Editing Regular polytope (section)

===Cross polytopes (orthoplexes)===
{| class="wikitable" align="right" style="border-width:30%;"
|+ Graphs of the 2-orthoplex to 4-orthoplex.
|align=center|[[Image:2-orthoplex.svg|80px]]
|align=center|[[Image:3-orthoplex.svg|80px]]
|align=center|[[Image:4-orthoplex.svg|80px]]
|-
| [[Square (geometry)|Square]]
| [[Octahedron]]
| [[16-cell]]
|-
| [[Image:Kvadrato.svg|80px]]
| [[Image:Octahedron.svg|80px]]
| [[Image:Schlegel wireframe 16-cell.png|80px]]
|}
{{main|Orthoplex}}

These are the '''cross polytopes''' or '''orthoplexes'''. Their names are, in order of dimensionality:

:0. Point
:1. Line segment
:2. Square (regular tetragon)
:3. Regular [[octahedron]]
:4. Regular hexadecachoron ([[16-cell]]) ''or'' 4-orthoplex
:5. Regular triacontakaiditeron ([[pentacross]]) ''or'' 5-orthoplex
:... An ''n''-orthoplex has ''2n'' vertices.

The process of making each orthoplex can be visualised on a graph: Begin with a point ''O''. Extend a line in opposite directions to points ''A'' and ''B'' a distance ''r'' from ''O'' and 2''r'' apart. Draw a line ''COD'' of length 2''r'', centred on ''O'' and orthogonal to ''AB''. Join the ends to form a [[Square (geometry)|square]] ''ACBD''. Draw a line ''EOF'' of the same length and centered on 'O', orthogonal to ''AB'' and ''CD'' (i.e. upwards and downwards). Join the ends to the square to form a regular [[octahedron]]. This process is repeated further using new lines to form higher-dimensional orthoplices.