Dear Dr Lehar,

We have received the reports from our advisors on your manuscript, "Double Conformal Mapping: A Finite Mathematics to Model an Infinite World.", which you submitted to Advances in Applied Clifford Algebras.

Based on the advice received, I have decided that your manuscript could be reconsidered for publication should you be prepared to incorporate major revisions. When preparing your revised manuscript, you are asked to carefully consider the reviewer comments which can be found below, and submit a list of responses to the comments. You are kindly requested to also check the website for possible reviewer attachment(s).


Dear Author,

Even though both reviews, which have reached us about your revised version are very negative (one previous reviewer refused to review the manuscript again, therefore we referred the manuscript to a new impartial reviewer #4), we still want to give you a final opportunity to take the constructive criticism by the reviewers (especially that of #4) into account and do a major revision/rewriting of your paper. But in the current unaltered form the paper is not fit for publication in AACA.

Associate Editor.

Reviewer #2: The answers given by the author don't bring any new element that can change my opinion. This paper which consists mostly of personal reflections doesn't contain mathematical results of any kind that can justify the publication in AACA and more generally a publication in a mathematical journal.

Reviewer #4: AACA-D-15-00168R1

The editor has asked me to give my opinion since the two other reviewers' opinions are hard to reconcile. Reading the documents in the case, I am rather surprised at the author's reaction to the remarks of the second reviewer.

An author writes so that his/her ideas may be understood and perhaps appreciated by others. The editor asks reviewers to check whether the ideas are clearly presented, correct and suitable for the journal.

When a reviewer has detailed questions, comments and remarks, the author learns that his presentation can apparently be improved. Where the reviewer has not understood what the author meant, very often the writing may be to blame; or the author may not have realized the full scope of his audience and can get more people on board by taking the ourisde comments seriously; or the reviewer is a good representation of the readership, but the paper was simply submitted to the wrong journal. This is all part of getting a paper into publishable form, on the right platform.

In the current case, the author appears to take the view 'take it or leave it'. That is not helpful; and neither is Reviewer 1s judgement to 'take it' as a whole. For to my mind, there is a lot that could be improved in this paper.

Let me now start my review.


It is not clear whether this paper is a mathematical model of perception/cognition, or a perception/cognition model of mathematics. The former would conceivably fit in the scope of AACA, if it uses Clifford/geometric algebra. The latter is more suitable for some journal on the philosophy of mathematics ( or the Mathematical Intelligencer). So I read it as the former; I am an AACA reviewer.

The author points out that infinities are hard to compute with in the brain, and that therefore representations which can represent infinities in a finite manner, but faithfully, are interesting. This skips the question of why the brain would have to deal with infinities: infinitely far things cannot be reached, grasped, etc., so there is no evolutionary pressure to evolve the brain to cope with them. Only when we make mathematical abstractions do we encounter infinities. But let us assume this point; the resulting question is worth exploring.

The author then find inspiration in 'Hestenes' conformal model',  which is a stereographic projection in a well-chosen space. In the purpose of making infinities finitely represented, this is not Hestenes' idea, such representations are found in 19th century mathematics. What is recently new in the work by Hestenes and around the same time Anglès is that the stereographic projection takes place in a space of 2 extra dimensions, and that this renders conformal transformations in 3D to be orthogonal transformations of this space, which by means of Clifford algebra can be represented as spinors (versors, rotors), which are structure-preserving and therefore very attractive for computational implementations.

The author uses a very impoverished version of this chain of ideas, only the original stereographic mapping, and then only qualitatively. So calling it 'Hestenes conformal model'  throughout is naive; that applies to the full chain only. It may be where you first learned of the stereographic trick, but no matter.

The mapping is introduced in Fig 1 Fig 5b. These figures are NOT correct (or at least misleading). The stereographic mapping employed therein is in a space of only 1 more dimension. That is not the Hestenes/Angles mapping. (I find Fig 4A, 5A and 5B very hard to read, the 3D nature of these is not rendered very skillfully. Use Perwass' software instead! But the figures are 3D, whereas they should be 4D to do the Hestenes/Anglès mapping for 2D space). Since formulas are never given, we as readers now no longer precisely understand what is going on. But in Hestenes' model we still have points at infinity as a sphere of directions (they are of the form n ^u); in your picture they all become the north pole, a single point. The extra dimension is very essential.

The advantage of CGA is that these kinds of figures are so easily converted into algebra. For the AACA audience, there is no objection to doing this; in fact, it will help them understand better what is happening. This really MUST be done to have this paper in AACA; it is a journal on Clifford/geometric algebra, so use that tool.

Doing so will also resolve some things that puzzle me in the pictorial depiction. The chosen 2D renderings of something that is a 3D world are rather clumsy for conveying the 3D nature of the Bubble, and the CGA sphere. I first interpreted the pictures as representing the usual 'visual sphere' surrounding an observer (perhaps misled by the puzzle of  Fig 4C and its resolution).  Perhaps it would have been better to work out a 2D example in the drawings; but here, again, formulas would help all AACA readers to understand what is really going on. And even for 2D reality, you would need to draw the 4D space in which the Hestenes stereographic projection lives, which will be a challenge.

 When the mapping 'back out' again is done, we find some rare formulas, but no explanation of how to read then. The inverse mapping of 1/x is the mapping 1/x, since (1/(1/x)) =x; it is not the mapping x^2 of Figure 12. (The product of 1/x with x^2 equals x, but these are mappings and should use mapping composition.) Then an atan is thrown in in Fig 13 for some desirable effect; but is the resulting mapping still conformal? Is it actually important to have the mappings be conformal (i.e. locally shape preserving)? For what externally desirable properties? The demand that infinity becomes finitely tractable is not sufficient to specify the mapping - conformal works, and rather nicely, but so do other mappings.

By the way, there is clear difference between 'reflection in a sphere' and 'inversion in a sphere'. In CGA, one always does inversions, but some authors have called them reflections (because of the analogy of -sx/s with Euclidean reflection formulas). A reflection in a polished sphere produces an image that depends on the location of the observer; inversion does not. It is not clear which of the two Figure 7 depicts. Again, a formula would help, and is easy to give in CGA.

So, lots of questions, all would be resolvable in a more exact, precise, and conventional description of this new model. Your chosen audience would be able to follow and assess the paper better if you cast it in that form, so why not do it?

On the non-Euclidean section, I (like reviewer 2) do not understand why you take the time to explain the history to an AACA audience. Moreover, I do not see whether your space is truly non-Euclidean, or merely a non-linear mapping of an essentially Euclidean geometry. It all depends on the local metric, and the geodesics of the space (it would be helpful to use the concept of 'geodesic' in the clumsy explanation at the top of page 7; really, you do not need to explain these things to this audience!). Anyway, the conformal conversions between such spaces are well-known, and much better illustrated in the book by Needham.

I miss references, too. There has been work on possible mental representations of space by Mach, on the brain as a geometry engine by Koenderink, on the metric of perceptual space by Koenderink, on the possibility of computing GA with neurons by Hestenes. All these appear to me relevant to what the author is trying to do, and some can be seen as prior art. It is customary in a scientific paper to discuss that, and why it does not satisy the author, and what is going to be better this time around. All we mainly find as references to the perceptual modelling side are self-references (perhaps they contain the references I mentioned, I did not check, but repeat some of the background here).

Finally, something about the general impression of the paper. There are a lot of properties that set off the 'crackpot alarm' in a reader: the nonstandard affiliation of the author; the heavy self-referencing; the metamathematical drive; the Use of Capitals in weird Places; the clumsy home-made drawings; the lack of precision by formulas; the naive attribution of historically well-known concepts to where one first heard them. Some of those are characteristics shared with Grassmanns work, so one should not always dismiss such papers off-hand; but most readers will decide that their time is better spent reading something else. If you want to be read, present your ideas in a more acceptable and precise form; especially for this AACA audience. This is not an inspirational talk at a conference; this could be a scientific publication - but not in this form on this platform.

So, what do I recommend based on this assessment? I think the remarks above would require a complete rewriting of the paper; this is beyond a major revision which can be compared to its original. Put the present form of the paper on your website, by all means (I see you already have done so) -it has the hallmarks of a web page. Then start anew and write down your ideas for an AACA audience in a form that they can absorb; with a clear specification of the desiderata of you mapping, why the CGA model satisfies those, and how to use it. See if you still stand by the atan when you have done all that, and/or whether it is still conformal, and whether than matters.

In the present form, you will find few understanding readers, and receive little feedback that might help you in your quest. That would be a missed opportunity; there may be something here that is worth getting precise.

Author's Response