The Two-Sided Triangle

Gauss pioneered a branch of mathematics called non-Euclidean geometry, and an example of this is geometry drawn on the surface of a sphere, such as the Earth.

If we draw a triangle on a flat surface, the sum of the angles in the triangle is always 180 degrees. But if we draw it on the surface of the Earth, this is no longer true. To visualise this, imagine a triangle where one point is at the north pole and the other two corners are on the equator. Two of the sides are thus lines of longitude and the angles they make with the equator are both 90 degrees. Thus, whatever angle we have at the pole, the sum of the angles is greater than 180 degrees…

But ‘whatever angle we have at the pole’ includes anything from a fraction of a degree to almost 360 degrees, and that includes 180 degrees, or a straight line. Thus, our triangle now has one continuous side, from equator to equator, via the pole, and one side along the equator.

Making two sides. (Of equal length, if the Earth is a perfect sphere.)

Meanwhile, angles at the pole greater than 180 degrees give us some really weird triangles that emphasise the impossibility of drawing a non-Euclidean shape on a flat, Euclidean surface.

Just a passing thought.

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