![]() As the nautilus matures its body moves forward, sealing the camerae behind it with a new septum. The phragmocone is divided into camerae by septa, all of which are pierced in the middle by a duct, the siphuncle. The shell is internally divided into chambers, the chambered section being called the phragmocone. The osmena pearl, contrarily to its name, is not a pearl, but a jewelry product derived from this part of the shell. The innermost portion of the shell is a pearlescent blue-gray. The nautilus shell is composed of 2 layers: the outer layer is a matte white, while the inner layer is a striking white with iridescence. The shell is coiled, calcareous, nacreous and pressure resistant (imploding at a depth of about 800 m). The animal can withdraw completely into its shell, closing the opening with a leathery hood formed from two specially folded tentacles. In practice, I had to shuffle them around a bit before finding their optimal locations.Nautiluses are the sole cephalopods whose bony structure of the body is externalized as a shell. In principle, the pole X and the points Y,Z define the logarithmic spiral unambiguously. I had Geogebra compute a,b from the points where the shell curve cuts the positive x-axis for the first time (Y, t=0) and for the third time (Z, t=4π). The equation r(t)=a exp(bt) implies that a=r(0) and that b can be determined from, e.g., r(4π)=a exp(4πb). I then defined a vector from X to the origin, translated pic1 by this vector (pic1' has the origin as pole), rotated pic1' (pic1'' shows the shell with its endpoint on the x-axis) and made pic1' invisible.ĥ. (This angle reflects how the shell happened to be placed on the scanner, and doesn't mean anything.) I had Geogebra compute the angle α by which the image should be rotated around X in order to give the left endpoint of the shell the ordinate of X. So one should regard the pole, critical as it is, as a fuzzy spot subject to some manoeuvrability. Also, moving A and B along the shell may lead to different locations of the pole. But the thickness of the walls of the nautilus allows some variation in choosing points "on the curve" and deciding when the vector points at a point "on the curve". If the shell curve were a geometric object and a perfect logarithmic spiral, this method would unambiguously lead from any two points A,B to the pole X. Then we have XA/XC=XC/XB, which is characteristic of the pole X. Move X around until the vector points exactly at a point, say C, of the shell curve. Choose any two points A,B on the shell curve and have Geogebra display a vector with starting point X, bisecting the angle formed by XA and XB, and with the geometric mean of these radii as length. Two expert authors locate the pole differently (compare p.386, fig.11įortunately, Geogebra allows a new high-precision technique to pinpoint the pole, which is as follows. The pole are described, to which added a new one. The axis of coiling (the pole of the logarithmic spiral) matters a lot,īecause we're dealing with an exponential, and a small imprecision ![]() Left corner at (0,0) and its lower right corner at (10,0).Ĥ. I then cropped the image to the sole shell ( here) and pasted the cropped png image as pic1 into a new Geogebra sheet, with its lower The plane of the cut), I decided not to interfere with the scanned (thickness of the Nautilus lines, possible and unknown inclination of That the most distorted square (top right) had deviations of less thanĠ.5 mm in 50, i.e., less than 1%. (Arbitrary choices, toĪttach the image to the axes while zooming and moving it.) I had Geogebraĭraw perfect squares on top of my scanned pin-holed squares, and found Image into a Geogebra worksheet, with the lower left corner at (0,0) and Using Preview.app I transformed the tiff file into PNG, then pasted the I continued with the first one, because this one has more fine structure near the center.ģ. Results (TIFF files, 600 dpi) are here and here. These squares allow to detectĪnd quantify any deformations caused by the scanning process. Strips of graph paper on each of which I had marked by pin-holes theĮdges of two squares of 50 mm by 50 mm. I put each half (in no particular position) on my CanoScan 9000F scanner, between two I bought two halves of a Nautilus Pompilius shell (€ 11.95 a piece) at De Schelpenshop.Ģ.
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