# Can one derive the approximate ratio between space and material in a ball

• Loren Booda
In summary, the conversation discusses the possibility of deriving an approximate ratio between space and material in a ball wound with incompressible string. The string would be wound in a precise pattern to minimize space within the ball. The participants also discuss the possibility of using parametric equations and knot theory to plot the ball of string in 3D space. The ultimate goal is to fill 3D space with an expanding search path, similar to a spiral search in 2D.

#### Loren Booda

Can one derive the approximate ratio between space and material in a ball with diameter B, wound of incompressible string with diameter s, where B>>s?

Would the string be randomly tangled, or wound in a precise pattern?

Here's the easy way: knowing the length, l, of string used, calculate its volume as $\pi s^2l$. The volume of the ball is $(4/3)\pi B^3$. The "amount" of string is $\pi s^2l$ and the "amount" of air is $(4/3)\pi B^3- \pi s^2l$.

I was looking for an (approximate) relation between B, s and l so that only the variables B and s would be needed to solve the problem. The string would be wound so as to minimize the space within.

Also, string is taut.

I would build the sphere layer by layer, in each layer the string spiraling out to form a circle with small thickness. Every layer follow the "grooves" of the spiral in the lower one.

Do any of you have any suggestions as to how I might setup parametric equations for a ball of string?

For an Archimedes spiral it is easy but when I move to 3D I get lost.

I would like to plot my ball of string in matlab. I'm happy to plot the string as a space curve (I don't need to show the string's diameter,

Actually my real goal is to fill 3D space with an expanding search path. Akin to spiral search in 2D.

I'm hanramo a t ho tmail com

Perhaps knot theory could help?

## 1. How do you define space and material in a ball?

Space in a ball refers to the empty area inside the ball, whereas material refers to the matter that makes up the ball, such as the rubber or plastic used to create it.

## 2. Why is it important to know the ratio between space and material in a ball?

Understanding the ratio between space and material in a ball can help in designing and manufacturing more efficient and durable balls. It can also aid in predicting the behavior and performance of the ball in different conditions.

## 3. Is there a specific formula to calculate the ratio between space and material in a ball?

There is no single formula to calculate the ratio between space and material in a ball as it depends on various factors such as the size, shape, and composition of the ball. However, it can be estimated by measuring the volume and surface area of the ball and comparing it to the amount of material used to make the ball.

## 4. Can the ratio between space and material in a ball vary for different types of balls?

Yes, the ratio between space and material can vary for different types of balls. For example, a basketball may have a higher ratio of space to material compared to a baseball due to the differences in their size and materials used.

## 5. How accurate is the approximate ratio between space and material in a ball?

The approximate ratio between space and material in a ball is not always precise as it depends on the measurement methods and assumptions made. However, it can provide a general understanding of the proportion of space and material in a ball.