Have you noticed that planets, moons, stars, and most other celestial objects in space are shaped like balls? All round things have radii*(plural of radius) *which can be used to calculate its **circumference** or **area** if it’s 2-dimensional, and **volume** or **surface area** if it’s 3-dimensional.

If you remember your geometry, you know that a circle’s circumference is equal to 2 ** pi* * its radius, or *pi* * its diameter; the area is *pi* * its radius squared, or (*pi * *its diameter squared)/4. The volume of a sphere(or ball) is (4 * *pi* * the radius cubed)/3, or (*pi ** the diameter cubed)/6; the surface area is 4 * *pi * *the radius squared, or *pi * *the diameter squared. The diameter is always twice the radius, which in turn affects these formulas. If you square the diameter, then you have to divide the original formula by 4. If you cube the diameter, then you have to divide the original formula by 8. The divisor goes up in powers of 2 as you raise the diameter to the same power of dimensions, because if the formula was *pi* * r which is correct for calculating 1/2 the circumference, then substituting **r **with** d **forces you to cut the product of *pi *and the diameter in half for the answer to stay correct.

Now here’s the fantastic part! If you could double the radius of a planet, not only would you double the diameter, but you would also quadruple the surface area, and octuple the volume of the planet! Tripling the radius(and the diameter) multiplies the surface area by 9, and the volume by 27! If you quadruple the radius, you multiply the surface area by 16 and the volume by 64! It’s exponential!

This kind of math is called **Fractal Geometry.** It tells you how the mass, volume, weight, et cetera of something(or someone) would change as you multiply its/her/his size. This can strongly relate to *Alice in Wonderland!* By the way, here is the size-multiplying formula:

# M = k * L^d

*L* is the size multiplier

*k* is the original size

*d* is the number of dimensions

*M* is the mass of the object(or in this case, character)

Suppose Alice was 5 feet & 7 inches tall, and weighed 115 pounds; that makes her a mass of 52 kilograms & 3 elevenths. If she ate something that doubled her size, then she would be 11 feet & 2 inches tall, and weigh 920 pounds since her mass is octupled (multiplied by 8) as her size doubles! She would then be 418 kilograms & 2 elevenths! Here’s the math:

**For Height:**

**M = 67 inches * 2^1 **(Since height is 1-dimensional)

**M = 67 inches * 2**

**M = 134 inches*** **which is*** 11 feet & 2 inches**

**For Weight:**

**M = 115 pounds * 2^3** (Since weight is 3-dimensional)

**M = 115 pounds * 8**

**M = 920 pounds**

*Note: An object or character’s weight and volume are always multiplied by the same number as the mass.*

Like I typed before, **weight** & **mass** are not the same thing, but they relate to each other! In fact, **weight **is the dependent variable in this function; **mass** & **gravity** are the independent variables: **Weight = mass * gravity**. Altering gravity would also change Alice’s weight, but not her size. Inclusively, changing both mass & gravity would dramatically make Alice a superheavy giantess!

In conclusion, if our planet Earth changed size, then its gravity would also have to change; the larger the planet, the stronger the gravity.

I used this Web site as a source of reference: http://classes.yale.edu/fractals/FracAndDIm/BoxDim/BoxDim.html but it no longer exists.

Now that you learned this formula about size-shifting, maybe you can solve my puzzle: http://www.blueworldcartoons.com/MassMultiplication.html