Abstract

The rainbow has been the subject of discussion across a variety of historical periods and cultures, and numerous optical explanations have been suggested. Here, we further explore the scientific treatise De iride [On the Rainbow] written by Robert Grosseteste in the 13th century. Attempting to account for the shape of the rainbow, Grosseteste bases his explanation on the optical properties of transparent cones, which he claims can give rise to arc-shaped projections through refraction. By stating that atmospheric phenomena are reducible to the geometric optics of a conical prism, the De iride lays out a coherent and testable hypothesis. Through both physical experiment and physics-based simulation, we present a novel characterization of cone–light interactions, demonstrating that transparent cones do indeed give rise to bow-shaped caustics—a nonintuitive phenomenon that suggests Grosseteste’s theory of the rainbow is likely to have been grounded in observation.

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References

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  1. H. E. Smithson, P. S. Anderson, G. Dinkova-Bruun, R. A. E. Fosbury, G. E. M. Gasper, P. Laven, T. C. B. McLeish, C. Panti, and B. Tanner, “A color coordinate system from a 13th century account of rainbows,” J. Opt. Soc. Am. A 31, A341–A349 (2014).
    [Crossref]
  2. I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
    [Crossref]
  3. R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990).
    [Crossref]
  4. B. S. Eastwood, “Grosseteste’s ‘quantitative’ law of refraction: a chapter in the history of non-experimental science,” J. Hist. Ideas 28, 403–414 (1967).
    [Crossref]
  5. R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State University, 2001).
  6. C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, 1959).
  7. A. M. Sayili, “The Aristotelian explanation of the rainbow,” Isis 30, 65–83 (1939).
    [Crossref]
  8. D. C. Lindberg, The Beginnings of Western Science: the European Scientific Tradition in Philosophical, Religious and Institutional Context, c.600 B.C. to A.D. 1450 (University of Chicago, 1992).
  9. C. Kendall and F. Wallis, Bede: On the Nature of Things and on Times (Liverpool University, 2010), Translated Texts for Historians.
  10. N. El-Bizri, “Grosseteste’s meteorological optics: explications of the phenomenon of the rainbow after Ibn al-Haytham,” in Robert Grosseteste and the Pursuit of Religious and Scientific Learning in the Middle Ages, J. P. Cunningham and M. Hocknull, eds. (Springer, 2016), vol. 18, pp. 21–39.
  11. D. C. Lindberg, “Roger Bacon’s theory of the rainbow: progress or regress?” Isis 57, 235–248 (1966).
    [Crossref]
  12. C. B. Boyer, “Robert Grosseteste on the rainbow,” Osiris 11, 247–258 (1954).
    [Crossref]
  13. B. S. Eastwood, “Robert Grosseteste’s theory of the rainbow: a chapter in the history of non-experimental science,” Archives Internationales d’Histoire des Sciences 19, 313–332 (1966).
  14. A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100-1700 (Oxford University, 1953).
  15. G. E. M. Gasper, T. C. B. McLeish, C. Panti, and H. E. Smithson, eds., Colour and the Refraction of Rays: Robert Grosseteste’s On Colour and On the Rainbow (Oxford University).
  16. D. Lindberg, Theories of Vision—from al-Kindi to Kepler (University of Chicago, 1976).
  17. A. M. Smith, From Sight to Light: The Passage from Ancient to Modern Optics (University of Chicago, 2015).
  18. D. B. Harden, “Anglo-Saxon and later medieval glass in Britain: some recent developments,” Medieval Archaeol. 22, 1–24 (1978).
    [Crossref]
  19. Corning Museum of Glass, 2002, http://www.CMoG.org/ (accessed: Febuary 2017).
  20. H. E. Smithson, “All the colours of the rainbow: Robert Grosseteste’s three-dimensional colour space,” in Robert Grosseteste and the Pursuit of Religious and Scientific Learning in the Middle Ages, J. P. Cunningham and M. Hocknull, eds. (Springer, 2016), Vol. 18, pp. 59–84.
  21. Blender Online Community, Blender—A 3D modelling and Rendering Package (Blender Foundation, 2016).
  22. J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.
  23. C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.
  24. E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
    [Crossref]
  25. M. Daimon and A. Masumura, “Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region,” Appl. Opt. 46, 3811–3820 (2007).
    [Crossref]
  26. Y. Petrov, Optometrika, MATLAB Central File Exchange, https://www.mathworks.com/matlabcentral/fileexchange/45355-optometrika (2014, retrieved January 2017).
  27. H. E. Smithson, G. Dinkova-Bruun, G. E. Gasper, M. Huxtable, T. C. McLeish, and C. Panti, “A three-dimensional color space from the 13th century,” J. Opt. Soc. Am. A 29, A346–A352 (2012).
    [Crossref]

2014 (1)

2012 (2)

H. E. Smithson, G. Dinkova-Bruun, G. E. Gasper, M. Huxtable, T. C. McLeish, and C. Panti, “A three-dimensional color space from the 13th century,” J. Opt. Soc. Am. A 29, A346–A352 (2012).
[Crossref]

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

2007 (1)

2002 (1)

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

1990 (1)

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990).
[Crossref]

1978 (1)

D. B. Harden, “Anglo-Saxon and later medieval glass in Britain: some recent developments,” Medieval Archaeol. 22, 1–24 (1978).
[Crossref]

1967 (1)

B. S. Eastwood, “Grosseteste’s ‘quantitative’ law of refraction: a chapter in the history of non-experimental science,” J. Hist. Ideas 28, 403–414 (1967).
[Crossref]

1966 (2)

B. S. Eastwood, “Robert Grosseteste’s theory of the rainbow: a chapter in the history of non-experimental science,” Archives Internationales d’Histoire des Sciences 19, 313–332 (1966).

D. C. Lindberg, “Roger Bacon’s theory of the rainbow: progress or regress?” Isis 57, 235–248 (1966).
[Crossref]

1954 (1)

C. B. Boyer, “Robert Grosseteste on the rainbow,” Osiris 11, 247–258 (1954).
[Crossref]

1939 (1)

A. M. Sayili, “The Aristotelian explanation of the rainbow,” Isis 30, 65–83 (1939).
[Crossref]

Anderson, P. S.

Antal, G.

C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.

Badler, N. I.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

Boyer, C. B.

C. B. Boyer, “Robert Grosseteste on the rainbow,” Osiris 11, 247–258 (1954).
[Crossref]

C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, 1959).

Chalmers, A.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

Crombie, A. C.

A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100-1700 (Oxford University, 1953).

Csonka, F.

C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.

Daimon, M.

Dinkova-Bruun, G.

Eastwood, B. S.

B. S. Eastwood, “Grosseteste’s ‘quantitative’ law of refraction: a chapter in the history of non-experimental science,” J. Hist. Ideas 28, 403–414 (1967).
[Crossref]

B. S. Eastwood, “Robert Grosseteste’s theory of the rainbow: a chapter in the history of non-experimental science,” Archives Internationales d’Histoire des Sciences 19, 313–332 (1966).

El-Bizri, N.

N. El-Bizri, “Grosseteste’s meteorological optics: explications of the phenomenon of the rainbow after Ibn al-Haytham,” in Robert Grosseteste and the Pursuit of Religious and Scientific Learning in the Middle Ages, J. P. Cunningham and M. Hocknull, eds. (Springer, 2016), vol. 18, pp. 21–39.

Ferwerda, J.

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

Fletcher, R. L.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

Fosbury, R. A. E.

Fraser, A. B.

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State University, 2001).

Gasper, G. E.

Gasper, G. E. M.

Gutierrez, D.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Harden, D. B.

D. B. Harden, “Anglo-Saxon and later medieval glass in Britain: some recent developments,” Medieval Archaeol. 22, 1–24 (1978).
[Crossref]

Holod, R.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

Huxtable, M.

Jarosz, W.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Jensen, H. W.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Kelemen, C.

C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.

Kendall, C.

C. Kendall and F. Wallis, Bede: On the Nature of Things and on Times (Liverpool University, 2010), Translated Texts for Historians.

Kider, J. T.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

Laven, P.

Lee, R. L.

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State University, 2001).

Lindberg, D.

D. Lindberg, Theories of Vision—from al-Kindi to Kepler (University of Chicago, 1976).

Lindberg, D. C.

D. C. Lindberg, “Roger Bacon’s theory of the rainbow: progress or regress?” Isis 57, 235–248 (1966).
[Crossref]

D. C. Lindberg, The Beginnings of Western Science: the European Scientific Tradition in Philosophical, Religious and Institutional Context, c.600 B.C. to A.D. 1450 (University of Chicago, 1992).

Masumura, A.

McLeish, T. C.

McLeish, T. C. B.

Munoz, A.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Panti, C.

Rashed, R.

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990).
[Crossref]

Reinhard, E.

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

Sadeghi, I.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Sayili, A. M.

A. M. Sayili, “The Aristotelian explanation of the rainbow,” Isis 30, 65–83 (1939).
[Crossref]

Seron, F.

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

Shirley, P.

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

Smith, A. M.

A. M. Smith, From Sight to Light: The Passage from Ancient to Modern Optics (University of Chicago, 2015).

Smithson, H. E.

Stark, M.

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

Szirmay-Kalos, L.

C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.

Tanner, B.

Wallis, F.

C. Kendall and F. Wallis, Bede: On the Nature of Things and on Times (Liverpool University, 2010), Translated Texts for Historians.

Yu, N.

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

ACM Trans. Graph. (2)

I. Sadeghi, A. Munoz, P. Laven, W. Jarosz, F. Seron, D. Gutierrez, and H. W. Jensen, “Physically-based simulation of rainbows,” ACM Trans. Graph. 31, 1–12 (2012).
[Crossref]

E. Reinhard, M. Stark, P. Shirley, and J. Ferwerda, “Photographic tone reproduction for digital images,” ACM Trans. Graph. 21, 267–276 (2002).
[Crossref]

Appl. Opt. (1)

Archives Internationales d’Histoire des Sciences (1)

B. S. Eastwood, “Robert Grosseteste’s theory of the rainbow: a chapter in the history of non-experimental science,” Archives Internationales d’Histoire des Sciences 19, 313–332 (1966).

Isis (3)

R. Rashed, “A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses,” Isis 81, 464–491 (1990).
[Crossref]

A. M. Sayili, “The Aristotelian explanation of the rainbow,” Isis 30, 65–83 (1939).
[Crossref]

D. C. Lindberg, “Roger Bacon’s theory of the rainbow: progress or regress?” Isis 57, 235–248 (1966).
[Crossref]

J. Hist. Ideas (1)

B. S. Eastwood, “Grosseteste’s ‘quantitative’ law of refraction: a chapter in the history of non-experimental science,” J. Hist. Ideas 28, 403–414 (1967).
[Crossref]

J. Opt. Soc. Am. A (2)

Medieval Archaeol. (1)

D. B. Harden, “Anglo-Saxon and later medieval glass in Britain: some recent developments,” Medieval Archaeol. 22, 1–24 (1978).
[Crossref]

Osiris (1)

C. B. Boyer, “Robert Grosseteste on the rainbow,” Osiris 11, 247–258 (1954).
[Crossref]

Other (15)

Y. Petrov, Optometrika, MATLAB Central File Exchange, https://www.mathworks.com/matlabcentral/fileexchange/45355-optometrika (2014, retrieved January 2017).

Corning Museum of Glass, 2002, http://www.CMoG.org/ (accessed: Febuary 2017).

H. E. Smithson, “All the colours of the rainbow: Robert Grosseteste’s three-dimensional colour space,” in Robert Grosseteste and the Pursuit of Religious and Scientific Learning in the Middle Ages, J. P. Cunningham and M. Hocknull, eds. (Springer, 2016), Vol. 18, pp. 59–84.

Blender Online Community, Blender—A 3D modelling and Rendering Package (Blender Foundation, 2016).

J. T. Kider, R. L. Fletcher, N. Yu, R. Holod, A. Chalmers, and N. I. Badler, “Recreating early Islamic glass lamp lighting,” in Proceedings of the 10th International Conference on Virtual Reality, Archaeology and Cultural Heritage (VAST) (Eurographics Association, 2009), pp. 33–40.

C. Kelemen, L. Szirmay-Kalos, G. Antal, and F. Csonka, “A simple and robust mutation strategy for the metropolis light transport algorithm,” in Computer Graphics Forum (Wiley, 2002), vol. 21, pp. 531–540.

A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100-1700 (Oxford University, 1953).

G. E. M. Gasper, T. C. B. McLeish, C. Panti, and H. E. Smithson, eds., Colour and the Refraction of Rays: Robert Grosseteste’s On Colour and On the Rainbow (Oxford University).

D. Lindberg, Theories of Vision—from al-Kindi to Kepler (University of Chicago, 1976).

A. M. Smith, From Sight to Light: The Passage from Ancient to Modern Optics (University of Chicago, 2015).

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Pennsylvania State University, 2001).

C. B. Boyer, The Rainbow: From Myth to Mathematics (Thomas Yoseloff, 1959).

D. C. Lindberg, The Beginnings of Western Science: the European Scientific Tradition in Philosophical, Religious and Institutional Context, c.600 B.C. to A.D. 1450 (University of Chicago, 1992).

C. Kendall and F. Wallis, Bede: On the Nature of Things and on Times (Liverpool University, 2010), Translated Texts for Historians.

N. El-Bizri, “Grosseteste’s meteorological optics: explications of the phenomenon of the rainbow after Ibn al-Haytham,” in Robert Grosseteste and the Pursuit of Religious and Scientific Learning in the Middle Ages, J. P. Cunningham and M. Hocknull, eds. (Springer, 2016), vol. 18, pp. 21–39.

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Figures (10)

Fig. 1.
Fig. 1. Interpretation of the geometric optics described in the De iride. Rays from the sun undergo three refractive events: the interface between the air and the dome of the cloud, within the cloud’s drops, and upon re-entering the air. These events collect the radiance into the form of a bow. In this depiction a single ray from the sun is shown.
Fig. 2.
Fig. 2. Cone-shaped glass beakers dated to the same period as Grosseteste from (a) Central Europe and (b) the Middle East. (c) Modern conical wine glass filled with water, illuminated by a candle. A bow-shaped caustic can be seen to the right of the stem. The bow features a dispersive spectrum with the correctly ordered colors for a rainbow.
Fig. 3.
Fig. 3. Parameters to describe the scene for both physical experiment and rendered simulations. The light source could emit either collimated light or divergent light.
Fig. 4.
Fig. 4. Physical experiment of cone–light interactions, α = 13.5 ° . Upper panel: illuminated by divergent light. Lower panel: illuminated using a collimating lens. Light elevation angles of (a)  ϵ = 25 ° , (b)  ϵ = 35 ° , and (c)  ϵ = 45 ° .
Fig. 5.
Fig. 5. Light entering the top of the cone is responsible for the formation of the colorful bow seen projected onto the screen, collimated light, ϵ = 35 ° . (a) Unobstructed illumination, (b) blocking the side of the cone, and (c) blocking the top of the cone.
Fig. 6.
Fig. 6. Images of caustics resulting from cone–light interactions simulated using the Luxrender engine in Blender. Upper panel: a simulation of the physical experiment photographed in Fig. 4, featuring an extended source of divergent light. Lower panel: a scene lit by collimated light to simulate an observation in the open air illuminated by the sun.
Fig. 7.
Fig. 7. Parameter space of cone-produced caustics obtained from Luxrender simulations, for scenes containing a divergent light source. The region where arcs are produced is shaded gray for emphasis.
Fig. 8.
Fig. 8. Light entering the top of the cone is collected into the bow-shaped caustic seen in the simulations. (a) Divergent light of ϵ = 45 ° . (b) The same scene, but the portion of the cone’s side facing the light source is opaque.
Fig. 9.
Fig. 9. Simulated dispersive effects, after rendering images for 12 different wavelengths of light, ϵ = 45 ° , α = 27 ° . On the left a composite image of 12 renders for light of wavelengths between 380 and 710 nm is shown. The plot shows the distribution of pixel intensities for each wavelength through the center of the rendered image.
Fig. 10.
Fig. 10. (a) Photograph of physical experiment and (b) Optometrika ray-tracing showing the bow-shaped caustic formed by internal reflection off the back face of the cone. (c) Rays projected onto the horizontal screen in the Optometrika simulation. ϵ = 45 ° .

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n 2 ( λ ) = 1 + B 1 λ 2 λ 2 C 1 + B 2 λ 2 λ 2 C 2 + B 3 λ 2 λ 2 C 3 ,

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