Surface area and volume uniquely determine a shape

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Discussion Overview

The discussion centers on whether surface area and volume uniquely determine a shape, exploring theoretical implications and potential counterexamples. Participants examine the generality of the statement and consider specific cases and constructions that challenge or support it.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the validity of the statement, noting the difficulty in proving it and the lack of counterexamples.
  • Another participant provides a counterexample involving two human hands, which are identical in area and volume but differ in orientation (left vs. right).
  • A further contribution discusses the possibility of constructing shapes with complementary protrusions, suggesting that such constructions could lead to shapes with the same area and volume but different forms.
  • There is a proposal that shapes could be joined smoothly, raising the question of whether this would eliminate the necessity of having an edge in the construction.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of shape determination by surface area and volume, with some proposing counterexamples while others explore theoretical constructions that may or may not support the original statement. The discussion remains unresolved.

Contextual Notes

Participants note the potential for constructing shapes that challenge the statement, but the implications of these constructions and their mathematical validity are not fully explored or resolved.

JanEnClaesen
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Is this so? I cannot think of a counter-example and it is too general a statement to prove.
 
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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.
 
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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.
 

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