File:Versor action on Hurwitz quaternions.svg
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Summary[edit]
DescriptionVersor action on Hurwitz quaternions.svg |
English: Left quaternion multiplication as an action on S3 of unit quaternions. This depiction of S3 shows Hurwitz quaternions and the 2-sphere of unit 3-vectors (quaternions with zero scalar part), visible as the yellow plane. The action shown is v ↦ 1 + i + j + k/2v.
The projection of S3 to the affine 3-space is quasi-stereographic with the pole at −i + j + k/√3, but generally irregular. It should be a stereographic projection on the 2-sphere of 3-vectors, it is a gnomonic projection in four overlapping 9- and 7-points regions around 1, 1 + i + j + k/2, −1 + i + j + k/2, and −1, and it is unspecified in the surrounding space of S3 (from where some curved arrows are mapped). Because of this piecewise compromise projection octahedral cells of the polychoron look distorted. 24 unit Hurwitz quaternions form vertices of a 24-cell and 4 cyclic orbits (grey, blue, purple, orange) of 6 elements each. All 24 elements are depicted (as balls), but only 59 polychoron’s edges of 96 total are visible as straight line segments, and yet 13 show as curved lines and arrows. 24 edges are completely missing, although 2 of them can be inferred if one closes white and orange cycles. The red roundabout is the invariant great circle of 3-vectors: see w:Talk:Versor for details. The 2-sphere of 3-vectors features a rhombic dodecahedron (green edges, yellow faces) formed by the polychoron’s faces and cells respectively. 9 its faces of 12 are visible: in six of them the invariant circle passes, whereas three inner rhombi near the vertical axis look as a Reuleaux triangle (due to the stereographic projection most “straight” segments on S2 project to circle arcs of various curvature). |
Date | |
Source | Own work |
Author | Incnis Mrsi |
Other versions |
action of the same versor, but in the adjoint representation. |
SVG development InfoField |
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 12:55, 8 October 2016 | 600 × 690 (12 KB) | Incnis Mrsi (talk | contribs) | clean SVG, <title>s | |
19:34, 28 January 2014 | 600 × 690 (11 KB) | Incnis Mrsi (talk | contribs) | better: rhombic dodecahedron feature, slightly improved geometry, more 1 arrow | ||
15:18, 28 January 2014 | 600 × 690 (10 KB) | Incnis Mrsi (talk | contribs) | changed magenta to purple, more edges of the 24-cell and some details about its intersection with S², but generally still far from perfect | ||
19:24, 27 January 2014 | 600 × 690 (9 KB) | Incnis Mrsi (talk | contribs) | upgraded to a usable version | ||
22:17, 26 January 2014 | 640 × 690 (6 KB) | Incnis Mrsi (talk | contribs) | {{Information |Description ={{en|1=(Uploaded an alpha version; see w:Talk:Versor. The geometry is imprecise and possibly incorrect in several places.) Left multiplication as an action on unit w:Hurwitz quaternions and also some 3-vectors (q... |
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Width | 600 |
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Height | 690 |