Pentatope (Simplex) |
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A Simplex can be described as the 4-space member of a class of
regular pyramids. In general these pyramids are formed by selecting
a point in the next higher space that is equidistant from all
the vertices in the pyramid of the current space and connecting
that point to all the vertices of the current vertices. The table
below describes the characteristics of pyramids in the first 4
dimensions. |
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Description |
| 1 | | | | |
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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, selecting a point outside the original 0 space and connecting it to
the vertex of the 0-space figure forms a line. This pyramid has
2 vertices and a single edge. |
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| 3 | 3 | 1 | | |
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In 2 space, selecting a point outside the space of the line which is equidistant from
the end points of the line and connecting it to the vertices of the line forms a
triangle. This pyramid has 3 vertices, and is enclosed by
3 edges. (Although this figure is an equilateral triangle, its image has
been skewed to match one of the faces in the next figure.) |
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| 4 | 6 | 4 | 1 | |
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In 3 space, selecting a point which is outside the 2 space of the triangle and
equidistant from its 3 vertices and connecting it to the vertices of the triangle
forms a Tetrahedron. This pyramid has 4 vertices, 6 edges,
and is enclosed by 4 triangular faces. Again the image is skewed. |
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| 5 | 10 | 10 | 5 | 1 |
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In 4 space, selecting a point outside the 3 space of the tetrahedron which is equidistant from
all the vertices of the tetrahedron and connecting that point to all the vertices of the Tetrahedron
forms a Simplex. This pyramid has 5 vertices, 10 edges, 10 faces, and is
enclosed by 5 tetrahedra. |
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This progression may be extended into as may spaces as the viewer
cares to. For example a 5 space Simplex will have 6 vertices,
(1 more than the number of dimensions in that figure's space).
The number of edges will be the number of vertices plus the number
of edges from the pyramid in the previous space (5+10=15). The
number of faces will be the number of edges plus the number of
faces from the pyramid in previous space (10+10=20). The number
of volumes would be the number of faces plus the number of volumes
from the pyramid in the previous space (10+5=15). The number of
hyper-volumes would be the number of volumes plus the number of
hyper-volumes in the pyramid of the previous space (5+1). |
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