Is a Snail Shell Archimedean or Logarithmic?
Snail shells, hurricanes and galaxies all spiral, but not all spirals are the same curve. Fit an Archimedean and a logarithmic spiral to a photograph of a real spiral and see which one actually matches.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Spirals appear throughout nature and engineering: snail shells, hurricane cloud bands, spiral galaxies, coiled hosepipes. Not all of these are the same mathematical curve. An Archimedean spiral has turns that are evenly spaced (its radius grows linearly with angle); a logarithmic spiral grows in radius by the same factor with every turn (its radius grows exponentially with angle), which is why it appears in shells that grow by adding material without changing shape. This exploration fits both spiral types, in polar form, to a photograph of a chosen object, and uses the better-fitting one to say something about how that object grew.
Guiding Questions
- Collect examples of spirals — shells, plants, galaxies, a coiled hose. Photograph or sketch one and mark points along it in polar coordinates.
- Write the equations of an Archimedean spiral and a logarithmic spiral in polar form, and plot both over your photograph.
- Linearise the logarithmic spiral's equation (taking logarithms) and use a best-fit line to estimate its growth parameter. Which spiral type fits your object better?
- Estimate the length of your fitted spiral. Its exact arc length is beyond this course, so approximate it by summing many short straight segments along the curve.