Apples and Lemons in a Doughnut

A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let {r} be the radius of the circle and {R} the distance from the axis to the centre of the circle, with {R>r}.

Generating a ring torus by rotating a circle of radius {r} about an axis at distance {R>r} from its centre.

If the axis of revolution is tangent to the circle ({R=r}), the generated surface is called a horn torus. If the axis intersects the circle ({0<R<r}), the generated surface is a spindle torus. For the special case  {R=0}, with the circle rotated about a diameter, we get a sphere.

Axis and generating circles for a horn torus (left), a spindle torus (centre) and a sphere (right).

Cutaway of a horn torus (left), a spindle torus (centre) and a sphere (right).

The calculation of the volume and surface area of a ring torus is a straightforward exercise in undergraduate calculus or, even more simply, an application of Pappus’s centroid theorems:

  1. The area of a surface of revolution generated by rotating a plane curve about an external axis in the same plane is equal to the product of the arc length of the curve and the distance travelled by the centroid of the curve.
  2. The volume of a solid of revolution generated by rotating a plane figure about an external axis in the same plane is equal to the product of the area of the figure and the distance travelled by its centroid.

For the torus, this gives

\displaystyle \begin{array}{rcl} S &= (2\pi R)\times(2\pi r) &= 4\pi^2 Rr \\  V &= (2\pi R)\times(\pi r^2) &= 2\pi^2 Rr^2 \end{array}

These values are the same as those for a cylinder of length {2\pi R} and radius {r}. The deficits in surface area and volume on the inside of the torus exactly cancel the excesses on the outside. For a horn torus, with {R = r}, we get the limiting values {S = (2\pi r)^2} and {V = 2\pi^2 r^3}.

Apples and Lemons

For a spindle torus, the surface intersects itself and care is required: a blind application of the theorems would give zero area and volume for a sphere! We note that the spindle torus has two distinct parts, one shaped like an apple and one like a lemon:

Cross-section of a spindle torus. The two parts generate an apple and a lemon.

The lemon is generated by rotating an arc of angle {2\alpha} less than {\pi} about its chord, with {\cos\alpha = R/r}. The surface area is given by

\displaystyle S = 2\pi r^2 \int_{-\alpha}^{\alpha} (\cos\phi-\cos\alpha)\mathrm{d}\phi \,,

and the volume is given by

\displaystyle V = \pi r^3 \int_{-\alpha}^{\alpha} (\cos\phi-\cos\alpha)^2 \cos\phi\,\mathrm{d}\phi \,.

These integrals can be evaluated analytically, giving

\displaystyle S = 4\pi r^2 [ \sin\alpha - \alpha\cos\alpha ]  \qquad\qquad \qquad\qquad       (1) 

\displaystyle  V = \textstyle{\frac{4}{3}}\pi r^3 [ \sin^3\alpha - \textstyle{\frac{3}{4}} \cos\alpha(2\alpha - \sin 2\alpha ) ]      \qquad   (2)

For the limiting case {\alpha=\frac{\pi}{2}}, these yield {S = 4\pi r^2} and {V = \frac{4}{3}\pi r^3}, the values for a sphere.

The apple is generated by rotating an arc of angle {2\alpha} greater than {\pi} about its chord. The formulas (1) and (2) are still valid. The surface area and volume of a lemon (with {\alpha<\pi/2}) and apple (with {\alpha>\pi/2}) are shown in this Figure.

The surface area (left) and volume (right) of a lemon or apple as a function of the angle {\alpha} ({r=1}). For the lemon, {\alpha<\pi/2}; for the apple, {\alpha>\pi/2}.}

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Fusiforms

Apples and lemons are special cases of orbifolds, topologically equivalent to a sphere but with two singular points. The apple and lemon together make up a self-intersecting spindle torus, the volume generated by rotation of the full circle, with the apple as the outer shell of the torus and the lemon as its inner shell.

The spindle-shaped solid that we call a lemon is also known as a fusiform. Wide in the middle and tapered at the ends, a fusiform may be fat like a lemon or, for a smaller opening angle, slim like a cigar (see Figure at top of this article).

Many aquatic animals have a fusiform shape. Some bacteria, such as the fusobacteria are of this shape, and a form of aneurysm in blood vessels also bears the name. Biological cells and some muscles in the body are fusiform in shape. The fusiform gyrus is part of the temporal lobe of the brain.


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