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

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• SamRoss
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.

#### SamRoss

Gold Member
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?

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?

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?

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.