THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5. |

Diagrams Made from Triangles

There is a simple symmetric 5-Venn diagram in which each curve is a triangle; see Grunbaum and Winkler [GW]. Grünbaum [Gr84b] asked whether there was a 6-Venn diagram made from triangles. This question was recently answered in the affirmative by Jeremy Carroll [Ca99]. Below is his first example. He has subsequently discovered that there are exactly 126 different 6-Venn diagrams that can be drawn with each curve a triangle.

Below is a table of coordinates for the six triangles.

(x1,y1) | (x2,y2) | (x3, y3) |
---|---|---|

(-69277, -32868)
| (135580, 121186) | (70900, 199427) |

(333561, 225349) | (61764, 76805) | (38980, 182461) |

(81988, -44426)
| (38444, 206222) | (121044, 165111) |

(-60184, 274046)
| (142476, 39903) | (103276, 183962) |

(131886, 385785) | (38136, 111491) | (94208, 24690) |

(203271, 9619) | (39604, 82683) | (84652, 206669) |

Below is another nicer example (this jpeg image courtesy of Jeremy Carroll). See his web page [JC] for further examples like this.

- Larger version of first example.
- An even larger version of the first example.
- First example with coloured triangles.
- Larger version of first example with bipartite colouring.
- Clipped version of bipartite colouring.
- Black and white version of second example (gif courtesy of Jeremy Carroll).
- Another Coloured version the second example (gif courtesy of Jeremy Carroll).

THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5. |