Editing Regular polytope (section)

===Measure polytopes (hypercubes)===
{|class="wikitable" align="right" style="border-width:30%;"
|+ Graphs of the 2-cube to 4-cube.
|align=center|[[Image:Cross graph 2.svg|80px]]
|align=center|[[Image:Cube graph ortho vcenter.png|80px]]
|align=center|[[Image:Hypercubestar.svg|80px]]
|-
| [[Square (geometry)|Square]]
| [[Cube]]
| [[Tesseract]]
|-
| [[Image:Kvadrato.svg|80px]]
| [[Image:Hexahedron.svg|80px]]
| [[Image:Schlegel wireframe 8-cell.png|80px]]
|}
{{main|Hypercube}}

These are the '''measure polytopes''' or '''hypercubes'''. Their names are, in order of dimension:

:0. Point
:1. Line segment
:2. [[Square (geometry)|Square]] (regular tetragon)
:3. [[Cube]] (regular hexahedron)
:4. [[Tesseract]] (regular octachoron) ''or'' 4-cube
:5. [[Penteract]] (regular decateron) ''or'' 5-cube
:... An ''n''-cube has ''2<sup>n</sup>'' vertices.

The process of making each hypercube can be visualised on a graph: Begin with a point ''A''. Extend a line to point ''B'' at distance ''r'', and join to form a line segment. Extend a second line of length ''r'', orthogonal to ''AB'', from ''B'' to ''C'', and likewise from ''A'' to ''D'', to form a [[Square (geometry)|square]] ''ABCD''. Extend lines of length ''r'' respectively from each corner, orthogonal to both ''AB'' and ''BC'' (i.e. upwards). Mark new points ''E'',''F'',''G'',''H'' to form the [[cube]] ''ABCDEFGH''. This process is repeated further using new lines to form higher-dimensional hypercubes.