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What is the Hypertetrahedron?
The hypertetrahedron is the four-dimensional tetrahedron. Other names
are simplex or pentatope.
Tetrahedrons in Perspective
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The four-dimensional tetrahedron belongs to the
world of thoughts. You can approach it through analogy considerations.
You go from the tetrahedron to the hypertetrahedron. This procedure is
legitimized by the "permanent principle", which is often used in mathematics
to get from the known to the unknown.
. .. |
If you connect the corners of a tetrahedron (1) with a fifth point
(2), then you get the perspective view of a hypertetrahedron (3). |
While a quadrilateral with the two diagonals can be seen as a perspective
view of a tetrahedron, a pentagon with the diagonals represents a model
of the hypertetrahedron.
Characteristics
of the Hypertetrahedron top
The hypertetrahedron has 5 corners (1 tetrahedron and the fifth point)
and 10 edges (1 tetrahedron with 6 edges and 4 connecting lines to the
fifth point).
The hypertetrahedron has 10 triangles.
The tetrahedron is covered by four triangles. In the same way five tetrahedrons
form the hypertetrahedron.
If you know the 3D-view, you can look at the hypertetrahedron three-dimensionally,
too.
4 tetrahedrons, 6 triangles and 4 edges meet at each corner.
3 tetrahedrons and 3 triangles meet at each edge.
2 tetrahedrons meet at each triangle.
Projections top
If you give the top view of a tetrahedron (=central projection) (1) and
a fifth point (red) inside (2) or outside (3), you get two models of the
hypertetrahedron. All points must have the same distances from each other
in "reality".
The hypertetrahedron 2 is a triangle with certain lines inside, the
hypertetrahedron 3 a quadrilateral. The model above is a pentagon.
You recognize in all models of the hypertetrahedrons, that four edges
meet at each corner.
Nets top
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If you spread out the tetrahedron, you get its net. Together the four
triangles have 4*3=12 sides. 2*3=6 sides (red) are bound. If you build
a tetrahedron, you must stick the remaining 6 sides in pairs. |
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If you spread out the hypertetrahedron, you get 5 tetrahedrons as its
net. Together the five tetrahedrons have 5*4=20 triangles. 2*4=8
triangles are bound. If you build a hypertetrahedron, you must stick the
remaining 12 triangles in pairs. |
Formulas top
The considerations above get a safer foundation
by formulas.
"Tetrahedrons" with the dimensions 1, 2 and 3
have the following characteristics.
You would have to determine the data of the hypertetrahedron
in the last line. Dimension=4, corners=5 is clear. The following law describes
how the sequences of the edges and triangles must be continued.
If you give n=4 into the terms of the chart, then
you get the following data for the hypertetrahedron.
The results for the 5-dimensional tetrahedron are:
Hypertetrahedron on
the Internet top
German
Marco Möller
Polytope
English
Eric W. Weisstein, (MathWorld)
Pentatope,
Simplex
Paul Bourke
Regular
Polytopes
NN
Pentatope,
Wikipedia
Pentatope,
References top
(1) Fritz Reinhardt, Heinrich Soeder: dtv-Atlas zur Mathematik I (Seite
172), München 1977
Feedback: Email address on my main page
This
page is also available in German
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2001 Jürgen Köller
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