Biology Reference
In-Depth Information
at closely spaced intervals during the development of an organ such as a leaf or petal. The
complexity of the images is reduced, often simply to an outline of the organ as a whole or,
for more detailed studies, a mesh that shows the outline of each cell. At each stage,
researchers attempt to identify a mathematical transformation that will alter the shape of
a particular cell, or the whole organ, to that of the next stage. The transformation will include
some function that causes growth and perhaps also one that causes rotation. An easy method
of visualizing such a transformation is to imagine that the organ has been drawn on a rubber
sheet, and that the sheet is stretched and perhaps twisted so that the drawing is distorted in
exactly the right way to resemble the next stage of development. This type of transformation
is illustrated using the familiar object of a mature leaf in
Figure 22.14
. Transformations may
also include cell division (roughly, drawing a new line on the rubber sheet). Often, the trans-
formations required will be different in different parts of the organ and will also change
through time.
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The actual process of determining these changing transformations can be
FIGURE 22.14
Modelling shape change by mathematical transformations, illustrated in this case by applying
transformations to an image of an adult leaf (for the sake of familiarity). The outline of the simple leaf in the
centre, with a square grid, can be transformed to approximate the outline of a Persimmon leaf (top left) or Laurel
leaf (top right) by stretching the coordinate system proximo-distally and shrinking it laterally. The outline can
also be transformed into the approximate outline of a Bay leaf (bottom left) by a transformation that varies
linearly with space, or a nettle leaf (bottom right) by applying more complex transformation that includes shear-
distortion of the grid as well as linearly-changing compression. In all cases the grid lines are those of the initial
picture. Please note that this figure is included simply to illustrate the mathematical idea of spatial trans-
formation; no claim is being made about the real plants and, in particular, even where these transformations
generate correct approximate outlines they do not generate the correct pattern of leaf veins or other detailed
structures.
