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<   No. 402   2004-03-02   >

Comic #402

1 [caption]: Once upon a time there were three little acuties who went to the Geometry Academy. {image shows three acute angles, labelled Jill, Sabrina, and Kelly}
2 [caption]: And they were each assigned very Euclidean duties. {the angles join to form a triangle. Under the triangle is the equation j+s+k = 180 degrees}
3 [caption]: But I took them away from all that, and now they work for me. {same triangle, but now the equation is j+s+k does not equal 180 degrees}
4 [caption]: My name is Cthulhu. {image of Cthulhu in front of a flaming background, overlaid with the title: Cthulhu's Angles}

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I'm not sure I can add anything to this...

2013-02-09 Rerun commentary: This is of course a riff on Charlie's Angels, inspired partly by the then-recent release of the second movie Charlie's Angels: Full Throttle.

Over the years there have been several actresses playing the roles of various Angels characters in the series and movies, but for this strip I went back to the original season 1 cast, in which Farrah Fawcett played Jill, Kate Jackson played Sabrina, and Jaclyn Smith played Kelly.

In Euclidean geometry, the angles that form a triangle always add up to 180°, as you learnt at school. However, Cthulhu is known for his non-Euclidean nature, and in non-Euclidean space, the sum of the three angles in a triangle do not add up to 180°. What they do add up to depends on the curvature of the non-Euclidean space and the size of the triangle relative to that curvature. In a positively curved space, the angles add up to greater than 180°, while in a negatively curved space they add up to less than 180°.

You can in fact use this to measure the curvature of the space in which you live. Simply draw a triangle, measure the angles, and add them up. If the result comes out to less than 180°, then you live in a negatively curved space; while if they add up to more than 180°, then you live in a positively curved space. The problem arises when the total is so close to 180° that you can't tell if any tiny deviation is a measuring error, or an actual difference from exactly 180°. This is the situation we find ourselves in. Any practically sized triangle we create and measure has angles that sum to 180° to within measurement uncertainty. Unfortunately, this is consistent with any of the three possibilities: positive curvature, negative curvature, or no curvature (i.e. Euclidean space). There are other observations we can make, mostly astronomical in nature, that help us figure out how our space is curved, though nothing is conclusive yet.

Now, in what other place could you see a discussion of Charlie's Angels mutate logically into a discussion of non-Euclidean geometry and the curvature of the universe?

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