We have just obtained a set of coordinates for the vertices of a regular three-dimensional octahedron thought of as the middle slice of a hypercube in four-dimensional space. It is also easy to give a three-dimensional coordinate description for the octahedron by taking advantage of the fact that the octahedron is the dual of the cube: the vertices of a regular octahedron can be obtained as the centers of the six square faces of a cube. If we choose coordinates for the vertices of the cube, we can figure out the coordinates of the centers of square faces and obtain the coordinates for the vertices of the octahedron. In the previous paragraph, we studied a cube with vertices having all coordinates 0 or 1, but in considering the dual, it turns out to be more convenient to start with a cube centered at the origin having all coordinates -1 or 1.

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Coordinates for the octahedron. |

What about the other regular polyhedra in three-space? We can exploit
the symmetry of the icosahedron to come up with some fairly
satisfactory coordinates, involving just one irrational number. And a
very important number it is. We start by observing that for every edge
of the icosahedron there is a parallel edge on the opposite side. We
may situate the icosahedron in the cubical box so that all 12 vertices
are on the boundary of the box and in fact so that the intersection
with the boundary of the box consists of six edges parallel to the
coordinate axes. As provisional coordinates for these segments, we may
choose (±1, 0, ±*t*), (0,
±*t*, ±1), and (±*t*,
±1, 0), where *t* is a number to be chosen later. In
general, the polyhedron with these 12 vertices will have edges of two
lengths: 2*t* for the edges on the boundary of the box and
(1 + t^{2} + (1-*t*)^{2})^{1/2} for the other
edges. In order for the icosahedron to be regular, the lengths of all
these segments should be equal. This condition leads to the algebraic
equation *t*^{2} + *t* - 1 = 0, with positive
solution *t* = (-1 + 5^{1/2})/2. This important number appears in
all sorts of contexts involving the concepts of growth and form. It is
called the *golden ratio*, and it expresses the ratio of a side of a
regular pentagon to one of its diagonals. We should not be surprised
to see it appear in conjunction with the regular icosahedron since the
vertex configuration of each corner of this object is a regular
pentagon.

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Choosing different values of |

It is possible to get a set of coordinates of the regular dodecahedron
by taking the centers of the faces of the icosahedron given above, but
we can find the coordinates more directly by exploiting another
relationship with the cube. We start with a particular diagonal of a
pentagonal face of the dodecahedron, then choose diagonals in the
adjacent faces so that the three diagonals meeting at a vertex are
mutually perpendicular and all of the same length. If we continue this
procedure, we obtain 12 diagonals fitting together to form the edges
of a cube inscribed in the dodecahedron, with one edge for each of the
12 faces of the dodecahedron. By starting with different diagonals of
the original pentagon, we end up with five different cubes; in this
collection of cubes, each of the 60 diagonals of the dodecahedron is
used exactly once. By choosing the 8 vertices of the cube first, we
may use the symmetry of this figure to find coordinates for the other 12 vertices of the dodecahedron. If the vertices of the cube are (±1, ±1, ±1), then the others are of the form (±*t*, 0, ±1/*t*), (0, ±1/*t*, ±*t*), and (±1/*t*, ±*t*, 0), where *t* is the same number appearing in the coordinates for the icosahedron.

Coordinates for Regular Polytopes | ||

Table of Contents | ||

Coordinates for Hypercube Slices |