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The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). It is an analog of the planar triangle and solid tetrahedron. It is an example of a n-simplex. Its Schläfli symbol is {3,3,3}. Image File history File links Download high resolution version (640x640, 38 KB)Four symmetry views of the 5-cell polytope, edges drawn with an orthographic projection (z-w axes ignored). ...
For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
The symmetry group of an object (e. ...
In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...
In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ...
Geometry (Greek γεωμετÏία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...
A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...
For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
The pentachoron consists of five cells, all tetrahedra, and is self-dual. Its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
In mathematics, duality has numerous meanings. ...
In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ...
In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...
One of its possible projections into 2 dimensions is the pentagram inscribed inside a pentagon. A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ...
Both the vertex-first and cell-first parallel projection of the pentachoron into 3 dimensions have a tetrahedral envelope. The closest or farthest vertex of the pentachoron, respectively, projects to the center of the tetrahedron. The farthest/closest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the center.
The edge-first and face-first projections of the pentachoron into 3 dimensions have a triangular dipyramidal envelope. Two of the cells project to the upper and lower halves of the dipyramid, while the remaining 3 project to 3 non-regular tetrahedral volumes arranged around the central axis of the dipyramid at 120 degrees to each other. In geometry, the triangular dipyramid is a polyhedron made entirely out of 6 faces, which are all equilateral triangles, 9 edges, and 5 vertexes. ...
External links
- Eric W. Weisstein, Pentatope at MathWorld.
- Diagrams of 5-cell projections
| Convex regular 4-polytopes | | pentachoron | tesseract | 16-cell | 24-cell | 120-cell | 600-cell | | {3,3,3} | {4,3,3} | {3,3,4} | {3,4,3} | {5,3,3} | {3,3,5} | |
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