To describe our central image in technical language, we have to define a central curve X(t) over a parameter t. The curve we chose is an arithmetic spiral lifted onto a cone, with equation
X(t) = (tcos(t), tsin(t), pt).
The radius function r(t) that describes the expanding cone is given by 0.8e0.6t.
- As t goes from 0 to p/2, we show a number of points on the curve. From p/2 to p, we show the curve itself, actually represented by a thin tube.
- From p to 3p/2, we show a "normal strip" of the form
X(t) + rP(t)where r goes from -r(t) to r(t). Here P(t) corresponds to the principal normal, a unit vector perpendicular to T(t), the unit tangent vector of the curve and lying in the plane determined by the velocity and the acceleration of the curve.
- From 3p/2 to 2p, the figure is an expanding cone, or "cornucopia". The equation of the circular slice for each t is
X(t) + r(t)(cos(u)P(t) + sin(u)B(t))
where B(t) is the unit binormal vector perpendicular to T(t) and P(t), and where u is a parameter that goes from 0 to 2p.
Thus the image grows not only in size, but but also in dimension.
If we continue to the next section of the curve, from 2p to 5p/2, the slices will be two-dimensional spheres, given by
X(t) = r(t)(cos(v)cos(u)P(t) + cos(v)sin(u)B(t) + sin(v)W(t))
where W(t) is a unit vector in 4-space perpendicular to T(t), P(t), and B(t). We can project this slice into our 3-space by pushing W(t) down to T(t), and this is the view we show on the previous page.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.