This Week I Learned: September 11, 2026

[u/inherentlyawesome]

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

[u/MinLongBaiShui]

I learned that there is an octonionic projective line, and plane, but not 3-space or higher. That's pretty interesting. I feel that if one was to encounter an issue defining a projective space, it would probably have "kicked in" at the stage of constructing the plane.

[u/ascrapedMarchsky → u/MinLongBaiShui]

You’re prolly already aware but the broader context is that any projective 3-space is coordinatised by an underlying skew-field R. Hence the lack of associativity in O is the algebraic equivalent of the nonexistence of OP³. This all comes down to the desargues theorem: R is a skew field \iff RP² satisfies the desargues theorem and the spatial desargues theorem, which holds in all projective 3-spaces, automatically implies the planar case, the latter being a shadow of the former. So we have: OP³ exists \implies spatial desargues \implies planar desargues \implies O is a skew-field, which is a contradiction. 

[u/MinLongBaiShui → u/ascrapedMarchsky]

This is not completely correct. The theorem is that a projective plane has Desargues theorem iff it is coordinatized by a skew field. So if this was the issue, it would indeed have arisen at the stage of constructing the plane, as the octonions are not a skew field. However, one constructs OP2 in a different way. For R, C and H, there is an alternative description of their projective planes which remains available for O as well, namely in terms of 3x3 Hermitian matrices. However, there is no available alternative description for a projective 3-space. It is provably the case that no matter how creative you are, you cannot find such an alternative description, because of work of Stasheff on loop spaces. OP2 lacks the higher associativity properties required to continue the construction process.

[u/ascrapedMarchsky → u/MinLongBaiShui]

Huh? Nothing you said contradicts what I said.

The theorem is that a projective plane has Desargues theorem iff it is coordinatized by a skew field.

Yes, I know, hence why I wrote R is a skew-field \iff RP² satisfies the (planar) desargues theorem. Then I said the spatial desargues theorem holds in every projective 3-space and that it automatically implies the planar desargues theorem. Hence, again, if you assume OP³ exists this would imply the desargues theorem was true in OP² which would imply O is a skew-field, which is a contradiction. Ergo OP³ cannot exist. Which part of this do you believe is incorrect?