Pentachoron

from Wikipedia, the free encyclopedia
Regular Pentachoron
(5-cell)
Schlegel wireframe 5-cell.png
Flail diagram
(corners and edges)
Type Regular poly choir
family simplex
Cells 5 ( 3.3.3 )Tetrahedron.png
Surfaces 10 {3}
edge 10
Corners 5
Corner figure ( 3.3.3 )
Schläfli icon {3,3,3}
Coxeter-Dynkin diagram CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
Symmetry group A 4 , [3,3,3]
properties convex
Corner figure: tetrahedron

A pentachōron ( that , from ancient Greek πεντα- penta- , prefix form of πέντε pénte 'five', and χῶρος chōros 'space'; also 5-cell , pentatope , four-dimensional hyperpyramid or four- dimensional hyperpyramid called) is a four-dimensional hyperpyramid as "a tetrahedron" Base ”, or a 4- simplex , the simplest polychoron (four-dimensional figure). It consists of five tetrahedral cells and is the analog of the triangle (2-simplex) and the tetrahedron (3-simplex).

The regular Pentachoron is one of the six regular, convex Polychora (the six Platonic solids in 4-dimensional space ) and is represented by the Schläfli symbol {3,3,3}.

geometry

A pentachoron consists of five cells, all of which are tetrahedral. Its corner figure is a tetrahedron. Its maximum intersection with three-dimensional space is the triangular prism.

photos

5-cell.gif
A 3-D projection of a 5-cell that performs a double rotation around two orthogonal planes. 		Cell5-4dpolytope.png
Four orthographic projections.

construction

A pentachoron can be constructed by adding a fifth corner to a tetrahedron that is the same distance from the other corners as the other corners from each other (essentially, a pentachoron is a four-dimensional pyramid with a tetrahedral base).

Projections

One possible projection of the pentachoron is a pentagram within a pentagon .

The corner-first and cell-first parallel projections of the pentachoron in three dimensions have a tetrahedral shell. The next or most distant cell is projected onto the tetrahedron itself, while the other four cells are mapped onto the four compressed tetrahedral regions that surround the center.

The edge-first and surface-first projection of the pentachoron in three dimensions have a double triangular pyramid-shaped envelope. Two of the cells are projected onto the upper and lower halves of the double pyramid, while the other three are projected onto three non-regular tetrahedral bodies around the central axis of the double pyramid at angles of 120 ° to each other.

  • Projections in the 3-dimensional space

  • Corner-first projection

  • Edge-first projection

  • Surface-first projection

  • Cell-first projection

Web links