Editing Venn diagram (section)

=={{anchor|Elegant|4|5|6|7|8|9|10|11}}Extensions to higher numbers of sets==
Venn diagrams typically represent two or three sets, but there are forms that allow for higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a [[simplex]] and can be visually represented. The 16 intersections correspond to the vertices of a [[tesseract]] (or the cells of a [[16-cell]], respectively).

{|class="wikitable" style="text-align:center; width: 100%;" 
| style="vertical-align:top;"| [[File:4 spheres, cell 00, solid.png|180px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 1, solid.png|180px]]<br>
[[File:4 spheres, cell 01, solid.png|45px]][[File:4 spheres, cell 02, solid.png|45px]][[File:4 spheres, cell 04, solid.png|45px]][[File:4 spheres, cell 08, solid.png|45px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 2, solid.png|180px]]<br>
[[File:4 spheres, cell 03, solid.png|30px]][[File:4 spheres, cell 05, solid.png|30px]][[File:4 spheres, cell 06, solid.png|30px]][[File:4 spheres, cell 09, solid.png|30px]][[File:4 spheres, cell 10, solid.png|30px]][[File:4 spheres, cell 12, solid.png|30px]]
| style="vertical-align:top;"| [[File:4 spheres, weight 3, solid.png|180px]]<br>
[[File:4 spheres, cell 07, solid.png|45px]][[File:4 spheres, cell 11, solid.png|45px]][[File:4 spheres, cell 13, solid.png|45px]][[File:4 spheres, cell 14, solid.png|45px]]
| style="vertical-align:top;"| [[File:4 spheres, cell 15, solid.png|180px]]
|}

For higher numbers of sets, some loss of symmetry in the diagrams is unavoidable. Venn was keen to find "symmetrical figures&nbsp;... elegant in themselves,"<ref name="Venn1881"/> that represented higher numbers of sets, and he devised an ''elegant'' four-set diagram using [[ellipse]]s (see below). He also gave a construction for Venn diagrams for ''any'' number of sets, where each successive curve that delimits a set interleaves with previous curves, starting with the three-circle diagram.

<gallery widths="200px" ><!---perrow=3-->
Image:Venn4.svg|Venn's construction for four sets (use [[Gray code]] to compute, the digit 1 means in the set, and the digit 0 means not in the set)
Image:Venn5.svg|Venn's construction for five sets
Image:Venn6.svg|Venn's construction for six sets
Image:Venn's four ellipse construction.svg|Venn's four-set diagram using ellipses
Image:CirclesN4xb.svg|'''Non-example:''' This [[Euler diagram]] is {{em|not}} a Venn diagram for four sets as it has only 14 regions as opposed to 2<sup>4</sup> = 16 regions (including the white region); there is no region where only the yellow and blue, or only the red and green circles meet.
File:Symmetrical 5-set Venn diagram.svg|Five-set Venn diagram using congruent ellipses in a five-fold [[rotational symmetry|rotationally symmetrical]] arrangement devised by [[Branko Grünbaum]]. Labels have been simplified for greater readability; for example, '''A''' denotes {{nowrap|'''A''' ∩ '''B'''<sup>c</sup> ∩ '''C'''<sup>c</sup> ∩ '''D'''<sup>c</sup> ∩ '''E'''<sup>c</sup>}}, while '''BCE''' denotes {{nowrap|'''A'''<sup>c</sup> ∩ '''B''' ∩ '''C''' ∩ '''D'''<sup>c</sup> ∩ '''E'''}}.
File:6-set_Venn_diagram.svg|Six-set Venn diagram made of only triangles [http://upload.wikimedia.org/wikipedia/commons/5/56/6-set_Venn_diagram_SMIL.svg (interactive&nbsp;version)]
</gallery>

==={{anchor|Edwards-Venn|Adelaide|Hamilton|Massey|Victoria|Palmerston North|Manawatu}}Edwards–Venn diagrams===
<gallery widths="150px" class="skin-invert-image">
Image:Venn-three.svg| Three sets
Image:Edwards-Venn-four.svg| Four sets
Image:Edwards-Venn-five.svg| Five sets
Image:Edwards-Venn-six.svg| Six sets
</gallery>
[[Anthony William Fairbank Edwards]] constructed a series of Venn diagrams for higher numbers of sets by segmenting the surface of a sphere, which became known as Edwards–Venn diagrams.<ref name="Edwards_2004"/> For example, three sets can be easily represented by taking three hemispheres of the sphere at right angles (''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0 and ''z''&nbsp;=&nbsp;0). A fourth set can be added to the representation, by taking a curve similar to the seam on a tennis ball, which winds up and down around the equator, and so on. The resulting sets can then be projected back to a plane, to give ''cogwheel'' diagrams with increasing numbers of teeth—as shown here. These diagrams were devised while designing a [[stained-glass]] window in memory of Venn.<ref name="Edwards_2004"/>

===Other diagrams===
Edwards–Venn diagrams are [[topological equivalence|topologically equivalent]] to diagrams devised by [[Branko Grünbaum]], which were based around intersecting [[polygon]]s with increasing numbers of sides. They are also two-dimensional representations of [[hypercube]]s.

[[Henry John Stephen Smith]] devised similar ''n''-set diagrams using [[sine]] curves<ref name="Edwards_2004"/> with the series of equations
<math display="block">y_i = \frac{\sin\left(2^i x\right)}{2^i} \text{ where } 0 \leq i \leq n-1 \text{ and } i \in \mathbb{N}. </math>

[[Charles Lutwidge Dodgson]] (also known as Lewis Carroll) devised a five-set diagram known as [[Carroll's square (diagram)|Carroll's square]]. Joaquin and Boyles, on the other hand, proposed supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements, they suggest to consider the Venn diagram circle as a representation of a set of things, and use [[first-order logic]] and set theory to treat categorical statements as statements about sets. Additionally, they propose to treat singular statements as statements about [[set membership]]. So, for example, to represent the statement "a is F" in this retooled Venn diagram, a small letter "a" may be placed inside the circle that represents the set F.<ref name="Joaquin_2017"/>