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Mathematics and Morphology: Exploring Transformations in Biology

Discover the interplay between mathematics and morphology in biology through homologous structures, transformations, and shape comparisons. Learn how insights from simple transformations provide clues to species relationships and growth forces. Explore the significance of homologies and the implications of transformations between species. Dive into the study of shape descriptions and the predictive power of morphing in biology.

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Mathematics and Morphology: Exploring Transformations in Biology

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Presentation Transcript

  1. Announcements • Mid-term given out the week after next. • Send powerpoint to me after presentation.

  2. Thompson: Comparison of Related Forms

  3. Key Points • Math is helpful for morphology. • Homologous structures necessary: correspondence. • Given these, compute transformations of plane. • Uses: • Nature of transformation gives clues to forces of growth. • Shapes related by simple transformation -> species are related. Many compelling examples. • Morph between species, predict intermediate species. • Can predict missing parts of skeleton.

  4. Math is helpful for morphology • Seems pretty obvious. • This was a radical view in biology.

  5. Homologies • Had a long tradition • Aristotle: Save only for a difference in the way of excess or defect, the parts are identical in the case of such animals as are of one and the same genus. • In biology, study of homologous structures in species preceded and provided background for Darwin. • Homologous structures explained by God creating different species according to a common plan. • Ontogeny provided clues to homology.

  6. Transformations • Given matching points in two images, we find a transformation of plane. • Homeomorphism (continuous, one-to-one) • This is underconstrained problem • Implicitly, seeks simple transformation. • Not well defined here, will be subject of much future research. • Intuitively pretty clear in examples considered.

  7. Cannon-bone of ox, sheep, giraffe Simplest, subset of affine

  8. Piecewise affine

  9. Logarithmically varying: eg., tapir’s toes

  10. Smooth: amphipods (a kind of crustacean).

  11. Descriptions of shape: Clues to Growth • Somewhat different topic, shape descriptions relevant even without comparison. • Introduces fourier descriptors. • Equal growth in all directions leads to circle (or sphere).

  12. No growth in one direction (as in a leaf on a stem), growth increases in directions away from this so r = sin(q/2).

  13. Asymmetric amounts of growth on two sides.

  14. Related Species

  15. Lack of transformation -> no straight line of descent.

  16. Invention of Morphing? • Given transformation between species, linearly interpolate intermediate transformations. • Intermediate morphs predict intermediate species.

  17. Pages 1070-71

  18. Figure 537

  19. Pages 1078-79

  20. Conclusions • Stress on homologies. • Shape comparison through non-trivial transformations. • Simplicity of transformation -> similarity of shape. • What is the simplest transformation? How do we find it? • Transformation may leave some deviations, how are these handled?

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