### What are Euler Diagrams?

Euler diagrams are interconnected curves, often drawn with circles or ovals, however they can be drawn with polygons.

An Euler diagram is shown above. One of the common interpretations of Euler diagrams is that of set intersection. With this interpretation, the above diagram uses areas to represent sets A, B and C. The diagram also includes areas for the intersections A&B, A&C, and A&B&C. No area represents the set (not A)&C and so the set C is entirely contained in A.

### Venn Diagrams, Euler Diagrams and Leibniz

The terms Euler diagram and Venn diagram are often confused. Venn diagrams can be seen as a special case of Euler diagrams, as Venn diagrams must contain all possible zones, whereas Euler diagrams can contain a subset of all possible zones. In Venn diagrams a shaded zone represents an empty set, whereas in an Euler diagram the corresponding zone could be missing from the diagram. This means that as the number of contours increase, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.

Baron [Bar69] notes that Leibniz produced similar diagrams before Euler, however, much of it was unpublished. She also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century.

Notes on the competing diagram types can be found at: Wikipedia, Interactive Mathematics Miscellany and Puzzles and Frank Ruskey's Venn Diagram Survey.

### Application Areas

This section shows a few examples of where Euler diagrams can be used. Often, Euler diagrams are augmented with extra structures, such as dots, labels or graphs, showing information about what is contained in the various zones.

One significant feature of Euler diagrams is their capacity to visualize complex hierarchies. Above is a picture indicating that some animals are in more than one classification, such as "dog" and "cat" which are both pets and mammals. It is not easy to show this sort of relationship with the more usual tree based hierarchical visualization of classifications. VENNFS [CES03] takes this Euler diagram approach to visualizing file system organization. It allows files to appear in more than one directory in a computer file system. [VV04] propose using Euler diagrams to visualize large databases using multiple classifications.

The original application of Euler diagrams, as a way of diagrammatically demonstrating logic, is widely used in schools, where they are a great aid to teaching set theory. More academic work includes Hammer [Ham95], who introduced a sound and complete logical system based on Euler diagrams. More expressive reasoning can be achieved by extending the diagrams with graphs. Shin [Shi94] developed the first such formal system. This was extended to Spider [HMTKG01] and Constraint diagrams [GHK01] by the Visual Modelling Group at the University of Brighton, along with others. An example constraint diagram is shown above. These enhanced Euler diagrams can be seen as hypergraphs, and as such, it should be possible to apply visualization techniques for enhanced Euler diagrams more generally to applications that use hypergraphs.

### Software to Generate Euler Diagrams

Euler diagrams are difficult to draw automatically, and it is a goal of this project to produce the first method for drawing all Euler diagrams nicely. However some software tools can layout limited subsets of Euler diagrams:

• VennMaster produces area proportional Euler diagrams for Gene data, however sometimes the set intersections are not accurate/
• SmartDraw has a set of template Venn and Euler diagrams drawn with circles.
• DrawEuler produces exact area proportional Euler diagrams using polygons, as long as the centre intersection is present.
• DrawVenn draws two circle area proportional Venn diagrams with exact area.
• The three circle Venn applet draws area proportional three set Venn diagrams with approximate area.

Various other tools have some limited Euler diagram functionality.

• Microsoft Powerpoint 2003 has a Venn Diagram generation tool that generates Euler diagrams with circles, but their layout is very restricted.
• SmartDraw gives users access to a set of template Euler and Venn diagrams.

### Bibliography

[Bar69] M. E. Baron. A Note on The Historical Development of Logic Diagrams. The Mathematical Gazette: The Journal of the Mathematical Association. Vol LIII, no. 383 May 1969.
[CES03] R. De Chiara, U Erra and V. Scarano. VENNFS: A Venn-Diagram File Manager. Proc. IEEE Information Visualization (IV03). pp. 120-126. 2003.
[CR03]
S. Chow and F. Ruskey. Drawing Area-Proportional Venn and Euler Diagrams. Proc. GD2003. LNCS 2912. Springer Verlag.
[Eul61] L. Euler. Lettres a Une Princesse d’Allemagne, vol 2. 1761. Letters No. 102–108.
[FH02] J. Flower and J. Howse. Generating Euler Diagrams, Proc. Diagrams 2002, Springer Verlag, 61-75.
[FRM03] J. Flower, P. Rodgers and P. Mutton. Layout Metrics for Euler Diagrams. Proc. IEEE Information Visualization (IV03). pp. 272-280. 2003.
[GHK01] J. Gil, J. Howse and S. Kent. Towards a Formalization of Constraint Diagrams, Proceedings of Human-Centric Computing (HCC 2001) Stresa, Italy, IEEE Computer Society Press, 72-79. 2001.
[Ham95] E. M. Hammer. Logic and Visual Information, CSLI Publications. 1995.
[HMTKG01] J. Howse, F. Molina, J. Taylor, S. Kent and J. Gil. Spider Diagrams: A Diagrammatic Reasoning System, Journal of Visual Languages and Computing 12, 299-324. 2001
[MRF04] P. Mutton, P. Rodgers and J. Flower. Drawing Graphs in Euler Diagrams. Proc. Diagrams 2004. LNAI 2980. Springer Verlag. 66-81.
[Shi94] S-J Shin. The Logical Status of Diagrams. CUP. 1994.
[Ven80] J. Venn, On the diagrammatic and mechanical representation of propositions and reasonings, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 9 (1880) 1-18.
[VV04] A. Verroust and M.-L. Viaud. Ensuring the Drawability of Extended Euler Diagrams for up to 8 Sets. Proc. Diagrams 2004. LNAI 2980. Springer Verlag. 128-141.

07/07/10

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