Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-18T07:30:46.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Is a straight line the shortest path?

Published online by Cambridge University Press:  08 February 2018

Jessica E. Banks
Jessica E. Banks*
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW e-mails: jessica.banks@lmh.oxon.org, jessica.banks@bristol.a.c.uk
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.

The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 553 , March 2018 , pp. 1 - 12
Copyright
Copyright © Mathematical Association 2018 

References

1. do Carmo, M. P., Differential geometry of curves and surfaces, Prentice-Hall (1976).Google Scholar
2. Spivak, M., A comprehensive introduction to differential geometry. Vol. III. (2nd edn.), Publish or Perish (1979).Google Scholar
3. Coxeter, H. S. M., Introduction to geometry (2nd edn.) Wiley (1989).Google Scholar
4. Henderson, D. W. and Taimina, D., Crocheting the hyperbolic plane, The Mathematical Intelligencer, 23(2) (2001) pp. 17–28.CrossRefGoogle Scholar
5. Taimina, D., Hyperbolic crochet – some fiber for thoughts about art, math, crochet, and all the various threads in our lives (2016), available at http://hyperbolic-crochet.blogspot.co.uk Google Scholar
6. Institute For Figuring, Hyperbolic space, accessed August 2017 at http://crochetcoralreef.org/about/hyperbolic_space.php Google Scholar
7. Weeks, J., How to sew a hyperbolic blanket, accessed August 2017 at http://www.geometrygames.org/HyperbolicBlanket/index.html Google Scholar
8. Joyce, D. E., Euclid's Elements, Clark University, Massachusetts (1998), available at http://aleph0.clarku.edu/∼djoyce/java/elements/ Google Scholar
9. Lewis, F. P., History of the parallel postulate, Amer. Math. Monthly, 27(1) pp. 16–23 (1920).CrossRefGoogle Scholar
10. Osserman, R., Poetry of the Universe, Anchor Books (1996).Google Scholar
11. O'Connor, J. J. and Robertson, E. F., Non-Euclidean geometry, MacTutor History of Mathematics archive (accessed 11 Jan 2017).Google Scholar
12. Hilbert, D. and Cohn-Vossen, S., Geometry and the imagination, Chelsea Publishing Company (1952).Google Scholar
13. Dhar, Anirbit, Geodesics on spheres are great circles (2010) available at http://mathoverflow.net/q/12200 Google Scholar