored) solids, then as thickened outlines:

And here is the compound being built up,
starting with the core icosahedron and adding one (or a few) tetrahedra at a time:

You will find everything you ever wanted to know about such things
in Regular Polytopes
by H. S. M. Coxeter
(a real classic, first published in 1947;
it has its own Wikipedia entry).
The Schläfli symbols for compounds (like the above)
are explained in Section 3.6.

The illustrations above are a selection of those from
Glimpses of Algebra and Geometry by Gabor Toth.
Only about fifteen percent of that book is devoted to such solids,
but the approach there seems more user-friendly than Coxeter.

Here is a question:
If the twenty vertices of the dodecahedron {5,3} can be divided into
the vertices of five tetrahedrons (each tetrahedron {3,3} having four vertices),
what can be done about dividing them into the vertices of cubes {4,3}?
Well, it turns out they can be divided also into the vertices of five cubes.
Since each cube has eight vertices, making a total of forty cube vertices,
each dodecahedron vertex is the vertex of two cubes.
Here is the picture:

To see the cubes, note that the white pentagonal face of the dodecahedron
has five diagonals (making a five-pointed star), each colored a different color.
Those five diagonals are edges of the five cubes.
In fact, the same is true of each of the twelve faces of the dodecahedron:
Each has five diagonals, making a total of sixty diagonals.
But since each inscribed cube has twelve edges,
the five cubes also have sixty edges.
How about that!
The Schläfli symbol for this is
The {5,3} denotes the outer dodecahedron,
the 5{4,3} denotes the five inscribed cubes,
while the 2 prefixing the {5,3} denotes, not that there are two of them,
but that each vertex of the {5,3} is also the vertex of two {4,3}s.
Another way of viewing this is that it gives the following (valid) equation:
2 × (20 vertices per {5,3}) = 40 = (5{4,3}) × (8 vertices per {4,3}).
Again, to read about the Schläfli symbols for compounds such as this,
see Section 3.6 of Coxeter.

Oh, and here are two representations of the
compound of five cubes,
that is, the 5{4,3} compound without the circumscribing {5,3}.
The one on the left is by Toth,
and colors the five cubes red, two shades of blue, green, brown;
the one on the right is by George Hart,
and colors the five cubes red, blue, green, yellow, brown.

Note how each vertex of the {5,3} dodecahedron has a pair of colors.
There are 5 × 4 = 20 possible ordered pairs (without repetition)
of the five colors.
Each such ordered pair appears on exactly one
of the 20 vertices of the {5,3} dodecahedron.
Reversing the order of the colors corresponds to going to the antipodal vertex;
e.g., in the right hand picture
the blue/green vertex is the antipode of the green/blue vertex.

Web Sites

Here are some web sites that look interesting:
“The Encyclopedia of Polyhedra” by George W. Hart
“A Ride Through the Polyhedra World” by Maurice Starck
“Platonic Solids in All Dimensions” by John Baez

or just see what Google finds for:
regular polyhedra

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Euler’s classification (1776) of isometries in dimensions ≤ 3.

Euler’s classification (1776)
of isometries in dimensions ≤ 3
Fixed Point?
Yes No
OP rotations ρ (dim ≥ 2) translations τ (dim ≥ 1)
screw (dim ≥ 3)
OR reflections σ (dim ≥ 1)
rotary reflections (dim ≥ 3)
glide reflections γ (dim ≥ 2)

  1. OP = orientation-preserving = even = direct = Isom+
    OR = orientation-reversing = odd =opposite = Isom-

  2. Each non-identity isometry is in precisely one of the above categories.

  3. Wikipedia discusses:
    Classification of Euclidean plane isometries and
    Overview of isometries in up to three dimensions.

  4. Reflections
    More precisely, we mean reflections in a hyperplane of codimension 1
    (thus in 2-space, in a line, in 3-space, in a plane)
    with respect to the orthogonal line to the hyperplane.

  5. Rotations in 2-space are with respect to a point,
    in 3-space with respect to a line.
    Thus rotations are always within a plane,
    about the orthogonal complement.
    Of course, in dimensions ≥ 4 things get more complicated,
    and are best discussed in terms of
    the decomposition of orthogonal operators studied in linear algebra.
    This link, if it is still active, gives the relevant theorem.
    There is also Wikipedia’s discussion of the orthogonal group,
    but that covers much more than is immediately relevant.

  6. Glide reflections:
    The direction of the glide is always within the reflecting hyperplane:
    in 2-space the reflecting line, in 3-space the reflecting plane.
    Glide reflections in 3-space are sometimes called
    glide plane operations”.

  7. Screw = rotary translation = glide rotation

  8. Inversions
    By inversion
    we mean what is sometimes called
    “central inversion” or “inversion in a point.”
    In 2-space, an inversion through a point
    is just a half-turn about the point, thus a rotation.
    In 3-space, for inversion through a point
    we may choose any line through that point,
    perform a half-turn rotation around that line
    combined with a reflection in
    the plane orthogonal to that line and through that point,
    thus we have a rotary reflection.
    This example shows that although each isometry
    is in one and only one of the above categories,
    its precise expression within that category is not unique.

  9. The familiar (from high school geometry) property of
    congruence of figures in the plane
    simply means that
    one figure is the image of the other under an isometry.

  10. There is doubt as to how much of that classification
    Euler really discovered.
    I’m giving him the benefit of the doubt,
    following Coolidge’s History of Geometrical Methods
    as cited in George Martin’s Transformation Geometry.

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