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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2015
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   Author: Mike Shulman
   Format: MarkdownItexI made [[projective plane]] less stubby.

   I made projective plane less stubby.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeMay 6th 2015
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   Author: Zhen Lin
   Format: MarkdownItexA colleague of mine worked on a synthetic theory of projective planes over local rings for his thesis. If I recall correctly, he needed to add a variant of Pappus's theorem as an axiom.

   A colleague of mine worked on a synthetic theory of projective planes over local rings for his thesis. If I recall correctly, he needed to add a variant of Pappus’s theorem as an axiom.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 6th 2015
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   Author: Mike Shulman
   Format: MarkdownItexIs there anything available about it?

   Is there anything available about it?

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 7th 2015
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   Author: DavidRoberts
   Format: MarkdownItexWasn't that presented at CT2014?

   Wasn’t that presented at CT2014?

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 7th 2015
   • (edited May 7th 2015)
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   Author: DavidRoberts
   Format: MarkdownItexThere's also the [result of Ruth Moufang](http://en.wikipedia.org/wiki/Moufang_plane) about coordinatising projective planes satisfying (edit: _little_) Desargues with alternative division rings. She discovered $OP^2$ this way, I gather.

   There’s also the result of Ruth Moufang about coordinatising projective planes satisfying (edit: little) Desargues with alternative division rings. She discovered $OP^2$ this way, I gather.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeMay 8th 2015
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   Author: Mike Shulman
   Format: MarkdownItexOne thing I don't yet understand: a projective plane without any Desargues' theorem can be constructed/coordinatized using a "ternary ring" (which isn't actually a ring at all, but an algebraic gadget whose basic operation is ternary, thought of as like $(a,b,c) \mapsto a b + c$). Is the construction of $O P^2$ a special case of that?

   One thing I don’t yet understand: a projective plane without any Desargues’ theorem can be constructed/coordinatized using a “ternary ring” (which isn’t actually a ring at all, but an algebraic gadget whose basic operation is ternary, thought of as like $(a,b,c) \mapsto a b + c$). Is the construction of $O P^2$ a special case of that?

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 8th 2015
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   Author: DavidRoberts
   Format: MarkdownItex$OP^2$ is a Moufang plane, so little Desargues' holds ([Theorem 3.8 here](http://math2.uncc.edu/~frothe/3181alleuclid1_3.pdf)). This is strictly weaker than the usual Desargues' theorem. Thanks for pointing out [planar ternary rings](http://en.wikipedia.org/wiki/Planar_ternary_ring). My [officemate](http://blogs.adelaide.edu.au/maths-learning/) as a PhD student was a finite geometer, so I learned a bit by osmosis about non-Desarguesian planes, but not about these.

   $OP^2$ is a Moufang plane, so little Desargues’ holds (Theorem 3.8 here). This is strictly weaker than the usual Desargues’ theorem. Thanks for pointing out planar ternary rings. My officemate as a PhD student was a finite geometer, so I learned a bit by osmosis about non-Desarguesian planes, but not about these.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 8th 2015
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   Author: David_Corfield
   Format: MarkdownItexDid John Baez's account cover this? Here he is on [Octonionic Projective Geometry](http://math.ucr.edu/home/baez/octonions/node8.html).

   Did John Baez’s account cover this? Here he is on Octonionic Projective Geometry.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 8th 2015
   • (edited May 8th 2015)
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   Author: DavidRoberts
   Format: MarkdownItexJB says _in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions_ Here's Moufang's paper he cites: >Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207-222. doi:[10.1007/BF02940648](http://dx.doi.org/10.1007/BF02940648)

   JB says

   in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions

   Here’s Moufang’s paper he cites:

   Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207-222. doi:10.1007/BF02940648

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 8th 2015
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    Author: Mike Shulman
    Format: MarkdownItexRight, I know all that. Since $O P^2$ is non-Desarguesian, we can construct from it a ternary ring that coordinatizes it --- but my question is, can the *original* construction of $O P^2$ from $O$ be obtained by way of regarding $O$ as a ternary ring? I presume so, but in that case it surprises me that hardly anyone seems to mention this when discussing $O P^2$, so I end up with doubt.

    Right, I know all that. Since $O P^2$ is non-Desarguesian, we can construct from it a ternary ring that coordinatizes it — but my question is, can the original construction of $O P^2$ from $O$ be obtained by way of regarding $O$ as a ternary ring? I presume so, but in that case it surprises me that hardly anyone seems to mention this when discussing $O P^2$, so I end up with doubt.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 8th 2015
    • (edited May 8th 2015)
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexFrom the wikipedia article on Moufang planes I linked to, we find characterisation of them: >Some ternary ring of the plane is an alternative division ring. >P is isomorphic to the projective plane over an alternative division ring. >Also, in a Moufang plane: >Any two ternary rings of the plane are isomorphic. Since O, as a ternary ring, is an alternating ring, any other ternary ring coordinatising $OP^2$ is isomorphic to O. When you say the original construction, I guess one would have to dig up either a translation of Moufang's construction, or read it in German. My guess is that the ternary ring version reproduces Moufang's result in the alternating ring case in general, not just for O.

    From the wikipedia article on Moufang planes I linked to, we find characterisation of them:

    Some ternary ring of the plane is an alternative division ring.

    P is isomorphic to the projective plane over an alternative division ring.

    Also, in a Moufang plane:

    Any two ternary rings of the plane are isomorphic.

    Since O, as a ternary ring, is an alternating ring, any other ternary ring coordinatising $OP^2$ is isomorphic to O.

    When you say the original construction, I guess one would have to dig up either a translation of Moufang’s construction, or read it in German. My guess is that the ternary ring version reproduces Moufang’s result in the alternating ring case in general, not just for O.

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMay 8th 2015
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    Author: Mike Shulman
    Format: MarkdownItexOh good.

    Oh good.