Abstract

The function of the central core in lenses of certain schizochroal-eyed trilobites is unknown. To understand the possible optical function(s) of this central core, we performed computational ray-tracing on the lens in the schizochroal compound eyes of a Silurian Dalmanites trilobite. We computed the intensity of light focused by the lens versus the distance from the lower lens surface along the optical axis as functions of the refractive indices nlu and ncc of the lower lens unit and the central core. We determined those values of nlu and ncc that ensure that the studied central-cored trilobite lens is monofocal, bifocal, or trifocal. The sharpness (as the measure of the correction for spherical aberration) of these focal points was quantitatively studied. We show here that one of the possible optical functions of the central core could be the correction for spherical aberration, independently of the number (1, 2, or 3) of focal points. Another possible optical function of the core could be to ensure bifocality of the lens. In this case the peripheral lens region could have a given focal length and the central lens region could possess a longer or shorter focal length, if the refractive index ncc of the core is smaller or larger than the refractive index nlu of the upper lens unit. Finally, trifocality of the lenses can be considered only as a theoretical option, but by no means an optically optimally functioning possibility.

© 2012 Optical Society of America

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References

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  3. E. N. K. Clarkson, “The visual system of trilobites,” Palaeontology 22, 1–22 (1979).
    [CrossRef]
  4. E. N. K. Clarkson, “The eye, morphology, function and evolution,” in Treatise on Invertebrate Paleontology, part O, Trilobita, Revised, R. L. Kaesler et al., eds. (University of Kansas, 1997), pp. 114–132.
  5. R. Levi-Setti, Trilobites, 2nd ed. (University of Chicago, 1993).
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    [CrossRef]
  7. E. N. K. Clarkson, “The evolution of the eye in trilobites,” Fossils Strata 4, 7–31 (1975).
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  9. R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “The eye: paleontology,” in Frontiers of Biology—Italian Encyclopedia. Part I. Origin and Evolution of Life. Section 7. Construction of the Organism, D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (2002), pp. 379–395.
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    [CrossRef]
  11. T. McCormick and R. A. Fortey, “Independent testing of a paleobiological hypothesis: the optical design of two pelagic trilobites reveals their relative palaeobathymetry,” Paleobiology 24, 235–253 (1998).
  12. J. Miller and E. N. K. Clarkson, “The post-ecdysial development of the cuticle and the eye of Phacops rana milleri,” Philos. Trans. R. Soc. Lond. Ser. B 288, 461–480 (1980).
    [CrossRef]
  13. E. N. K. Clarkson, “Fine structure of the eye in two species of Phacops (Trilobita),” Palaeontology 10, 603–616 (1967).
  14. G. Lindström, “Researches on the visual organs of the trilobites,” Kongliga Svenska Vertenskaps Akademiens Handlingar 8, 1–89 (1901).
  15. E. N. K. Clarkson and R. Levi-Setti, “Trilobite eyes and the optics of Des Cartes and Huygens,” Nature 254, 663–667 (1975).
    [CrossRef]
  16. G. Horváth, “Geometric optics of trilobite eyes: a theoretical study of the shape of aspherical interface in the cornea of schizochroal eyes of phacopid trilobites,” Math. Biosci. 96, 79–94 (1989).
    [CrossRef]
  17. G. Horváth and E. N. K. Clarkson, “Computational reconstruction of the probable change of form of the corneal lens and maturation of optics in the post-ecdysial development of the schizochroal eye of the Devonian trilobite Phacops rana milleri Stewart 1927,” J. Theor. Biol. 160, 343–373 (1993).
    [CrossRef]
  18. G. Horváth, “The lower lens unit in schizochroal trilobite eyes reduces reflectivity: on the possible optical function of the intralensar bowl,” Hist. Biol. 12, 83–92 (1996).
    [CrossRef]
  19. J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
    [CrossRef]
  20. M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
    [CrossRef]
  21. R. Feist, “The effect of paedomorphosis in eye reduction on patterns of evolution and extinction in trilobites,” in Evolutionary Change and Heterochrony, K. J. McNamara, ed. (Wiley, 1995), pp. 225–244.
  22. E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
    [CrossRef]
  23. G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
    [CrossRef]
  24. V. B. Meyer-Rochow, “Structure and function of the larval eye of the sawfly, Perga,” J. Insect Physiol. 20, 1565–1591 (1974).
    [CrossRef]
  25. E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
    [CrossRef]
  26. D. Fordyce and T. W. Cronin, “Trilobite vision: a comparion of schizochroal and holochroal eyes with compound eyes of modern arthropods,” Paleobiology 19, 288–303 (1993).
  27. B. Schoenemann, “Trilobite eyes and a new type of neural superposition eye in an ancient system,” Palaeontographica A 281, 63–91 (2007).
  28. X. G. Zhang and E. N. K. Clarkson, “The eyes of Lower Cambrian eodiscid trilobites,” Palaeontology 33, 911–933 (1990).
  29. J. Gál, G. Horváth, and E. N. K. Clarkson, “Reconstruction of the shape and optics of the lenses in the abathochroal-eyed trilobite Neocobboldia chinlinica,” Hist. Biol. 14, 193–204 (2000).
    [CrossRef]
  30. Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
    [CrossRef]
  31. E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).
  32. M. F. Land and D.-E. Nilsson, Animal Eyes (Oxford University, 2002), p. 221.
  33. H. Hinton and G. Jarman, “Physiological colour change in the Hercules beetle,” Nature 238, 160–161 (1972).
    [CrossRef]
  34. M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
    [CrossRef]

2010 (1)

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

2008 (1)

E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).

2007 (2)

B. Schoenemann, “Trilobite eyes and a new type of neural superposition eye in an ancient system,” Palaeontographica A 281, 63–91 (2007).

M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
[CrossRef]

2006 (1)

E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
[CrossRef]

2005 (1)

A. T. Thomas, “Developmental palaeobiology of trilobite eyes and its evolutionary significance,” Earth Sci. Rev. 71, 77–93 (2005).
[CrossRef]

2000 (2)

J. Gál, G. Horváth, and E. N. K. Clarkson, “Reconstruction of the shape and optics of the lenses in the abathochroal-eyed trilobite Neocobboldia chinlinica,” Hist. Biol. 14, 193–204 (2000).
[CrossRef]

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

1999 (1)

E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
[CrossRef]

1998 (1)

T. McCormick and R. A. Fortey, “Independent testing of a paleobiological hypothesis: the optical design of two pelagic trilobites reveals their relative palaeobathymetry,” Paleobiology 24, 235–253 (1998).

1997 (1)

G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
[CrossRef]

1996 (1)

G. Horváth, “The lower lens unit in schizochroal trilobite eyes reduces reflectivity: on the possible optical function of the intralensar bowl,” Hist. Biol. 12, 83–92 (1996).
[CrossRef]

1993 (2)

D. Fordyce and T. W. Cronin, “Trilobite vision: a comparion of schizochroal and holochroal eyes with compound eyes of modern arthropods,” Paleobiology 19, 288–303 (1993).

G. Horváth and E. N. K. Clarkson, “Computational reconstruction of the probable change of form of the corneal lens and maturation of optics in the post-ecdysial development of the schizochroal eye of the Devonian trilobite Phacops rana milleri Stewart 1927,” J. Theor. Biol. 160, 343–373 (1993).
[CrossRef]

1990 (1)

X. G. Zhang and E. N. K. Clarkson, “The eyes of Lower Cambrian eodiscid trilobites,” Palaeontology 33, 911–933 (1990).

1989 (1)

G. Horváth, “Geometric optics of trilobite eyes: a theoretical study of the shape of aspherical interface in the cornea of schizochroal eyes of phacopid trilobites,” Math. Biosci. 96, 79–94 (1989).
[CrossRef]

1985 (1)

R. A. Fortey, “Pelagic trilobites as an example of deducing the life habits of extinct arthropods,” Trans. R. Soc. Edinburgh Earth Sci. 76, 219–230 (1985).
[CrossRef]

1980 (1)

J. Miller and E. N. K. Clarkson, “The post-ecdysial development of the cuticle and the eye of Phacops rana milleri,” Philos. Trans. R. Soc. Lond. Ser. B 288, 461–480 (1980).
[CrossRef]

1979 (1)

E. N. K. Clarkson, “The visual system of trilobites,” Palaeontology 22, 1–22 (1979).
[CrossRef]

1975 (2)

E. N. K. Clarkson, “The evolution of the eye in trilobites,” Fossils Strata 4, 7–31 (1975).

E. N. K. Clarkson and R. Levi-Setti, “Trilobite eyes and the optics of Des Cartes and Huygens,” Nature 254, 663–667 (1975).
[CrossRef]

1974 (1)

V. B. Meyer-Rochow, “Structure and function of the larval eye of the sawfly, Perga,” J. Insect Physiol. 20, 1565–1591 (1974).
[CrossRef]

1973 (1)

K. M. Towe, “Trilobite eyes: calcified lenses in vivo,” Science 179, 1007–1009 (1973).
[CrossRef]

1972 (2)

H. Hinton and G. Jarman, “Physiological colour change in the Hercules beetle,” Nature 238, 160–161 (1972).
[CrossRef]

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[CrossRef]

1967 (1)

E. N. K. Clarkson, “Fine structure of the eye in two species of Phacops (Trilobita),” Palaeontology 10, 603–616 (1967).

1901 (1)

G. Lindström, “Researches on the visual organs of the trilobites,” Kongliga Svenska Vertenskaps Akademiens Handlingar 8, 1–89 (1901).

Buschbeck, E.

E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
[CrossRef]

Clarkson, E.

E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
[CrossRef]

Clarkson, E. N. K.

E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
[CrossRef]

G. Horváth and E. N. K. Clarkson, “Computational reconstruction of the probable change of form of the corneal lens and maturation of optics in the post-ecdysial development of the schizochroal eye of the Devonian trilobite Phacops rana milleri Stewart 1927,” J. Theor. Biol. 160, 343–373 (1993).
[CrossRef]

X. G. Zhang and E. N. K. Clarkson, “The eyes of Lower Cambrian eodiscid trilobites,” Palaeontology 33, 911–933 (1990).

J. Miller and E. N. K. Clarkson, “The post-ecdysial development of the cuticle and the eye of Phacops rana milleri,” Philos. Trans. R. Soc. Lond. Ser. B 288, 461–480 (1980).
[CrossRef]

E. N. K. Clarkson, “The visual system of trilobites,” Palaeontology 22, 1–22 (1979).
[CrossRef]

E. N. K. Clarkson and R. Levi-Setti, “Trilobite eyes and the optics of Des Cartes and Huygens,” Nature 254, 663–667 (1975).
[CrossRef]

E. N. K. Clarkson, “The evolution of the eye in trilobites,” Fossils Strata 4, 7–31 (1975).

E. N. K. Clarkson, “Fine structure of the eye in two species of Phacops (Trilobita),” Palaeontology 10, 603–616 (1967).

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “The eye: paleontology,” in Frontiers of Biology—Italian Encyclopedia. Part I. Origin and Evolution of Life. Section 7. Construction of the Organism, D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (2002), pp. 379–395.

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “Paleontologia dell’occhio (Paleontology of the eye),” in Frontiere della Vita—Enciclopedia Italiana (Frontiers of Biology—Italian Encyclopedia). I. Origine ed evoluzione della vita. (Origin and Evolution of Life) 7. La construzione degli organismi (Construction of the Organism), D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (1998), pp. 365–379(in Italian). http://www.treccani.it

E. N. K. Clarkson, “The eye, morphology, function and evolution,” in Treatise on Invertebrate Paleontology, part O, Trilobita, Revised, R. L. Kaesler et al., eds. (University of Kansas, 1997), pp. 114–132.

Cronin, T. W.

D. Fordyce and T. W. Cronin, “Trilobite vision: a comparion of schizochroal and holochroal eyes with compound eyes of modern arthropods,” Paleobiology 19, 288–303 (1993).

Egri, Á.

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

Ehmer, B.

E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
[CrossRef]

Feist, R.

R. Feist, “The effect of paedomorphosis in eye reduction on patterns of evolution and extinction in trilobites,” in Evolutionary Change and Heterochrony, K. J. McNamara, ed. (Wiley, 1995), pp. 225–244.

Fordyce, D.

D. Fordyce and T. W. Cronin, “Trilobite vision: a comparion of schizochroal and holochroal eyes with compound eyes of modern arthropods,” Paleobiology 19, 288–303 (1993).

Fortey, R. A.

T. McCormick and R. A. Fortey, “Independent testing of a paleobiological hypothesis: the optical design of two pelagic trilobites reveals their relative palaeobathymetry,” Paleobiology 24, 235–253 (1998).

R. A. Fortey, “Pelagic trilobites as an example of deducing the life habits of extinct arthropods,” Trans. R. Soc. Edinburgh Earth Sci. 76, 219–230 (1985).
[CrossRef]

Gál, J.

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

Haiman, O.

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

Hinton, H.

H. Hinton and G. Jarman, “Physiological colour change in the Hercules beetle,” Nature 238, 160–161 (1972).
[CrossRef]

Horváth, Á.

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

Horváth, G.

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).

E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
[CrossRef]

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
[CrossRef]

G. Horváth, “The lower lens unit in schizochroal trilobite eyes reduces reflectivity: on the possible optical function of the intralensar bowl,” Hist. Biol. 12, 83–92 (1996).
[CrossRef]

G. Horváth and E. N. K. Clarkson, “Computational reconstruction of the probable change of form of the corneal lens and maturation of optics in the post-ecdysial development of the schizochroal eye of the Devonian trilobite Phacops rana milleri Stewart 1927,” J. Theor. Biol. 160, 343–373 (1993).
[CrossRef]

G. Horváth, “Geometric optics of trilobite eyes: a theoretical study of the shape of aspherical interface in the cornea of schizochroal eyes of phacopid trilobites,” Math. Biosci. 96, 79–94 (1989).
[CrossRef]

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “The eye: paleontology,” in Frontiers of Biology—Italian Encyclopedia. Part I. Origin and Evolution of Life. Section 7. Construction of the Organism, D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (2002), pp. 379–395.

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “Paleontologia dell’occhio (Paleontology of the eye),” in Frontiere della Vita—Enciclopedia Italiana (Frontiers of Biology—Italian Encyclopedia). I. Origine ed evoluzione della vita. (Origin and Evolution of Life) 7. La construzione degli organismi (Construction of the Organism), D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (1998), pp. 365–379(in Italian). http://www.treccani.it

Hoy, R.

E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
[CrossRef]

Jarman, G.

H. Hinton and G. Jarman, “Physiological colour change in the Hercules beetle,” Nature 238, 160–161 (1972).
[CrossRef]

Kriska, G.

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

Land, M. F.

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
[CrossRef]

M. F. Land and D.-E. Nilsson, Animal Eyes (Oxford University, 2002), p. 221.

Lee, M.

M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
[CrossRef]

Levi-Setti, R.

E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).

E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
[CrossRef]

E. N. K. Clarkson and R. Levi-Setti, “Trilobite eyes and the optics of Des Cartes and Huygens,” Nature 254, 663–667 (1975).
[CrossRef]

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “Paleontologia dell’occhio (Paleontology of the eye),” in Frontiere della Vita—Enciclopedia Italiana (Frontiers of Biology—Italian Encyclopedia). I. Origine ed evoluzione della vita. (Origin and Evolution of Life) 7. La construzione degli organismi (Construction of the Organism), D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (1998), pp. 365–379(in Italian). http://www.treccani.it

R. Levi-Setti, Trilobites, 2nd ed. (University of Chicago, 1993).

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “The eye: paleontology,” in Frontiers of Biology—Italian Encyclopedia. Part I. Origin and Evolution of Life. Section 7. Construction of the Organism, D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (2002), pp. 379–395.

Lindström, G.

G. Lindström, “Researches on the visual organs of the trilobites,” Kongliga Svenska Vertenskaps Akademiens Handlingar 8, 1–89 (1901).

McCormick, T.

T. McCormick and R. A. Fortey, “Independent testing of a paleobiological hypothesis: the optical design of two pelagic trilobites reveals their relative palaeobathymetry,” Paleobiology 24, 235–253 (1998).

Meyer-Rochow, V. B.

V. B. Meyer-Rochow, “Structure and function of the larval eye of the sawfly, Perga,” J. Insect Physiol. 20, 1565–1591 (1974).
[CrossRef]

Miller, J.

J. Miller and E. N. K. Clarkson, “The post-ecdysial development of the cuticle and the eye of Phacops rana milleri,” Philos. Trans. R. Soc. Lond. Ser. B 288, 461–480 (1980).
[CrossRef]

Nilsson, D.-E.

M. F. Land and D.-E. Nilsson, Animal Eyes (Oxford University, 2002), p. 221.

Owen, A. W.

M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
[CrossRef]

Pix, W.

G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
[CrossRef]

Schoenemann, B.

B. Schoenemann, “Trilobite eyes and a new type of neural superposition eye in an ancient system,” Palaeontographica A 281, 63–91 (2007).

Thomas, A. T.

A. T. Thomas, “Developmental palaeobiology of trilobite eyes and its evolutionary significance,” Earth Sci. Rev. 71, 77–93 (2005).
[CrossRef]

Torney, C.

M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
[CrossRef]

Towe, K. M.

K. M. Towe, “Trilobite eyes: calcified lenses in vivo,” Science 179, 1007–1009 (1973).
[CrossRef]

Whittington, H. B.

H. B. Whittington, Fossils Illustrated 2—Trilobites (Boydell, 1992).

Zhang, X. G.

X. G. Zhang and E. N. K. Clarkson, “The eyes of Lower Cambrian eodiscid trilobites,” Palaeontology 33, 911–933 (1990).

Arthropod Struct. Dev. (1)

E. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system,” Arthropod Struct. Dev. 35, 247–259 (2006).
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Earth Sci. Rev. (1)

A. T. Thomas, “Developmental palaeobiology of trilobite eyes and its evolutionary significance,” Earth Sci. Rev. 71, 77–93 (2005).
[CrossRef]

Fossils Strata (1)

E. N. K. Clarkson, “The evolution of the eye in trilobites,” Fossils Strata 4, 7–31 (1975).

Fundam. Appl. Nematol. (1)

E. N. K. Clarkson, R. Levi-Setti, and G. Horváth, “The eyes of trilobites: the oldest preserved visual system + Los ojos de los trilobites: el sistema visual más antiguo conservado (in Spanish),” Fundam. Appl. Nematol. 13, 1–70 (2008).

Hist. Biol. (3)

J. Gál, G. Horváth, and E. N. K. Clarkson, “Reconstruction of the shape and optics of the lenses in the abathochroal-eyed trilobite Neocobboldia chinlinica,” Hist. Biol. 14, 193–204 (2000).
[CrossRef]

G. Horváth, E. N. K. Clarkson, and W. Pix, “Survey of modern counterparts of schizochroal trilobite eyes: structural and functional similarities and differences,” Hist. Biol. 12, 229–263 (1997).
[CrossRef]

G. Horváth, “The lower lens unit in schizochroal trilobite eyes reduces reflectivity: on the possible optical function of the intralensar bowl,” Hist. Biol. 12, 83–92 (1996).
[CrossRef]

J. Insect Physiol. (1)

V. B. Meyer-Rochow, “Structure and function of the larval eye of the sawfly, Perga,” J. Insect Physiol. 20, 1565–1591 (1974).
[CrossRef]

J. Theor. Biol. (1)

G. Horváth and E. N. K. Clarkson, “Computational reconstruction of the probable change of form of the corneal lens and maturation of optics in the post-ecdysial development of the schizochroal eye of the Devonian trilobite Phacops rana milleri Stewart 1927,” J. Theor. Biol. 160, 343–373 (1993).
[CrossRef]

Kongliga Svenska Vertenskaps Akademiens Handlingar (1)

G. Lindström, “Researches on the visual organs of the trilobites,” Kongliga Svenska Vertenskaps Akademiens Handlingar 8, 1–89 (1901).

Math. Biosci. (1)

G. Horváth, “Geometric optics of trilobite eyes: a theoretical study of the shape of aspherical interface in the cornea of schizochroal eyes of phacopid trilobites,” Math. Biosci. 96, 79–94 (1989).
[CrossRef]

Nature (2)

E. N. K. Clarkson and R. Levi-Setti, “Trilobite eyes and the optics of Des Cartes and Huygens,” Nature 254, 663–667 (1975).
[CrossRef]

H. Hinton and G. Jarman, “Physiological colour change in the Hercules beetle,” Nature 238, 160–161 (1972).
[CrossRef]

New Phytologist (1)

Á. Egri, Á. Horváth, G. Kriska, and G. Horváth, “Optics of sunlit water drops on leaves: conditions under which sunburn is possible,” New Phytologist 185, 979–987 (2010).
[CrossRef]

Palaeontographica A (1)

B. Schoenemann, “Trilobite eyes and a new type of neural superposition eye in an ancient system,” Palaeontographica A 281, 63–91 (2007).

Palaeontology (4)

X. G. Zhang and E. N. K. Clarkson, “The eyes of Lower Cambrian eodiscid trilobites,” Palaeontology 33, 911–933 (1990).

M. Lee, C. Torney, and A. W. Owen, “Magnesium-rich intralensar structures in schizohroal trilobite eyes,” Palaeontology 50, 1031–1038 (2007).
[CrossRef]

E. N. K. Clarkson, “Fine structure of the eye in two species of Phacops (Trilobita),” Palaeontology 10, 603–616 (1967).

E. N. K. Clarkson, “The visual system of trilobites,” Palaeontology 22, 1–22 (1979).
[CrossRef]

Paleobiology (2)

T. McCormick and R. A. Fortey, “Independent testing of a paleobiological hypothesis: the optical design of two pelagic trilobites reveals their relative palaeobathymetry,” Paleobiology 24, 235–253 (1998).

D. Fordyce and T. W. Cronin, “Trilobite vision: a comparion of schizochroal and holochroal eyes with compound eyes of modern arthropods,” Paleobiology 19, 288–303 (1993).

Philos. Trans. R. Soc. Lond. Ser. B (1)

J. Miller and E. N. K. Clarkson, “The post-ecdysial development of the cuticle and the eye of Phacops rana milleri,” Philos. Trans. R. Soc. Lond. Ser. B 288, 461–480 (1980).
[CrossRef]

Prog. Biophys. Mol. Biol. (1)

M. F. Land, “The physics and biology of animal reflectors,” Prog. Biophys. Mol. Biol. 24, 75–106 (1972).
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Science (2)

E. Buschbeck, B. Ehmer, and R. Hoy, “Chunk versus point sampling: visual imaging in a small insect,” Science 286, 1178–1180 (1999).
[CrossRef]

K. M. Towe, “Trilobite eyes: calcified lenses in vivo,” Science 179, 1007–1009 (1973).
[CrossRef]

Trans. R. Soc. Edinburgh Earth Sci. (1)

R. A. Fortey, “Pelagic trilobites as an example of deducing the life habits of extinct arthropods,” Trans. R. Soc. Edinburgh Earth Sci. 76, 219–230 (1985).
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Vis. Res. (1)

J. Gál, G. Horváth, E. N. K. Clarkson, and O. Haiman, “Image formation by bifocal lenses in a trilobite eye?” Vis. Res. 40, 843–853 (2000).
[CrossRef]

Other (7)

R. Feist, “The effect of paedomorphosis in eye reduction on patterns of evolution and extinction in trilobites,” in Evolutionary Change and Heterochrony, K. J. McNamara, ed. (Wiley, 1995), pp. 225–244.

M. F. Land and D.-E. Nilsson, Animal Eyes (Oxford University, 2002), p. 221.

H. B. Whittington, Fossils Illustrated 2—Trilobites (Boydell, 1992).

E. N. K. Clarkson, “The eye, morphology, function and evolution,” in Treatise on Invertebrate Paleontology, part O, Trilobita, Revised, R. L. Kaesler et al., eds. (University of Kansas, 1997), pp. 114–132.

R. Levi-Setti, Trilobites, 2nd ed. (University of Chicago, 1993).

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “Paleontologia dell’occhio (Paleontology of the eye),” in Frontiere della Vita—Enciclopedia Italiana (Frontiers of Biology—Italian Encyclopedia). I. Origine ed evoluzione della vita. (Origin and Evolution of Life) 7. La construzione degli organismi (Construction of the Organism), D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (1998), pp. 365–379(in Italian). http://www.treccani.it

R. Levi-Setti, E. N. K. Clarkson, and G. Horváth, “The eye: paleontology,” in Frontiers of Biology—Italian Encyclopedia. Part I. Origin and Evolution of Life. Section 7. Construction of the Organism, D. Baltimore, R. Dulbecco, F. Jacob, and R. Levi-Montalcini, eds. (2002), pp. 379–395.

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Figures (10)

Fig. 1.
Fig. 1.

(a) Main longitudinal section of the central-cored lens in a Silurian Dalmanites parallel to the optical axis of the lens (after Fig. 3 on p. 664 in [15]). Cc, central core; uu, upper unit; lu, lower unit. (b) Shape of the refractive surfaces of the Dalmanites lens in the coordinate system. f 1 ( r ) , outer lens surface; f 2 ( r ) , inner lens surface; f 3 ( r ) , interface between the upper and lower lens units; f 4 ( r ) , upper surface of the core; f 5 ( r ) , lower surface of the core; R , lens radius; n w , refractive index of seawater; n uu , refractive index of the upper lens unit; n lu , refractive index of the lower lens unit; n cc , refractive index of the central core; n bf , refractive index of body fluid; Z , optical axis of the lens coinciding with the axis of rotation symmetry.

Fig. 2.
Fig. 2.

Path of a light ray starting from point p 0 with direction e ̲ 0 , if there are two refracting surfaces, the vertical main longitudinal sections of which are described by the functions f ( r ) and g ( r ) . The refractive indices of the different optical media are n 0 , n 1 , and n 2 . The direction of the refracted ray is e ̲ 1 after refraction at point p 1 . The angles of incidence and refraction are α and β , respectively. The normal vectors of the refractive surfaces f ( r ) and g ( r ) are N ̲ f and N ̲ g at points p 1 and p 2 , respectively. e ̲ 0 , e ̲ 1 , N ̲ f , and N ̲ g are unit vectors ( e 0 = e 1 = N f = N g = 1 ).

Fig. 3.
Fig. 3.

(a) Division of a paraxial homogenous light beam into m = 10 zones, for example. (b) Ray-tracing through the main longitudinal section of a central-cored trilobite lens to calculate the intensity along the optical axis below the lens. The darker the infinitesimal cylindrical cells (with radius ρ , and height 2 ρ ) along the optical axis, the more rays pass through them.

Fig. 4.
Fig. 4.

Examples for the relative intensity i = I / I beam as a function of the relative distance l = L / R from the bottom of the studied central-cored trilobite lens (Fig. 1). (a) One relative intensity peak, n lu = 1.545 , n cc = 1.66 . (b) Two peaks, n lu = 1.42 , n cc = 1.605 . (c) Three peaks, n lu = 1.64 , n cc = 1.645 . The inset in (a) defines the sharpness Q = h / w of a focal point, where h is the height and w is the width of the peak at i = 0.8 · h .

Fig. 5.
Fig. 5.

Computation of the intensity distribution below a central-cored trilobite lens in the x z plane divided into 600 × 4000 cells representing a matrix, the elements of which were initially set to 0. The elements hit by a ray refracted by the lens are increased by 1. As examples, two rays [(a), red; (b), blue] are here shown starting from the point p ̲ red = ( 0.75 R , 0 , R ) and p ̲ blue = ( 0 , 0.75 R , R ) , respectively, where R is the lens radius.

Fig. 6.
Fig. 6.

Left: relative intensity i as a function of the relative distance l from the bottom of the studied central-cored trilobite lens (Fig. 1) along the optical axis when the curve i ( l ) has only one pronounced peak, the position of which is marked by a thin vertical line. The values of the refractive indices n lu and n cc of the lower lens unit and the central core, respectively, are given in the insets. The labels R 1 , , R 16 correspond to the same labels in Fig. 9. Right: ray-tracing in the main longitudinal plane parallel to the optical axis of the lens.

Fig. 7.
Fig. 7.

Left: Relative intensity i as a function of the relative distance l from the bottom of the studied central-cored trilobite lens (Fig. 1) along the optical axis when the curve i ( l ) has two pronounced peaks, the positions of which are marked by thin vertical lines. The values of the refractive indices n lu and n cc of the lower lens unit and the central core, respectively, are given in the insets. The labels G 1 , , G 38 correspond to the same labels in Fig. 9. Right: Ray-tracing in the main longitudinal plane parallel to the optical axis of the lens.

Fig. 8.
Fig. 8.

Left: relative intensity i as a function of the relative distance l from the bottom of the studied central-cored trilobite lens (Fig. 1) along the optical axis when the curve i ( l ) has three pronounced peaks, the positions of which are marked by thin vertical lines. The values of the refractive indices n lu and n cc of the lower lens unit and the central core, respectively, are given in the insets. The labels B 1 , , B 38 correspond to the same labels in Fig. 9. Right: ray-tracing in the main longitudinal plane parallel to the optical axis of the lens.

Fig. 9.
Fig. 9.

The sides of the large rectangle represent the intervals of the refractive indices n lu and n cc of the lower lens unit and the central core in the studied central-cored trilobite lens (Fig. 1). A given point in the rectangle corresponds to a given value-pair ( n lu , n cc ) . The positions of the small white squares represent the values ( n lu , n cc ) for which the relative intensity i was computed versus the relative distance l from the bottom of the lens along its optical axis. The colors (red, green, blue) and labels (R, G, B) represent the number (1, 2, 3) of pronounced peaks of the curve i ( l ) , some of which are shown in Figs. 6, 7, and 8. The vertical and horizontal white dashed lines represent n lu = n cc = 1.66 . The yellow dashed line represents the situations where the detection of the furthest focal point become possible: below this line the furthest focal point is out of view.

Fig. 10.
Fig. 10.

Sharpness Q of a focal point [defined by Eq. (15) and Fig. 4(a)] as functions of the refractive indices n lu and n cc of the lower lens unit and the central core. (a)  Q ( n lu , n cc ) of the single peak of i ( l ) for values of ( n lu , n cc ) represented by red (medium gray in print) in Fig. 9. (b)  Q ( n lu , n cc ) of the first (closer to the lens) peak of i ( l ) for values of ( n lu , n cc ) represented by green (light gray in print) in Fig. 9. (c)  Q ( n lu , n cc ) of the second (further from the lens) peak of i ( l ) for values of ( n lu , n cc ) represented by green in Fig. 9. The darker the gray, the smaller the value of Q in such a way that the gray shades code the values of q = ( Q / Q max ) 1 / 3 , where Q max = 3096223 (black: Q = 0 , q = 0 % ; white: Q = Q max , q = 100 % ). In diagram (a), those areas are striped where the numbers of i ( l ) -peaks are 2 or 3 (coinciding with the green and blue (dark, upper right in print) regions in Fig. 9). In diagrams (b) and (c), those areas are striped where the numbers of i ( l ) -peaks are 3 or 1 (coinciding with the blue and red regions in Fig. 9).

Equations (17)

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f 1 ( r ) = A 1 + B 1 r 2 , f 2 ( r ) = A 2 + B 2 r 2 , f 3 ( r ) = A 3 + B 3 cos ( C 3 r ) , f 4 ( r ) = { A 4 ( 1 r 2 / B 4 2 ) 1 / 2 r B 4 0 otherwise , f 5 ( r ) = { A 5 ( 1 r 2 / B 5 2 ) 1 / 2 r B 5 0 otherwise , A 1 = 0.572039 · R , B 1 = 0.425220 / R , A 2 = 1.065815 · R , B 2 = 0.358080 / R , A 3 = 0.774962 · R , B 3 = 0.224472 · R , C 3 = 3.368190 / R , A 4 = 0.268429 · R , B 4 = 0.316354 · R , A 5 = 0.437346 · R , B 5 = B 4 ,
p ̲ ( t ) = p ̲ 0 + e ̲ 0 · t x ( t ) = x 0 + e 0 x · t , y ( t ) = y 0 + e 0 y · t , z ( t ) = z 0 + e 0 z · t ,
f ( x 0 + e 0 x · t , y 0 + e 0 y · t ) = z 0 + e 0 z · t .
N ̲ = e ̲ 1 × e ̲ 2 | e ̲ 1 × e ̲ 2 | , with e ̲ 1 = ( 1 , 0 , f ( x , y ) x ) , e ̲ 2 = ( 0 , 1 , f ( x , y ) y ) .
e ̲ new = e ̲ old n ( cos β cos α n ) · N ̲ ,
r = R / m ,
A k = ( k r ) 2 π [ ( k 1 ) r ] 2 π = ( 2 k 1 ) r 2 π = ( 2 k 1 ) R 2 π / m 2 , k = 1 , 2 , , m .
A beam = R 2 π ,
P beam = I beam · A beam ,
P k = P beam A k / A beam = I beam A k ,
Δ I k = P k / A cell , A cell = ρ 2 · π ,
Δ I k = ( I beam A k ) / ( ρ 2 π ) .
1.36 < n lu < 1.68 , 1.52 < n cc < 1.74 .
i ( l ) smoothed = k = α α i ( k ) · 1 2 π σ 2 e ( l k ) 2 2 σ 2 d k with σ = 0.02 · R ( = length of 10 cells ) and α = 5 · σ .
Q = h / w
i ( l i 1 ) < i ( l i ) and i ( l i + 1 ) < i ( l i ) ,
Q 4000 .

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