Cross Polytope |
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A Cross Polytope can be described as the 4-space member of a series
of dipyramids (2 pyramids joined at their bases). In general these dipyramids
are formed by selecting 2 points outside, and perpendicular to, the space of the
previous dipyramid, these new points being equidistant from all the vertices
of the previous dipyramid and connected to the vertices of the previous dipyramid.
(The 2 new points are not connected to each other.) Please see the table below. |
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Description |
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Beginning with 0 space, all figures may be represented by a point. |
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 | 2 | 1 |
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In 1 space, choose 2 points outside the 0 space, equidistant from
the original figure and connect them to the original figure forms a line. This
dipyramid has 2 vertices and a single edge. The original point is contained within the line and
therefore loses significance as a vertex. |
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 | 4 | 4 |
1 | | |
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In 2 space, choose 2 points outside the 1 space, equidistant
from the endpoints of the previous line, and connect them to the end points of the line forms a
square. This dipyramid has 4 vertices, and is enclosed by 4 edges. The original
line (the previous dipyramid) is contained within the new figure and therefore loses significance as an edge. |
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 | 6 | 12 |
8 | 1 | |
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In 3 space, choose 2 points outside the 2 space dipyramid,
equidistant from all the points in the previous square and connect them to the vertices of the square
forms an octahedron. This dipyramid has 6 vertices, 12 edges, and is enclosed by 8
triangular faces. The original square (the previous dipyramid) is contained within the new octahedron
and therefore loses significance as a bounding face. However, its edges still have significance as
edges of the octahedron. |
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 | 8 | 24 |
32 | 16 | 1 |
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In 4 space, choose 2 points outside the 3
space, equidistant from the vertices of the octahedron and connect them to the vertices of the octahedron
forms a Cross Polytope. This dipyramid has 8 vertices, 24 edges, 32 faces, and is enclosed
by 16 tetrahedrons. The original octahedron is completely contained within the new hyper-volume and therefore
loses its significance as a bounding volume. However its triangular faces become faces of a tetrahedra that
bounds this hyper-volume. |
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This progression may be extended into as may spaces as the viewer
cares to. For example the number of vertices of 5 space Cross
Polytope is 2 plus the number of vertices of the figure in the
next lower space (8+2=10). The number of edges is 2 times the
number of vertices plus the number of edges from the figure in
the next lower space (2*8+24=40). The number of faces is 2 times
the number of edges plus the number of faces from the figure in
the next lower space (2*24+32=80). The number of volumes is 2
times the number faces plus the number of volumes from the figure
in the next lower space (2*32+16=80). The number of hypervolumes
is 2 times the number of volumes from the figure in the next lower
space (2*16=32). You do not add the hypervolume form the previous
figure because it is absorbed by the new hyper-hypervolume that
is created. |
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