Iv never heard of that and don't think it's exactly mainstream either. The main attraction to GA is that it's coordinate free. But I guess it fails to provide an intrinsic differentual geometry

Well the redundancy part is quite subtle. For example in GA stokes theorem implies both divergence and rotation theorems, and at the same time applies to vector valued differential forms. So this is how redundancy is cut down.

So I would like a simple, coordinate free manifold geometry theory without embedding. Hestenes has the coordinate free but not intrinsic , and mainstream formulation has a lot of redundancy and coordinates though it's intrinsic.

This is exactly what I'm saying. He explicitly states his formulation is intrinsic yet I can't see how. At no poit did I say intrinsic geometry is redundant if anything, sometimes it's the other way around. I said that the mainstream intrinsic formulation has a lot of redundancy. Hestenes managed to remove this redundancy of structure in which he failed to retain the intrinsic formulation despite claiming otherwise.

I don't see how this answers the problem. The main point hestenes tried to make is that exactly what you seem to be talking about is just a redundancy ridden formalism that he tries to simplify in what he calls geometric algebra. Also geometric algebra is coordinate less, so it doesn't use coordinates. Thaugh his calculations are coordinate less, they seem to unequivocally imply an imbedding. I don't see how this is not the case, that is the question

That's not coordinate free..

OK good answer, still, I would love if there was a way to somehow define an infinite series that would compute the area or circumference of a circle without using sine, unless one can define and relate sine to geometry without coordinates.

What do you mean I'm not asking about how to tech these concepts?

this is exactly what the question is about, new approach to circ functions and pi and lim nsin(x/n). As long as it's not usi g coordinates or ortho normal basis

Yea I wish to avoid adopting a basis or anything like that.

It doest matter what I consider coordinate free(thaugh I explained it clearly in all my posts those lines if for some reason that is of interest here.. I staded it perfectly unambiguously here too look at the description:"without in any way introducing coordinates",...

@Yemon Choi we are some thousands of years past archimedes, regardless, evidently now, one has to define angle either using arclengt or arch area of a circle, I just want to show that both are proportional to pi as given by some infinite series one calculates it with

In the answer it's stated that trig functions are defined as matrices of the homomorphism. So instead of matrix algebras you use Clifford algebras, certain spinors as their elements..

@Alexandre Eremenko Also there are ways to have a rotation group representation without matrices or hyper complex numbers. Every rotation is a composition of two reflections, and in some systems, variations of Clifford algebra where this ratation groups are equivalent to spinors in a completely coordinate free way.

I agree that distinction is blurry, but does that mean that in the theory of pure geometry the ratio of circle length and radius is an undecidable question?