Lesson plan
Hypercone
of a spherical cone's generating lines (red), parallels (green) and hypermeridians (blue). Because stereographic projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone. The hypermeridians generates a set of concentric spheres. In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation : x^2 + y^2 + z^2 - w^2 = 0. It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named "spherical cone" because its intersections with hyperplanes perpendicular to the w-axis are spheres. A four-dimensional right hypercone can be thought of as a sphere which expands with time, starting its expansion from a single point source, such that the center of the expanding sphere remains fixed. An oblique hypercone would be a sphere which expands with time, again starting its expansion from a point source, but such that the center of the expanding sphere moves with a uniform velocity.
Article summary
of a spherical cone's generating lines (red), parallels (green) and hypermeridians (blue). Because stereographic projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone. The hypermeridians generates a set of concentric spheres. In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation : x^2 + y^2 + z^2 - w^2 = 0. It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named "spherical cone" because its intersections with hyperplanes perpendicular to the w-axis are spheres. A four-dimensional right hypercone can be thought of as a sphere which expands with time, starting its expansion from a single point source, such that the center of the expanding sphere remains fixed. An oblique hypercone would be a sphere which expands with time, again starting its expansion from a point source, but such that the center of the expanding sphere moves with a uniform velocity.
Key claims
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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.