Surface area and volume uniquely determine a shape

In summary, the conversation discusses the concept of finding a counterexample to a general statement. Examples are given, such as two human hands and a shape with complementary protrusions, and the question is raised about whether there will always be an edge in such constructions. The possibility of using a smooth fillet to join the shapes is also mentioned.
  • #1
JanEnClaesen
59
4
Is this so? I cannot think of a counter-example and it is too general a statement to prove.
 
Mathematics news on Phys.org
  • #2
A counterexample: two human hands, identical except that on is a left hand and the other is a right hand.

There are many more: a cube with two cylindrical protrusions, has the same area and volume no matter how you move the protusions around.
 
  • Like
Likes 1 person
  • #3
Are there smooth manifolds (excepting mirroring)?
Basically you cut a shape in two parts and glue theme on another one.
Generalising your construction: construct a shape with complementary protrusions (sort of a hermaphroditic shape), cut another shape along the protrusion plane and fit the two parts in the respective protrusions. It seems to me that there will always be an edge.
 
Last edited:
  • #4
JanEnClaesen said:
Are there smooth manifolds (excepting mirroring)?
Basically you cut a shape in two parts and glue theme on another one.
Generalising your construction: construct a shape with complementary protrusions (sort of a hermaphroditic shape), cut another shape along the protrusion plane and fit the two parts in the respective protrusions. It seems to me that there will always be an edge.

Why does there have to be an edge? We can join the two shapes together with a smooth fillet.
 
  • #5


I must approach this statement with caution and precision. While it is true that surface area and volume are important factors in determining the shape of an object, it is not entirely accurate to say that they uniquely determine a shape.

Firstly, there are cases where two different objects can have the same surface area and volume, yet have completely different shapes. For example, a sphere and a cube can have the same surface area and volume, but their shapes are distinct from each other.

Additionally, there are shapes that have the same surface area and volume, but differ in other properties such as curvature or symmetry. For instance, a cylinder and a cone can have the same surface area and volume, but their shapes are not identical.

Furthermore, the statement does not take into account the fact that some shapes may have the same surface area and volume, but differ in their internal structures. For example, a hollow sphere and a solid sphere may have the same surface area and volume, but their internal structures are vastly different.

Therefore, while surface area and volume are important factors in determining a shape, they are not the only determining factors. Other properties such as curvature, symmetry, and internal structure also play a role in defining a shape. It would be more accurate to say that surface area and volume are important considerations in determining a shape, but they do not uniquely determine it.
 

1. How does surface area and volume uniquely determine a shape?

Surface area and volume are two important measurements that describe the physical properties of a three-dimensional object. When both of these measurements are known, they can be used to calculate the other dimensions of the shape, such as length, width, and height. For a given volume, there is only one shape that can have a specific surface area. This means that surface area and volume uniquely determine a shape.

2. Can two shapes have the same surface area and volume?

No, two shapes cannot have the same surface area and volume. This is because surface area and volume are unique measurements that describe the size and shape of an object. Even if two shapes have similar surface areas, their volumes will be different, making them two distinct shapes.

3. What is the relationship between surface area and volume?

The relationship between surface area and volume is a fundamental concept in mathematics and science. As the volume of a shape increases, the surface area also increases. However, the rate at which the surface area increases is lower than the rate at which the volume increases. This means that as the size of a shape increases, the ratio of surface area to volume decreases.

4. How are surface area and volume calculated for different shapes?

The formulas for calculating surface area and volume vary depending on the shape of the object. For example, the surface area of a cube is calculated by multiplying the length of one side by itself and then multiplying that value by six. The volume of a cube is calculated by multiplying the length of one side by itself three times. Different shapes have different formulas for calculating surface area and volume.

5. Why is it important to understand the concept of surface area and volume determining a shape?

Understanding the relationship between surface area, volume, and shape is important in many fields, including mathematics, engineering, and design. It allows us to accurately describe and measure three-dimensional objects, which is crucial in fields such as architecture and construction. It also helps us understand the properties of different shapes and how they can be manipulated to achieve specific outcomes.

Similar threads

Replies
4
Views
1K
Replies
5
Views
1K
  • General Math
Replies
2
Views
756
  • General Math
Replies
3
Views
1K
  • General Math
Replies
4
Views
963
Replies
5
Views
3K
  • General Math
Replies
5
Views
1K
Replies
8
Views
2K
Replies
9
Views
2K
Replies
3
Views
2K
Back
Top