depression near Greenland – logarithmic spirals in nature

Logarithmic spirals frequently occur in nature.  Is this such a manifestation?

The general equation for a logarithmic spiral is as follows.

r=ae^{b\theta}

Changing the variables a and b produces spirals of different qualities.  The a  is really an enlargement scale factor but the b controls how the spiral grows per revolution.

If b=0.2 is more representative of an ammonite then b=0.75 seems to be our Greenland depression.

Wouldn’t it be nice to take the equations for atmospheric fluid dynamics and show this explicitly.  Unfortunately this is beyond the scope of this blog: here we have circular motion on the surface of a rotating sphere in an elliptical orbit … . The way to go with this would be to take the basic baratropic equations and then perform a scale analysis to disregard ‘small’ terms.  At this point the pure mathematician get frustrated with approximations and goes into a sulk.  We are left with contemplating the beautiful images though…

third century wisdom of Diophantus gives us the essential truth about teaching and learning mathematics

The Google facsimile of the 1621 edition of Diophantus’s Arithmetica lays out, in original Greek and Latin translation, the essentials for success in teaching and learning mathematics.  Also included is the translation of the second half by Heath.

When I first studied and learned to love mathematics I was beset by friends and family who stridently claimed that this field of study was closed to them because of some biological, or other, reason.  This always seemed to undervalue my own effort in getting off the first page.

Teachers of mathematics cannot be permitted to admit that nature is able to restrict the ability to learn the subject but the oriental wisdom of the student making their way to the master is a precondition for success when things seem difficult; in tales of martial arts the student must somehow prepare themselves to learn.

So,

  1.  Mathematics appears rather difficult if one is not familiar with it.
  2. Students need energy, eagerness and enthusiasm.
  3. When such motivation is backed up by good teaching rapid learning results.

In school we struggle because of item 1 and bear our hearts to gain item 2, being so often rebuffed.  I suppose that we get paid for 1 and 2 because, when item 3 is achieved the job is it’s own reward.

Conic sections in the sand give shipwrecked philosophers good hope

 

Conic sections in the sand give shipwrecked Socratic philosopher, Aristippus, good hope.

I wonder if this ancient tale, in which the mathematics of planetary motion some 2000 years prior to it’s application in space travel, is used to signify the presence of intelligent life and therefore hope of salvation has some echo in our modern era.

What obscure and impenetrable modern theorems could be left on a 21st century beach to do the same job?