# Octonions and the Standard Model

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Summary:
A survey of attempts to connect the octonions to the Standard Model - mainly explaining the math and having fun, not advocating any particular theory.
I've slowly been writing a thread on octonions and particle physics, just to explain some facts in a self-contained way, with all the proofs. I don't know where this will lead. I'm certainly not presenting a theory of physics, much less advocating one. Mainly it's just fun.
• Octonions and the Standard Model 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under SU(3).
• Octonions and the Standard Model 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Octonions and the Standard Model 3. How a lepton and a quark fit together into an octonion - at least if we only consider them as representations of SU(3), the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group SU(3).
• Octonions and the Standard Model 4. Introducing the exceptional Jordan algebra: the 3×3 self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the 2×2 adjoint octonionic matrices form precisely the Standard Model gauge group.
I didn't give the proof of that result. After a long break I thought it would be good to go about it quite slowly, introducing lots of nice background material:
• Octonions and the Standard Model 5. How to think of the 2×2 self-adjoint octonionic matrices as 10-dimensional Minkowski space, and pairs of octonions as left- or right-handed Majorana-Weyl spinors in 10 dimensional spacetime.

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rc1, mattt, vanhees71 and 12 others

ohwilleke
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Parts 1-3 seem to map only to QCD and not to the SM more generally.

It sounds from your summary like Part 4 picks up the weak force and QED. Is that correct?

The SM is, of course, formulated in three dimensional space plus one dimensional time. How do the ten dimensions of Part 5 or the seven dimensions of Part 3 reduce to the four dimensions of the SM?

Are there any other differences from the ordinary SM? For example, does this approach have right handed neutrinos?

fresh_42
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A question from a mathematical point of view: The octonions are closely related to ##G_2##, as Lie group as well as Lie algebra via their automorphisms, resp. derivations. If I understood it right, then ##G_2## is a GUT candidate. Do you say anything about this connection, and if, in which part?

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##\mathrm{G}_2## is the automorphism group of the octonions. As I explain in part 3, ##\mathrm{G}_2## contains ##\mathrm{SU}(3)## in a way that's nicely suited to explain how the strong force interacts with a lepton and a quark (and indeed, leptons and quarks come in matched pairs). But ##\mathrm{G}_2## is not a candidate for a grand unified gauge group because it doesn't contain the Standard Model gauge group.

As I explain in part 4, Dubois-Violette and Todorov showed how to get the Standard Model gauge group from the octonions - or more precisely, the exceptional Jordan algebra, which is built using octonions. This is nicely compatible with what I just said about ##\mathrm{G}_2## and the strong force, but it goes further.

None of this stuff adds up to a theory of physics, at present.

vanhees71, berkeman, Dale and 1 other person
You may be interested to know that the group of gauge symmetries of the standard model can be associated with the compactification of an 8-dimensional space with a neutral metric. Please see the post #11 in the thread Geometry of matrix Dirac algebra
By the way, the prospects for building new physics are clearly visible there.

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One of the most interesting functions on the exceptional Jordan algebra is the determinant. The linear transformations that preserve this form the exceptional Lie group ##\mathrm{E}_6##. I just wrote a blog article explaining how to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and spinors in 10d Minkowski spacetime.

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Auto-Didact, arivero and fresh_42
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Parts 1-3 seem to map only to QCD and not to the SM more generally.

Yes, and it's exaggeration to say it gets QCD. It just shows:

1) we get SU(3), the strong force gauge group, by taking the symmetries of the octonions that fix a unit imaginary octonion;

2) the resulting representation of SU(3) on the octonions matches what we get from one quark and one lepton.

This is far from QCD: I'm not talking about a Lagrangian yet. It's just a mathematical fact that one could try to exploit. After all, it's a mystery why quarks and leptons come in matched pairs, with quarks having 3 colors and leptons being colorless. I'm just saying this mathematical structure shows up when you take the octonions and pick a unit imaginary octonion.

It sounds from your summary like Part 4 picks up the weak force and QED. Is that correct?

Again this is an exaggeration: all we get is the Standard Model gauge group arising in a mathematically natural way. We get it from taking the exceptional Jordan algebra and doing two things: 1) picking a copy of 10-dimensional Minkowski spacetime inside the exceptional Jordan algebra, and 2) picking a unit imaginary octonion. The symmetries of the exceptional Jordan algebra that preserves these two choices are the Standard Model gauge group.

The SM is, of course, formulated in three dimensional space plus one dimensional time. How do the ten dimensions of Part 5 or the seven dimensions of Part 3 reduce to the four dimensions of the SM?

Forget the 7-dimensional stuff, that was just for kicks. The important thing is in Part 5 we see a very precise mathematical analogy at work:

4d Minkowski spacetime : 10 Minkowksi spacetime :: complex numbers : octonions

Speaking roughly: picking a unit imaginary octonion not only picks out a copy of the complex numbers in the octonions, it lets us break an octonion into a quark and a lepton, and gives a copy of 4-dimensional Minkowski spacetime sitting inside 10-dimensional Minkowski spacetime.

Are there any other differences from the ordinary SM? For example, does this approach have right handed neutrinos?

I am not discussing a theory of physics. I'm discussing some mathematical facts that might help someone come up with a nice theory of physics.

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Auto-Didact and ohwilleke
fresh_42
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I know that my comment now is scientifically irrelevant, but whenever I read about possible GUT related structures, it appears to me as if people blow up the groups to an extent such that any symmetry group can be embedded: if we make ##n## large enough then ##SU(n)## will contain enough copies of the smaller ones.

Sorry, this might have been a bit off topic here. It's because I stranded on your articles:
https://golem.ph.utexas.edu/category/2009/02/the_algebra_of_grand_unified_t.html
https://golem.ph.utexas.edu/category/2009/03/the_algebra_of_grand_unified_t_1.html

Allow me a question from a layman here: Do we need the geometric structure of the root systems (i.e. the angles and lengths), or only their skeleton (i.e. possible paths from short to maximal roots)?

I am not discussing a theory of physics. I'm discussing some mathematical facts that might help someone come up with a nice theory of physics.
Semisimplicity is a true hurdle. I remember that I once asked why it is physically necessary, and received the answer: because of the subsequent metric from the Killing form. This sounded reasonable, but I'm not completely convinced. Are other symmetry groups thinkable? I mean Poincarré and Heisenberg play a role, but neither is semisimple.

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I know that my comment now is scientifically irrelevant, but whenever I read about possible GUT related structures, it appears to me as if people blow up the groups to an extent such that any symmetry group can be embedded: if we make ##n## large enough then ##SU(n)## will contain enough copies of the smaller ones.

Any particular compact Lie group is a subgroup of ##SU(n)## for sufficiently large ##n##. But people working on GUTs aren't just trying to find a huge group. That would be kind of self-defeating, since a huge group means lots of symmetries, and we don't see evidence for so much symmetry. What they're looking for is a group that's big enough for the known particles to naturally appear arise from one, or a few, irreducible representations of that group. In other words, they're seeking "unification".

About the best example is the Spin(10) theory, usually misnamed the SO(10) theory. All the fermions in a single generation fit neatly into one irreducible representation of Spin(10) - together with one extra particle, a "sterile" right-handed neutrino, meaning one that doesn't feel the electroweak force. And this way of packing all those fermions into a single irreducible representation only works because quarks have charges 2/3 and -1/3, and other facts that previously seemed quite remarkable. So it's nice.

John Huerta and I explained it here:

Allow me a question from a layman here: Do we need the geometric structure of the root systems (i.e. the angles and lengths), or only their skeleton (i.e. possible paths from short to maximal roots)?

We need the whole root system to do physics.

Semisimplicity is a true hurdle. I remember that I once asked why it is physically necessary, and received the answer: because of the subsequent metric from the Killing form. This sounded reasonable, but I'm not completely convinced. Are other symmetry groups thinkable? I mean Poincarré and Heisenberg play a role, but neither is semisimple.

There are pretty good arguments in favor of gauge groups being compact. There's no need for them to be semisimple, and indeed the gauge group of the Standard Model, usually taken as ##\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)##, is not semisimple.

It's a theorem that every gauge group is either a product of a semisimple compact group and an abelian compact group, or a quotient of some such group by a finite subgroup. For example ##\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)## is a product of the semisimple ##\mathrm{SU}(3) \times \mathrm{SU}(2)## and the abelian ##\mathrm{U}(1)##, and the "true" gauge group of the Standard Model is ##\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)## modulo a ##\mathbb{Z}/6## subgroup.

As you note, noncompact symmetry groups are important in physics, but their unitary representations tend to be infinite-dimensional, and while this is fine for some purposes it's not so good for others.

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rc1, Klystron, PeterDonis and 1 other person
arivero
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what I would like of a such model is an interpolation between QCDxEM and SU(3)xSU(2)xU(1)

Point being, mass of W and Z do such interpolation. If masses go to zero, the full unbroken group, with chiral representations and all that, appears. But if masses of W and Z go to infinity, then we get a "vector" (in the sense of non-axial or not needing complex representations) group: colour plus electromagnetism.

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ohwilleke
dlgoff
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Mainly it's just fun.

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I just wrote two more article in my series:
The first describes the symmetry group ##\mathrm{E}_6## of the exceptional Jordan algebra in terms of 10d Minkowski spacetime. The second starts giving a geometrical explanation of why this works. It involves the octonionic projective plane ##\mathbb{O}\mathrm{P}^2##, so as a warmup I look at the complex projective line ##\mathbb{C}\mathrm{P}^1##, which is isomorphic to the 'heavenly sphere': the starry sky you'd see all around you if you were floating in space.

##\mathrm{E}_6## acts on ##\mathbb{O}\mathrm{P}^2## much as the Lorentz group acts on the heavenly sphere.

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Greg Bernhardt, ohwilleke, fresh_42 and 1 other person
I just wrote another article in my series:
This starts giving a geometrical explanation of why we can describe the symmetry group ##\mathrm{E}_6## of the exceptional Jordan algebra in terms of 10d Minkowski spacetime. It involves the octonionic projective plane ##\mathbb{O}\mathrm{P}^2##, so as a warmup I look at the complex projective line ##\mathbb{C}\mathrm{P}^1##, which is isomorphic to the 'heavenly sphere': the starry sky you'd see all around you if you were floating in space.

##\mathrm{E}_6## acts on ##\mathbb{O}\mathrm{P}^2## much as the Lorentz group acts on the heavenly sphere.
John, I'm almost sure you're already aware of this, but your geometrical explanation here of ##\mathrm{E}_6## by analogy with the Lorentz group and the celestial sphere is a natural generalization of twistor theory. To top it off, you start your latest piece off, almost prophetically, with my favorite quote about mathematics by Atiyah, who as you probably know also had an unpublished physical model using octonions which he didn't publish due to his untimely death; is there by any chance something you aren't telling us?

In either case, it feels like Christmas has come early this year.

ohwilleke
Perhaps I lean too eagerly on the work of string theorists, but in seeking to apply this to physics, I would start by looking for a firm connection between the exceptional Jordan algebra (27 dimensions) and e.g. the bosonic string (26 dimensions). Or: Hisham Sati already suggested that there should be a 27-dimensional theory that can manifest as OP2 fibers (16 dimensions) over M-theory (11 dimensions). Can we interpret J3(O) as OP2 fibered over some 11-dimensional space? @kneemo wrote about similar things (including the twistorial aspects) fifteen years ago.

It is impossible to look without regret at the attempts to apply these complex algebraic structures to physics. New algebraic structures should not be the basis of new physics, they should only be a consequence of new physics. In turn, the new physics must be a consequence of the new geometry and the new dynamic principle. Excuse my categorical judgments, but someone had to say that.

weirdoguy and Motore
In turn, the new physics must be a consequence of the new geometry and the new dynamic principle.

And what does this "new geometry" and "new dynamic principle" even mean? And why "new geometry" is ok and "new algebraic structure" is not?

Auto-Didact and fresh_42
Gold Member
Mitchell Porter wrote:

Can we interpret J3(O) as OP2 fibered over some 11-dimensional space?

I don't think the exceptional Jordan algebra J3(O) (which I'm calling ##\mathfrak{h}_3(\mathbb{O})##) is ##\mathbb{O}\mathrm{P}^2## fibered over some 11-dimensional space, since ##\mathfrak{h}_3(\mathbb{O})## is just a vector space, hence contractible, while ##\mathbb{O}\mathrm{P}^2## is a compact manifold with a lot of interesting topology. That's not a proof yet, but I'd be really shocked to be wrong here.

Auto-Didact
Perhaps I lean too eagerly on the work of string theorists, but in seeking to apply this to physics, I would start by looking for a firm connection between the exceptional Jordan algebra (27 dimensions) and e.g. the bosonic string (26 dimensions). Or: Hisham Sati already suggested that there should be a 27-dimensional theory that can manifest as OP2 fibers (16 dimensions) over M-theory (11 dimensions). Can we interpret J3(O) as OP2 fibered over some 11-dimensional space? @kneemo wrote about similar things (including the twistorial aspects) fifteen years ago.
I was wondering how long the 26/27 dimensionality would bring out the string believers, but no harm
To each their own.
It is impossible to look without regret at the attempts to apply these complex algebraic structures to physics. New algebraic structures should not be the basis of new physics, they should only be a consequence of new physics. In turn, the new physics must be a consequence of the new geometry and the new dynamic principle. Excuse my categorical judgments, but someone had to say that.
I suspect you are essentially paraphrasing Archimedes who spoke about creative mathematics saying: “It is easier to supply the proof [i.e. the algebra] when we have previously acquired, by the method [i.e. through geometry and/or dynamics], some knowledge of the questions than it is to find it without any previous knowledge.” Given that the non-specific nature of algebra - i.e. that algebra seems to always work completely subject independent, even when we have no clue whatsoever what the subject itself is to which the algebra is applied - is its very strength, it is obvious how this can easily lead one down endless rabbit holes.

Although I somewhat share your reservations with respect to the role of algebra as opposed to the role of geometry and dynamics in creative mathematics, I can not fully share your sentiment. During the creative process, especially in mathematical physics, it is almost impossible to say beforehand whether the geometry, the dynamical principle or the algebra comes or should come first. This is because these things are all so intertwined and interdependent with both unclear borders as well as multiplicity.

An impression of the one (i.e. of some algebra, geometry or dynamics) on the trained mind can and often does naturally evoke the unconscious processing of one or both of the others. Just because by historical precedence this typically was geometry or dynamics does not imply that algebra itself can not be useful. That algebra essentially lacks direct specificity of course makes it a greater risk to take it as a starting point, but that doesn't imply that it should therefore be dismissed out of hand; this argument stands, even after string theory.

As long as one has a clear goal or vision in mind that one is trying to reach toward - be it through algebra, geometry or dynamics - there is reason for hope. The clearer this goal or vision can be stated, the easier it is to delineate it. In other words, the most important thing is to state clearly what one's vision is at the outset of the journey; only then can one say in the end if it was reached or if not, how far one was away from reaching it, what it would take to reach it in the future, make estimates about its general reachability or state definitively whether or not it is reachable at all. If string theory has failed it is precisely because the original vision, even to the main founders themselves, has remained to this very day unclear; if the founder doesn't have the vision in sight what hope do the followers have?

In any case, in the practice of theoretical and mathematical physics, what should and what should not be taken as a starting point is not something to be decided beforehand categorically; one works with what is given and then just does various mathematical experiments to see how far these can take one towards realizing one's own vision or proving it false. As Einstein said: "One should not reproach the theorist who undertakes such a task by calling him a fantast; instead, one must allow him his fantasizing, since for him there is no other way to his goal whatsoever. Indeed, it is no planless fantasizing, but rather a search for the logically simplest possibilities and their consequences."

@ Auto-Didact Thanks for the detailed comment. One can agree with many things in it, but nevertheless, if the Cayley algebra is "grounded", it would be easier to come up with the use of octonions in physics. Why not immediately tell the author that the algebra of complex numbers is the algebra of conformally linear mappings of the plane, the algebra of quaternions is the algebra of conformally linear mappings of a pair of planes and conformally linear mappings of the planes of this pair, and finally, that the algebra of octonions is the algebra of conformally linear mappings of double pairs, pairs of planes and the planes themselves. Then it would be clear that the octonions are somehow related to the 8-dimensional space, which could be tried to be related to the doublet of Minkowski spaces. However, these are my fantasies, speculations.

Auto-Didact
Gold Member
The octonions and exceptional Jordan algebra are grounded in geometry; I explain that in my paper
The algebra comes in when you want to do computations. Ever since Descartes - if not sooner - it's been clear that algebra is useful for working with geometry.

The octonions arise because in 8-dimensional Euclidean space - and only in 8 dimensions - there's a symmetry between vectors, right-handed spinor and left-handed spinors, called triality. This has big ramifications for 10-dimensional Minkowski spacetime: a vector is a self-adjoint ##2 \times 2## matrix of octonions, while a left-handed or right-handed Majorana-Weyl spinor is a pair of octonions. This underlies a lot of superstring theory, but I'm more interested in other applications - applications that connect more directly to the Standard Model.

The exceptional Jordan algebra can be seen as an "exceptional" 27-dimensional spacetime where the lightcone is defined by a cubic equation instead of a quadratic one. It contains lots of copies of 10-dimensional Minkowski spacetime, and if we pick one of those copies then a point in the exceptional Jordan algebra can be seen a vector in 10d Minkowski spacetime together with a right-handed Majorana-Weyl spinor and a scalar.

Here's the interesting part: the Standard Model gauge group is precisely the subgroup of symmetries of the exceptional Jordan algebra that preserve a copy of 10d Minkowski spacetime and preserve a copy of the complex numbers in the octonions.

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Auto-Didact
arivero
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Perhaps I lean too eagerly on the work of string theorists, but in seeking to apply this to physics, I would start by looking for a firm connection between the exceptional Jordan algebra (27 dimensions)

Well, it is about physics, to the real game is, as always, three generations of the standard model. That was the sexy aspect of the papers of Dubois-Violette, and the question could be how constrained the assignment of charges is.

@ Auto-Didact Thanks for the detailed comment. One can agree with many things in it, but nevertheless, if the Cayley algebra is "grounded", it would be easier to come up with the use of octonions in physics. Why not immediately tell the author that the algebra of complex numbers is the algebra of conformally linear mappings of the plane, the algebra of quaternions is the algebra of conformally linear mappings of a pair of planes and conformally linear mappings of the planes of this pair, and finally, that the algebra of octonions is the algebra of conformally linear mappings of double pairs, pairs of planes and the planes themselves. Then it would be clear that the octonions are somehow related to the 8-dimensional space, which could be tried to be related to the doublet of Minkowski spaces. However, these are my fantasies, speculations.
What would be clear depends on one's own personal mathematical temperament as well as on one's (not necessarily professional) experience and preference in mathematics. A century ago Hadamard discovered - by directly studying mathematicians themselves - that the two main temperaments among mathematicians are the geometric and the algebraic temperament. One's mathematical temperament is innate while one's mathematical character is pliable, at least it is in youth; it is a shame that such knowledge is not utilized in mathematics education, which explains the persistence of the problem to a certain extent.

As John mentioned, Descartes already solved this dispute between the primacy of algebra and geometry 400 years ago, yet until this very day you can still find naive mathematicians vehemently bickering about this. This is not only because they typically lack the vision and ability to be able generalize his work in any direction by themselves, but simply because they typically deeply believe contrary to Descartes that there is an objective foundational view which has primacy - i.e. there is a fundamental principle - namely the one that they personally subscribe to, whether it be geometry or algebra. To quote Poincaré: "fundamental principles are only conventions - adopted due to some convenience - and it is quite unreasonable to ask whether they are true or false as it is to ask whether the metric system is true or false."

john baez
The octonions
From this celebrated review (1445 citations according to Google Scholar), I have learned that ##\mathbb{O}\mathrm{P}^2## can be realized via equivalence classes of certain elements of ##\mathfrak{h}_3(\mathbb{O})##. The elements of each such equivalence class are simply real scalar multiples of each other. To me this says that there's a 17-dimensional submanifold of ##\mathfrak{h}_3(\mathbb{O})## which can be interpreted (with some extra structure) as ##\mathbb{O}\mathrm{P}^2## fibered along a line.

So I'd amend my original quest. Now the goal is to interpret the whole of ##\mathfrak{h}_3(\mathbb{O})## (with some extra structure) as a fibration, with the fibers along that line being ##\mathbb{O}\mathrm{P}^2##, but something else (deformations of ##\mathbb{O}\mathrm{P}^2##?) elsewhere. The goal being, to seek convergence with Hisham Sati's notion of a 27-dimensional theory which encompasses ##\mathbb{O}\mathrm{P}^2## fibered over M-theory.

Probably I should next study these copies of 10-dimensional Minkowski space inside ##\mathfrak{h}_3(\mathbb{O})##. Maybe ##\mathfrak{h}_3(\mathbb{O})## can be obtained from 17-dimensional fibers over one of these...

The algebra comes in when you want to do computations. Ever since Descartes - if not sooner - it's been clear that algebra is useful for working with geometry.

The octonions arise because in 8-dimensional Euclidean space - and only in 8 dimensions - there's a symmetry between vectors, right-handed spinor and left-handed spinors, called triality. This has big ramifications for 10-dimensional Minkowski spacetime: a vector is a self-adjoint matrix of octonions, while a left-handed or right-handed Majorana-Weyl spinor is a pair of octonions. This underlies a lot of superstring theory, but I'm more interested in other applications - applications that connect more directly to the Standard Model.
Isn't it possible to define abstract algebra as the algebra of a geometric object? For example, the algebra of complex numbers is easily defined as the algebra of linear vector fields on the Euclidean plane, whose Lie algebra is tangent to the hypercircles of the Euclidean plane. Similarly, the algebra of quaternions is defined as the algebra of linear vector fields of 4-dimensional Euclidean space, whose Lie algebra is tangent to the hyperspheres of the Euclidean space. If we continue the analogy to 8-dimensional Euclidean space, then we have to take the corresponding Clifford algebra. And what place does the octonion algebra occupy in this algebra?

Gold Member
I wrote another installment:
The introduction goes like this:

The Riemann sphere ##\mathbb{C}\mathrm{P}^1## leads a double life in physics. On the one hand it's the set of states of a complex qubit. In this guise, physicists call it the 'Bloch sphere'. On the other hand it's the set of directions in which you can look when you're an inhabitant of 4d Minkowski spacetime &mdash; as we all are. In this guise, it's called the 'celestial sphere'. But these two roles are deeply connected! In 4d spacetime a Weyl spinor is described by a complex qubit: that is, a unit vector in $\mathbb{C}^2$. A state - that is, a unit vector modulo phase - simply says which way the spinor is spinning. Its spin can point in any direction, and these directions are points in the celestial sphere.

Last time I started explaining how to generalize some of these ideas from ##\mathbb{C}\mathrm{P}^1## to what I'm really interested in, ##\mathbb{O}\mathrm{P}^2##. Again this has two roles. On the one hand it's the set of states of an octonionic qutrit. On the other hand it's the heavenly sphere in a 27-dimensional spacetime modeled on the exceptional Jordan algebra. This is a funny spacetime where the lightcone is described not by the usual sort of quadratic equation like

## t^2 - x^2 - y^2 - z^2 = 0 ##

but instead by a cubic equation.

It'll be easy for me to get lost in the pleasures of this geometry. But I have a concrete goal in mind. The symmetries of ##\mathbb{O}\mathrm{P}^2## form the group ##\mathrm{E}_6##, and I'm trying to use this to understand a fact about ##\mathrm{E}_6##. Namely, this Lie group has four Lie subgroups:
• the double cover of the Lorentz group in 10 dimensions
• translations in right-handed spinor directions
• translations in left-handed spinor directions
• 'dilations' (rescalings)
and these give Lie subalgebras whose direct sum, as vector spaces, is all of ##\mathfrak{e}_6##:

## \mathfrak{e}_6 \cong \mathfrak{so}(9,1) \oplus \mathbb{O}^2 \oplus (\mathbb{O}^2)^\ast \oplus \mathbb{R} ##

I proved these facts back in Part 7, but now I'm trying to understand them better. The duality between points and lines in projective plane geometry turns out to be the key!

Greg Bernhardt