The Shape of Things

Let's begin investigating relationships between shape and size by determining exact formulae for readily understood geometries. Consider first an object that is familiar to all, the lowly sphere. Spheres come in a wide variety of sizes, but all have the same shape which does not vary with viewpoint. All spheres have the following properties:

$$\Large d=2r$$

$$\Large C=2 \pi r$$

$$\Large A=4 \pi r^2$$

$$\Large V=\frac{4}{3} \pi r^3$$

In the foregoing, $$r$$ = radius (physical units of length e.g. cm), $$d$$ = diameter (physical units of length e.g. cm) $$C$$ = circumference (physical units of length e.g. cm), $$A$$ = area (physical units of length2 e.g. cm2), and $$V$$ = volume (physical units of length3 e.g. cm3).

A plot of circumference against radius results in a straight line through the origin with a slope of $$2 \pi$$ where $$\pi$$ is the miraculous irrational number, approximately 3.14159, that drops out of the sky whenever discussing circles or spheres. Since the intercept is 0, the relationship is characterized by the ratio, $$C/r \equiv 2\pi$$ , with no variation for all spheres. The ratio $$C/r$$ is dimensionless, having been derived from the ratio of 2 quantities with the same physical units, i.e. length. The ratio $$C/r$$ is undefined when $$r=0$$ but approaches the value $$2\pi$$ in the limit as $$r$$ (or $$C$$ ) approaches 0. The ratio $$C/r$$ is a shape factor for the sphere; it is absolutely independent of the size of the sphere, but is intrinsic to its shape.

The Similarity Principal

• For a set of geometrically similar objects (same shape but varying size), the ratio of any 2 characteristic lengths, 2 areas, or 2 volumes is a dimensionless constant that is particular to the shape and is independent of the objects' size in an absolute sense. The word characteristic is included because the lengths, areas, or volumes must be definable unambiguously in terms of the geometry. Note that each such ratio defines an aspect of shape; objects may be geometrically similar in one regard (one shape factor) but not another.

A number of other lengths, areas, and volumes can characterize a sphere. A length, for example, can be derived from the volume by determining the cube root, $$L_V=V^{1/3}=(4\pi/3)^{1/3}r$$ . Notice that no conceptual anomalies arise in determining the cube root of a volume. The latter has physical units of length cubed, i.e. L3, with the cube root having units of length, L1. The cube root of any volume is a length and there is an obvious physical interpretation in this case as well; the length obtained corresponds to the edge length of the cube that has the same volume as the original sphere.

In general terms, deriving a length (L1), area (L2), or volume (L3) from any other is a matter of raising the latter to the correct power or exponent. Obvious algebraic rules apply:

• Length from area: L1=(L2)1/2 exponent = 1/2
• Length from volume: L1=(L3)1/3 exponent = 1/3
• Area from length: L2=(L1)2 exponent = 2
• Area from volume: L2=(L3)2/3 exponent = 2/3
• Volume from length: L3=(L1)3 exponent = 3
• Volume from area L3=(L2)3/2 exponent = 3/2

Each of these derived lengths, areas, or volumes conforms to the original similarity principal. E.g. a length divided by the cube root of a volume is a constant that is particular to the shape. Applied specifically to the spherical geometry, certain derived characteristic lengths, areas, and volumes are as follows:

• Length from area: $$L_A=A^{1/2}=(4\pi)^{1/2}r$$
• Length from volume: $$L_V=V^{1/3}=(4\pi/3)^{1/3}r$$
• Area from length: $$A_L=r^2$$
• Area from volume: $$A_V=V^{2/3}=(4\pi/3)^{2/3}r^2$$
• Volume from length: $$V_L=r^3$$
• Volume from area: $$V_A=A^{3/2}=(4\pi)^{3/2}r^3$$

In the foregoing it is apparent that each formula is comprised of a non-dimensional shape factor that multiplies a dimensional component, r1, r2, or r3. i.e. a length, area, or volume. However a constant is obtained whenever we determine the ratio of 2 lengths, 2 areas, or 2 volumes that characterize the sphere. Each ratio obtained characterizes a specific aspect of the geometry that is absolutely independent of size.

Note that while each of the above formulas has been expressed in terms of the sphere radius, any one of them could be expressed directly in terms of any other one using simple algebra. Sphere area, for example, in terms of sphere volume is $$4\pi r^2=4\pi[3V/4\pi]^{2/3}$$ , where the previously determined expression for radius has been substituted, and simplifies to $$A=6^{2/3}\pi^{1/3}V^{2/3}$$ . Once again we have a non-dimensional constant ($$6^{2/3}\pi^{1/3}$$ ) that multiplies a dimensional component that has the correct physical units, i.e. V2/3 has units of area, L2.

For a set of geometrically similar objects, any characteristic length, area, or volume can be derived from any other by applying the correct exponent and multiplying by the shape factor.

$$\Large L^A=\left [ \frac{L_0^A}{(L_0^B)^{A/B}}\right] \left [(L^B)^{A/B} \right]$$

This equation generalizes the similarity principal where L is length; A and B can be 1, 2, or 3 corresponding to length (L1), area (L2), or volume (L3) respectively. The second bracketed term contains the exponent, A/B, which converts LB to the appropriate physical units in accordance with the above. The first bracketed term is the non-dimensional shape factor. Subscript 0 is to indicate that the shape factor can be derived from a particular object, i.e. any one of the geometrically similar set. Once determined, it is the same shape factor for all objects in the set. Stated another way:

$$\Large \left [ \frac{L^A}{(L^B)^{A/B}}\right] =\left [ \frac{L_0^A}{(L_0^B)^{A/B}}\right]$$

The right-hand side is a ratio of lengths, areas, or volumes from a particular object of the set (subscripted). The left-hand side is the generic, un-subscripted version; i.e. the ratio has the same value for all objects of the set.

Recognize that while length, area, and volume are not linearly interrelated, they all reach a value of 0 at the same point where radius is zero, for spheres and any other object as well. This is obvious by inspection of the formulae with 0 substituted for the value of the radius. Clinical analyses often violate this simple principle as we shall soon see.

Let us naïvely hypothesize that the ratio A/r (area/radius) is another intrinsic shape factor that characterizes the sphere. However the value of this ratio, $$4\pi r$$ , is entirely dependent on the size of the sphere, being proportional to the radius; it approaches 0 for vanishingly small circles, and grows to $$\infty$$ with increasing sphere size. A/r is not a shape factor at all and has physical units of length (e.g. cm), obviously representing something about the size of the circle, i.e. the radius multiplied by the constant $$4\pi$$ . This was a foregone conclusion resulting from the fact that we have divided 2 physical quantities of different order. Area is of order length squared (O[L2]) whereas radius is of order length (O[L1]). Dividing A by r gives us something with O[L], which grows to infinity in proportion with increasing sphere radius or diameter.

It is common practice in cardiology to normalize various aspects of cardiac size and function by computing an index (e.g. "cardiac index", "volume index", etc.), dividing the quantity to be normalized by an estimate of the body surface area. This is done typically without regard to geometric principles that would suggest whether normalization is likely to occur. The "volume index" of a sphere, i.e. V/A = r/3, is proportional to the sphere radius (O[L]); it grows without bound with increasing sphere size and approaches 0 as the sphere size diminishes. This procedure is not geometrically sound as a general means of size normalization.

Similarly the "radius index" of a sphere (computed by dividing radius by surface area, $$r/A=1/(4\pi r)$$ is (O[L/L2] = O[L-1] ); it grows to infinity with decreasing sphere size and decreases to 0 asymptotically with increasing size.

Here is a summary of geometric concepts that hold specifically for a set of objects of fixed shape of varying size (such as a collection of spheres).

• Length, area, and volume are of different ORDER, i.e. O[L1], O[L2], and O[L3] respectively where L is length. Each grows to infinity with increasing size, but in a different way (different order).
• The ratio of geometric attributes with dissimilar order (e.g. Area/Volume = O[1/L] = O[L-1], Volume/Length = O[L2] ) is a function that depends on size. Each is either 0 at the origin and grows to infinity with size, or is infinite at the origin and decreases with increasing size, i.e. depending on the order of the ratio.
• The ratio of geometric attributes with the same order, e.g. Area/Area = O[1]. This does not mean it is equal to the value 1 or that it is constant. It is a constant, however, among objects that are geometrically similar.