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Studies on the Paua, Haliotis iris Martyn in the Wellington district, 1945-46

Determination of Logarithmic Spiral

page 11

Determination of Logarithmic Spiral

The large shell of H. iris has a very small apical spire and an extremely large last whorl which is very depressed with a relatively enormous aperture.

Crofts (1929) suggests that the flattened shell of Haliotis has been evolved from a shell with a taller spiral because of the habit of squeezing into confined spaces between rocks. The shell of Haliotis grows in the form of a logarithmic spiral. The form of a single curve following a logarithmic spiral is given by the expression Equation describing a logarithmic spiral. where r is the radius of the shell from centre to circumference; θ is the angle of revolution which the spiral has described and α is the angle between the tangent of the curve and the radius vector of this curve, which remains constant. This is known as the constant angle of the curve and has been determined in many species of Haliotis. D'Arcy Thompson (1942) states that in Haliotis the constant angle (α) varies from about 70 degrees to 75 degrees while in the majority of gastropods it lies between 80 degrees to 85 degrees or even more.

To determine the constant angle of H. iris a photograph was taken through the axis of a shell 9.9cm in largest diameter. The curve made by the line of holes in the shell was taken as the logarithmic spiral. Plate I shows a photograph and a diagram can be drawn from it. The following expression of the formula given above was used in the actual determination of the angle.

Equation describing the constant angle for a shell in the form of a logarithmic spiral.

The result obtained gives α = 53 degrees 46′ which as far as the writer can ascertain is smaller than any other value of α obtained for a Haliotis. This means then, other things being equal, that H. iris has fewer whorls per unit height of shell than other species of Haliotis.

Moore (1936) found that the value of α in Purpura lapillus varied during the lifetime of the shell. The value of α calculated from a H. iris specimen 3.72cm in largest diameter was considerably higher in degree than in the case of the specimen 9.9cm in largest diameter. Therefore it appears probable that the value of α decreases as the shell grows.

Species Longest Diameter in Cms Constant Angle of Spiral (α)
H. australis 7.75 72 degrees 6′
H. iris 3.72 60 degrees 53′
H. iris 9.9 53 degrees 46′

The constant angle of the spiral of H. australis in the above table shows a large increase over the values given for H. iris and is higher than some other species of Haliotis, e.g. H. tuberculata where α = 69 degrees 48′ (Moore, 1936).

Crofts (1929) states that the shell of H. tuberculata is so flattened that the animal is unable to retract completely into the shell. From the value of the constant angle in H. iris given above it follows that the shell of H. iris is lower in relative height; but in contrast to H. tuberculata, H. iris can retract completely within the shell when disturbed.