Lesson plan
Hyperpyramid
In geometry, a hyperpyramid is a generalisation of the normal pyramid to dimensions. In the case of the pyramid one connects all vertices of the base (a polygon in a plane) to a point outside the plane, which is the peak. The pyramid's height is the distance of the peak from the plane. This construction gets generalised to dimensions. The base becomes a -polytope in a -dimensional hyperplane. A point called apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height. This construct is called a -dimensional hyperpyramid. A normal triangle is a 2-dimensional hyperpyramid, the tetrahedron or triangular pyramid is a 3-dimensional hyperpyramid, and the pentachoron or tetrahedral pyramid is a 4-dimensional hyperpyramid with a tetrahedron as base. The -dimensional volume of a -dimensional hyperpyramid can be computed as follows: V_n = \frac{A \cdot h}{n} Here denotes the -dimensional volume of the hyperpyramid, the -dimensional volume of the base and the height, that is the distance between the apex and the -dimensional hyperplane containing the base . For the formula above yields the standard formulas for the area of a triangle and the volume of a pyramid.
Article summary
In geometry, a hyperpyramid is a generalisation of the normal pyramid to dimensions. In the case of the pyramid one connects all vertices of the base (a polygon in a plane) to a point outside the plane, which is the peak. The pyramid's height is the distance of the peak from the plane. This construction gets generalised to dimensions. The base becomes a -polytope in a -dimensional hyperplane. A point called apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height. This construct is called a -dimensional hyperpyramid. A normal triangle is a 2-dimensional hyperpyramid, the tetrahedron or triangular pyramid is a 3-dimensional hyperpyramid, and the pentachoron or tetrahedral pyramid is a 4-dimensional hyperpyramid with a tetrahedron as base. The -dimensional volume of a -dimensional hyperpyramid can be computed as follows: V_n = \frac{A \cdot h}{n} Here denotes the -dimensional volume of the hyperpyramid, the -dimensional volume of the base and the height, that is the distance between the apex and the -dimensional hyperplane containing the base . For the formula above yields the standard formulas for the area of a triangle and the volume of a pyramid.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.