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Peer-reviewedCC BYThe Mathematical Intelligencer

Orange peels and Fresnel integrals

  • Laurent Bartholdi
  • André G. Henriques
Open at sourcePDFDOICite

Abstract

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.

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Cite

BibTeXRIS
BibTeX
@article{bartholdi2026orange,
  title = {{Orange peels and Fresnel integrals}},
  author = {Laurent Bartholdi and André G. Henriques},
  journal = {The Mathematical Intelligencer},
  year = {2026},
  month = nov,
  doi = {10.1007/s00283-012-9304-1},
  eprint = {1202.3033},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05849590v1},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Orange peels and Fresnel integrals
AU  - Laurent Bartholdi
AU  - André G. Henriques
PY  - 2026
DA  - 2026/11/03
JO  - The Mathematical Intelligencer
DO  - 10.1007/s00283-012-9304-1
UR  - https://hal.science/hal-05849590v1
AB  - There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.
SN  - 0343-6993
SN  - 1866-7414
ER  -