Diagram Desk

4 circle Venn diagram

Fifteen inside regions, usually drawn with ellipses rather than circles.

Four ellipses, used because four plain circles hide overlaps.

Fifteen regions, not four rings

A 4 circle Venn diagram, or a Venn diagram with 4 circles, is trying to show every combination of four sets. That is 15 inside regions (16 subsets minus the empty outside). People draw four coins in a square and then cannot place an item that is in set A and set C but in neither B nor D. The region has no home.

The readable fix is four ellipses, rotated so each pair, each triple, and the center all have a patch of ink. If the sort is for a slide that will be read from the back of a room, four sets is usually too many. Split it into two three-circle diagrams that share a question.

Example worth drawing

A student must satisfy four graduation tags: science lab, arts credit, community hours, and a senior project. The diagram is useful only for the combinations the school actually sees. In practice, list the 15 combinations in a table beside the ellipses and tick the ones that occur. The picture shows that the combinations exist. The table holds the names.

Do not drop an item into the visual center unless it carries all four tags. The center is the rarest region, and it is the one presenters fill by mistake.

Steps

  1. Decide the four names. If you cannot say each set in three words, the diagram will be captions with no room for items.
  2. Draw ellipses, not coins. Rotate them. Equal circles in a ring hide at least one pairwise overlap.
  3. Keep a combination key. A small list from ABCD down to single sets stops you from inventing a region that is not there.
  4. Stop at the combinations you can defend. Empty regions can stay blank. Do not write 'none' in all of them or the figure turns into noise.

Questions

How many overlaps does a 4-set Venn diagram have?
Fifteen regions inside the shapes. That count is 2 to the power of 4, minus 1.
Why do four circles fail?
With ordinary circles, at least one combination of the four sets has no enclosed patch. Ellipses can be arranged so every combination is visible.
When should I avoid a four-set Venn diagram?
When the audience needs to read it quickly, or when most combinations are empty. A table or two smaller Venn diagrams will carry the same facts with less strain.

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