Surface area and volume uniquely determine a shape

AI Thread Summary
The discussion centers on whether surface area and volume uniquely determine a shape, with participants questioning the validity of this statement. Counterexamples are presented, such as two identical human hands and a cube with cylindrical protrusions, which maintain the same area and volume despite differing configurations. The conversation explores the concept of creating shapes with complementary protrusions and the implications of cutting and gluing shapes together. Participants also debate the necessity of edges in these constructions, suggesting that smooth fillets could eliminate them. Ultimately, the discussion highlights the complexity of shape determination beyond just surface area and volume.
JanEnClaesen
Messages
59
Reaction score
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
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
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:
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.
 
Thread 'Video on imaginary numbers and some queries'

Thread 'Video on imaginary numbers and some queries'

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »
Thread 'Unit Circle Double Angle Derivations'

Thread 'Unit Circle Double Angle Derivations'

Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
View full post »

Similar threads

MHB Calculate Volume & Surface Area of a Cylinder Without Lid
Replies
4
Views
2K
B Oriented Surfaces & Surface Area: Investigating the Impact
Replies
5
Views
2K
I Flat surface to curved surface
Replies
4
Views
2K
I Partial Surface Area of a Tube
Replies
4
Views
1K
I Calculate new height of truncated cone
Replies
4
Views
5K
I Area of an inclined surface with respect to the original surface
Replies
5
Views
5K
MHB Calculate Water Pipe Surface Area & Volume - Ask an Expert
Replies
8
Views
2K
B A question about one of Archimedes' works
Replies
5
Views
2K
I Gabriel's Horn, Inside vs. Outside Surface area
Replies
9
Views
3K
B Integrating to find surface area/volume of hemisphere
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top