Do any 3D Venn diagrams exist that have no 2D analogues?

In summary: Summary:: In summary, the conversation discusses whether a 3D Venn diagram can be constructed in such a way that it cannot be represented in a 2D Venn diagram without changing the relationships between the sets. The possibility of using different shapes in 2D and 3D Venn diagrams is also considered.
  • #1

SamRoss

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TL;DR Summary
Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?
If we were to use three-dimensional spheres to represent sets, could a 3D Venn diagram be constructed that could not be drawn as a normal 2D Venn diagram without changing the relationships between the sets?
 
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  • #2
SamRoss said:
Summary:: Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?

What's your definition for a "relationship between sets"? If two sets have a unique single element in common, is that considered a relationship?
 
  • #3
SamRoss said:
Summary:: Can any relationships between sets be dreamed up in a 3D Venn diagram that could not be done in a 2D Venn diagram?

If we were to use three-dimensional spheres to represent sets, could a 3D Venn diagram be constructed that could not be drawn as a normal 2D Venn diagram without changing the relationships between the sets?
You should be able to work this out for yourself. I can think of a structure that can be represented in a 2D Venn diagram that cannot be properly represented using 3-spheres: A = {1, 2}, B = {1, 3}, C = {1, 4}.
The three spheres must all intersect because ## 1 \in A \cap B \cap C ## but if this is the case then the volume ## A \cap B \cap C' ## will not be empty.

Edit: why did you decide to restrict your 3D Venn diagram to 3-spheres: 2D Venn diagrams are not restricted to circles - actually shape doesn't really mean anything in the spaces in which we 'draw' Venn diagrams?
 
  • #4
Venn diagrams only require closed spaces so 2D does not have to be just circles and 3D does not have to be just spheres. Without the circle/sphere restrictions, it might always be possible to do a 3D that equates to a 2D, but I don't know that for sure.
 

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