The Mathematical Intelligencer

, Volume 40, Issue 2, pp 1–9 | Cite as

A Bug’s Eye View: The Riemannian Exponential Map on Polyhedral Surfaces

  • David Glickenstein

When a blind beetle crawls over the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved. I was lucky enough to notice what the beetle didn’t notice.

Albert Einstein—letter to his son Eduard (1922) [2].

What would happen if Einstein’s beetle saw in the same way he was able to move? If light traveled along a surface to the eye, we would not perceive things in quite the same way. We will attempt to conceive of what the world would look like if light traveled in straight lines on the surface of a polyhedron. We will become flatlanders, as in E. A. Abbott’s famous book [ 1], except that our Flatland has pockets of curvature that come from the failure of vertices of polyhedra to be flat.
Our assumption is that in every direction, we can look out along straight paths (geodesics) and see everything that reaches our eye along that path unless it is blocked by something in front of it. In this way, when we look out, we will see a vast plane of view in every...



The software used to generate the pictures in this article allows for interactive exploration of polyhedral surfaces, both embedded and not. It was developed by a number of graduate and undergraduate students, most notably graduate students Thomas (Danny) Maienschein and Joseph Thomas and undergraduate researchers Joseph Crouch, Mark Doss, Taylor Johnson, Kira Kiviat, Justin Lanier, Taylor Corcoran, Qiming Shao, Staci Smith, Jeremy Mirchandani, and Tanner Prynn. Much thanks to previous developers and collaborators Dan Champion, Yuliya Gorlina, Alex Henniges, Tom Williams, Mitch Wilson, Kurtis Norwood, and Howard Cheng. The software is available on github [3] and uses jReality. This work was supported by NSF DMS 0748283 and DMS 0602173. I am also grateful to the reviewer, who provided helpful feedback on which pictures were needed and suggestions on how to describe many of the phenomena.


  1. 1.
    E. A. Abbott. Flatland: A Romance of Many Dimensions. Reprint of the sixth edition, with a new introduction by Thomas Banchoff. Princeton Science Library, Princeton University Press, Princeton, NJ, 1991.Google Scholar
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    Alice Calaprice, ed. The Expanded Quotable Einstein. Collected and edited by Alice Calaprice, with a foreword by Freeman Dyson. Princeton University Press, Princeton, NJ, 2011.Google Scholar
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  4. 4.
    J. M. Lee. Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, 176, Springer, New York, 1997.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

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