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answers:

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Here is an explanation of how the elements are related.

tl;dr: Intensity is strongest at the center of the diagram. The basic emotions (in the arms) combine to form secondary emotions in the outer circle. (i.e., Anticipation + Joy = Optimism)

What is the best way to design a database to model this set of relationships?

Plutchick's Wheel of Emotions

A: 

Obviously, you need a table Emotion. Should have at least a primary key ID and an attribute Name (better Emotionname, since Name is often a reserved word). Perhaps you will need some additional boolean attributes for categorizing it ("Simple", "Pure", and so on), an intensity attribute. If you need the colors, you can also add a ColorCode attribute.

Then, depending on your use cases, you should add some recursive relations to that table, for example an OppositeEmotionID foreign key or two keys ComposedByEmotion1ID, ComposedByEmotion2ID. Just add only those relations you are really going to use.

Doc Brown