The ancient Greeks struggled with the concept of a horn angle, the "angle" formed by the intersection of two curves. The only information I find in Mathworld is that horn angles are examples of non-Archimedean geometry. Where can I find out more about them? What work has been done on them, and what is the current state of knowledge on them?

Since they are part of non-Archimedean geometry, can they be understood using the hyperreal numbers or some other non-standard model of the first-order theory of real closed fields?

Any help would be greatly appreciated. Thank You in Advance.

  • There's a notion of "intersection multiplicity" in algebraic geometry (and sometimes differential topology) that might be relevant. – Charles Staats Feb 22 '11 at 16:20
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    I don't know what non-Archimedean geometry is, but my guess is that it is a different use of "non-Archimedean" than that of non-standard analysis. Also, if the curves are differentiable, isn't the angle of intersection calculatable from the tangent vectors? – Jason Rute Feb 22 '11 at 20:34
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    I think this question needs some background, so I will try to give such to the best of my limited understanding. In the above context, "Archimedian" geometry is geometry based on structures, where the Archimedian property holds (e.g. the reals). That is, geometrically speaking, if you have 3 "collinear" points $A$,$B$ and $C$ (in that order), you can always find an integer $n$ such that $n\lambda(AB)\geq\lambda(AC)$ for a "Lebesgue"-measure $\lambda$, or better, $n d(A,B)\geq d(A,C)$ for a given distance function $d$. (to be continued) – M.G. Feb 22 '11 at 21:01
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    (continuation) In other words, the Archimedian property ensures the non-existence of non-trivial infinitesimals, and here is the point where non-standard models/analysis come in question as a language for geometry based on structures where the Archimedian property fails, thus "non-Archimedian" geometry. Hyperreals and geometry thereof are an example here. More generally, "non-Archimedian" geometry is geometry over "non-Archimedian" fields, i.e. fields that do not admit Archimedian valuation, see this paper of B. Conrad --> (continued) – M.G. Feb 22 '11 at 21:08
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    (continuation) Now, the question is what Horn Angles have to do with all that stuff. And the answer is that Horn Angles build a non-Archimedian system w.r.t. to certain ordering relation constituted by the size of the angle between the tangent vectores and certain radii, see e.g. "Foundations and fundamental concepts of mathematics" by Howard Whitley Eves, p. 107 (Problems). – M.G. Feb 22 '11 at 21:22

1) Horn angles and variations thereof appear to be well studied in conformal geometry, see e.g. the papers by Ladue, Mary Elizabeth "Conformal geometry of horn angles of higher order", Amer. J. Math. 65, (1943), 455–476, and "Trihornometry: A New Chapter of Conformal Geometry" by Edward Kasner, Proceedings of the National Academy of Sciences of the United States of America, Vol. 23, No. 6 (Jun. 15, 1937), pp. 337-341

2) You can find more about the theory of horn angles in the paper "The recent theory of the horn angle" by E. Kasner, Scripta Math. 11, (1945). 263–267, 53.0X. The paper is from 1945, but it should give you some idea about the directions of study in that area.

3) Your intuition is right as horn angles can be measured within the model of superreal numbers (in the sense of D. O. Tall), see the paper by J. Bair and V. Henry "Angles corniculaires et nombres superréels" (French) ("Horn angles and superreal numbers), Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 77–86.

  • Have the philosophical problems the Greeks struggled with been solved? Also, which superreal numbers are you talking about, Hugh Woodin's system or David Tall's system? – Keshav Srinivasan Feb 23 '11 at 3:27
  • It is the model of superreal numbers of David Tall. This is mentioned in my answer under 3). – M.G. Feb 23 '11 at 12:13

A quick perusal of Math Reviews leads one to suspect that not much research goes on. (A better perusal might show otherwise). For Euclid a "line" was a curve and an "angle" was a Horn Angle. But he actually only looked at angles formed by lines except for Book III proposition 16 where he essentially commented that the angle formed by a circle and a tangent line is positive but less than any linear angle. Maybe he did not do much more because Horn angles are pretty tough; Unlike linear angles, it is hard to know how to add them or even double one (in a geometrically meaningful way). I'm not even sure how to say when one contains another. In modern times Knasner studied them using somewhat advanced notions of differential geometry (the references above are good) with latest paper in 1951 (dealing with trisection of Horn Angles). After that there is a paper from 1968 and the mentioned one from 2008 (along with some historical papers).

I think that Horn Angles might inform non-standard models more than vice versa. One does not need non-standard models to talk about the order on rational functions (or more general classes of functions) based on their behavior as $x\to\infty$. It is essentially the same to look at behavior as $x\to 0^+$ and that is pretty much the order of Horn Angles.

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