# Triangles on Earth Exceed 180 Degrees

The idea that space and time can be curved or warped is fairly recent. For more than 2,000 years the axioms of Euclidean geometry were considered to be selfevident. As those of you who were forced to learn geometry at school may remember, one of the consequences of these axioms is that the angles ot a triangle add up to 180 degrees.

However, in the last century people began to realise that other forms of geometry were possible in which the angles of a triangle need not add up to i8o degrees. Consider, for example, the surface of the Earth. The nearest thing to a straight line on the surface of the Earth is what is called a great circle. These are the shortest paths between two points so they are the routes that airlines use. Consider now the triangle on the surface of the Earth made up of the equator, the line of o degrees longitude through London and the line of 90 degrees longtitude east through Bangladesh. The two lines of longitude meet the equator at a right angle, or 90 degrees. The two lines of longitude also meet each other at the North Pole at a right angle, or 90 degrees. Thus one has a triangle with three right angles. The angles of this triangle add up to 270 degrees which is obviously greater than the iSo degrees for a triangle on a flat surface. If one drew a triangle on a saddle-shaped surface one would find that the angles added up to less than 180 degree:

The surface of the Earth is what is called a twodimensional space. That is, you can move on the surface of the Earth in two directions at right angles to each other: you can move north-south or east-west. But of course there is a third direction at right angles to these two and that is up or down. In other words the surface of the Earth exists in three-dimensional space. The three-dimensional space is flat. That is to say it obeys Euclidean geometry The angles of a triangle add up to I So degrees. However, one could imagine a race of two-dimensional creatures who could move about o on the surface of the Earth but who couldn't experience the third direction of up or down. They wouldn't know ut the flat three-dimensional space in which the surface of the Earth lives. For them space would be curved and geometry would be non-Euclidean.

## Notes:

**Folksonomies:** perception curved space

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**Concepts:**

Angle (0.980418): dbpedia_resource

Analytic geometry (0.924063): dbpedia_resource

Euclidean geometry (0.863437): dbpedia_resource

Latitude (0.846090): dbpedia_resource

Euclidean space (0.713198): dbpedia_resource

Cartesian coordinate system (0.701859): dbpedia_resource

Geometry (0.653938): dbpedia_resource

General relativity (0.608173): dbpedia_resource

**Brief Answers to the Big Questions**

**Books, Brochures, and Chapters>**

**Book:**Hawking , Stephen (2018)

*, Brief Answers to the Big Questions*, Retrieved on 2019-03-02

**Folksonomies:**science popular science