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Euler, however, much of it was unpublished.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.
Links
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.
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