Logarithmic spirals frequently occur in nature. Is this such a manifestation?
The general equation for a logarithmic spiral is as follows.
Changing the variables and produces spirals of different qualities. The is really an enlargement scale factor but the controls how the spiral grows per revolution.
If is more representative of an ammonite then 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…
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.
- Mathematics appears rather difficult if one is not familiar with it.
- Students need energy, eagerness and enthusiasm.
- 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.
This preamble from Euclid’s Elements is where mathematics education still goes wrong, even after 2000 years or so.
Sound notions of what equality is are required as a bedrock of algebra. Dynamic approaches of teaching equation solving, in which terms and numbers move and change sign, seem to work at first but lack the simplicity of mathematical logic and create all sorts of problems when randomly, but apparently sensibly, applied.
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?
First definition of triangles from Euclid’s Elements Book I (Fundamentals of Plane Geometry Involving Straight Lines), Definition 20.
Intersection of perpendicular normals on one parabola makes another. Another A level further maths conic sections locus problem.
A level further mathematics question animated. Mid point of Normal x-intercept and Tangent y-intercept traces out another curve as original point moves round an ellipse.
Hyperbola – locus of points in which ratio of distance from a focus point to the perpendicular distance to a line (directrix) is constant and greater than 1. Staple diet of school Further Mathematics.
Ancient theorem of Euclid (Book III, Proposition 31) still a fundamental part of school maths. The angle in a semi-circle is a right angle.
Sneaky play with algebraic notation and function definition lays traps for scholarship students in school maths.