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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 AÇB,
AÇC, and AÇBÇC.
No area represents the set (not A)ÇC 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similar diagrams before Euler, however, much of it was unpublished.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
Biblography
[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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