Comments for Libertes Philosophica

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Fri, 13 Mar 2026 11:45:26 +0000

As mentioned above, I have added three new appendices to the following preprint:

https://arxiv.org/abs/1908.06172 .

They address the misinterpretations of its main results by Lasenby and others. In Appendix D, I prove once again that the algebra K^lambda is a normed division algebra, and the sphere it embeds is a 7-sphere. In particular, I demonstrate that the embedded surface cannot be interpreted as anything but a seven-dimensional sphere. Moreover, the so-called “nonzero zero divisors” in K^lambda cannot be normalized to give scalar lengths, and therefore they are not part of the *normed* algebra. They are excluded by the criterion of scalar-valued normalization explained in the paper.

Then, in Appendix E, I explicitly construct a linearly independent basis for its tangent space, demonstrating that this 7-sphere is parallelizable, which further consolidates that the surface embedded by the algebra is a genuine 7-sphere.

Finally, in Appendix F, I prove that there are no non-zero zero divisors in the *normed* algebra K^lambda.

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Wed, 18 Feb 2026 15:45:37 +0000

I am not concerned about either of the two issues you have raised because both are non-issues in my view. But I understand that sometimes issues like these are not easy to follow from my writings. If an expert on GA like Lasenby can stumble on some of them, then others may also have difficulty understanding them. Therefore, I plan to add a new appendix to one of my existing papers, where I will make my point of view clearer and also address the two issues you have raised.

Comment on Topologies of the 3- and 7-spheres by Sandra

Sandra — Wed, 18 Feb 2026 13:11:42 +0000

I think we are now repeating ourselves. From my side, the issue is straightforward: orthogonality makes QQ* a scalar, but it does not fix its numerical value. Imposing a fixed radius (whether called r or rescaled to 1) is therefore an additional scalar condition unless one proves that the radius is already determined by orthogonality. I do not see such a proof in the argument presented. At the very most, you can say that the Klambda subset is a set of infinitely many 6-spheres of all radii, like you can have infinitely many 3-spheres inside the quaternions.
On the algebraic side, normalization does not remove zero divisors. So from my perspective the structural questions remain.
In any case, I don’t think we’re going to reach agreement here, so I’ll leave it at that. I appreciate the exchange very much as usual Joy, and I hope that if I have more questions on other parts of your work I can still post them here.

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Wed, 18 Feb 2026 12:16:04 +0000

It is a compact 7-sphere of fixed radius r, embedded in R^8.

One can always choose units so that r reduces to 1. So, I do not agree with either you or Lasenby.

Comment on Topologies of the 3- and 7-spheres by Sandra

Sandra — Wed, 18 Feb 2026 12:10:05 +0000

It is you who set r=1 in the paper and called the resulting set the “unit 7-sphere.” If you do not fix r then the orthogonality locus is not a unit sphere at all but a noncompact 7 dimensional cone or an hypersurface in R8.

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Wed, 18 Feb 2026 11:51:28 +0000

Then don’t set r to 1 in QQ* = r^2 for the 7-sphere. There is no reason to set r equal to 1 in that case. This removes your argument.

See, for example, Eq. (31) in this paper: https://arxiv.org/abs/2211.09867

In any case, choosing units such that in those units a given quantity reduces to 1 is an elementary practice in physics.

Comment on Topologies of the 3- and 7-spheres by Sandra

Sandra — Wed, 18 Feb 2026 11:37:04 +0000

But qq* = r^2 is valid for the whole quaternion algebra. Its solution set is 4 dimensional, not 3 dimensional. In particular, it’s a single scalar constraint on an expression with 5 variables, q0, q1,q2,q3 and r. Or in vector notation,
if q = a + v, then qq* = a^2 + |v|^2 = r^2, which is an expression with variables (a, vx, vy, vz, r).

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Wed, 18 Feb 2026 09:52:19 +0000

You are mistaken about both issues. The orthogonality constraint reduces QQ* to a scalar radius, say r. But setting r = 1 does not impose an additional constraint on the resulting space. It just shrinks the radius of the 7-sphere from r to 1. Moreover, there is no reason to set r = 1 apart from convenience. We can just leave the radius to be r without changing anything fundamental. In other words, there is only one constraint, not two, which reduces the 8 dimensions of K to the 7 dimensions of the 7-sphere.

To convince yourself, consider the set of quaternions q with radius r, which is constructed by conjugation: qq* = r^2 (try this as an exercise). This is a 3-sphere of radius r, which is a three-dimensional space embedded in R^4. Now set r = 1. Does that reduce the dimensions of the space from 3 to 2? Of course not. Setting r = 1 simply shrinks the radius of the 3-sphere to 1, making it a *unit* 3-sphere. It simply shrinks the size of the sphere from r to 1 without changing its dimensions (i.e., the number of independent variables required to uniquely specify a point in the space are still 3, not 2). Lasenby has made a silly mistake, and you are misled by it.

The second issue is largely a verbal quibble. It is true that in the un-normed algebra K, one can construct non-zero zero divisors that cannot be normalized to scalar lengths. So what? To me, that is not a very interesting observation, because it holds only for the un-normed algebra K.

What is interesting is that the algebra K, once normed, has no non-zero zero divisors. In fact, all non-zero zero divisors reduce to ordinary zero divisors once all multivectors in K are normalized to scalar lengths. Therefore, K is a *normed* division algebra.

Comment on Topologies of the 3- and 7-spheres by Sandra

Sandra — Wed, 18 Feb 2026 08:18:48 +0000

Dear Joy,
I’ve read lasenby’s and your reply to it again. It’s clear lasenby makes conceptual mistakes in multiple points, but i don’t see how your reply dispels the doubts i raised earlier.
You say, correctly, that the orthogonality constraint is equivalent to
.
g h + u_x v_x + u_y v_y + u_z v_z = 0
.
But note that this constraint does not constrain the norm of Q. In fact, the norm of Q is a scalar, but any scalar. This means that when we set it to 1, we further impose
.
h^2 + ux^2 + uy^2 + uz^2 + g^2 + vx^2 + vy^2 + vz^2 = r^2
.
to be
.
h^2 + ux^2 + uy^2 + uz^2 + g^2 + vx^2 + vy^2 + vz^2 = 1.
.
This is an independent constraint from the other one. Two independent constraints, 8D ambient space, 6 dimensions.
.
As for the division algebra issue, I don’t think we’ll reach an agreement there. The S7 is not a vector space, so the second part of “division – algebra” is not satisfied. Klambda has zero divisors, so the first part of “division – algebra” can’t be satisfied. Since one is a subset of the other, there’s no union or other operation on the sets that allows for both parts to be true at the same time. It’s that simple.

Comment on Topologies of the 3- and 7-spheres by Joy Christian

Joy Christian — Mon, 26 Jan 2026 15:52:37 +0000

In topology and differential geometry, the normed division algebras are closely linked to parallelizable spheres S^0, S^1, S^3, and S^7, and are discussed interchangeably with them, because, clearly, only *normalized* multivectors with scalar norms can satisfy (or appear in) the scalar norm relation (2.27). So, the titles of some of my papers and statements about normed division algebras are neither unreasonable nor wrong. Moreover, it should be obvious to anyone that nonzero zero divisors play no role whatsoever in the 7-sphere framework I have proposed. They were dug up by Gill for disingenuous reasons. Therefore, all the talk about them is indeed a distraction, at least for me.

As for your continued interest in my work, I suggest that next time, if you read a critique of my work somewhere, then please first read my existing replies to it. I have published some two dozen papers and preprints, not to mention hundreds of blog posts and forum comments, over the period of two decades, where I have already provided all the answers to any conceivable questions one can have. It is not a good use of my time to keep repeating those answers for every new person who becomes interested in my work.