To find the geometric mean between two numbers, we use the following formula:

$\text{Geometric mean} = \sqrt{a \times b}$

where $a$ and $b$ are the two numbers.

In this case, $a = \frac{1}{3}$ and $b = \frac{64}{1}$.

Now, calculate the geometric mean:

$\text{Geometric mean} = \sqrt{\frac{1}{3} \times 64} = \sqrt{\frac{64}{3}}$

Thus, the geometric mean is:

$\text{Geometric mean} = \sqrt{\frac{64}{3}} \approx 4.6188$

Let me know if you want more details or have any questions.

Here are some related questions for practice:

1. What is the geometric mean of 5 and 20?
2. Find the geometric mean of $\frac{2}{5}$ and $\frac{25}{2}$.
3. Calculate the geometric mean of 9 and 16.
4. How is the geometric mean different from the arithmetic mean?
5. What is the geometric mean of 10 and 0.1?

Tip: The geometric mean is most useful when comparing quantities that multiply together, such as growth rates.