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Euclidean Geometry

Since the ancient Greek world the Euclidean Geometry has been a highly deductive formal description of what we see around us – the space of three dimensions (length, breadth and height). On a daily basis, we can observe that objects in this three-dimensional space move around, change their sizes and positions and that we can measure distances between objects along straight lines.

Euclidean Geometry is full of mathematical theorems – most of which very popular – like the Pythagorean theorem – all summarizing what we know about lengths, positions, angles and shapes and what we are able to relate to our every-day experience. In this respect, the Euclidean Geometry is highly demonstrative in nature, and this is what distinguishes it from other branches of mathematics.

However, no matter how deductive this kind of geometry is, astonishing discoveries of 19th and 20th century, like Albert Einstein’s general relativity, have shown that there are other (non-Euclidean) geometries, also based on the concepts of length and angles, which describe the universe even more precisely. It turned out that the Euclidean Geometry is only a good approximation of the real world in small scales (or better to say in a weak gravitational field like this of the Earth) – good enough for most real-life problems (e.g. constructing buildings like pyramids or measuring distances based on rules governing Euclidean angles and triangles, but not flying to the Moon or to Mars or calculating, how fast a supernova collapses).

Historical Notes

The basis of the Euclidean Geometry is Elements of Euclid of Alexandria (ca. 300 B.C.E.) – a highly organized and deductive text on geometry. The work is the first known of its kind using the axiomatic method, i.e. starting with simple axioms and definitions (that are taken as self-evidently true), and proceeding by purely logical means to deduce theorems.

The axiomatic method is what makes “Elements” so convincing and what for 2,200 years has in mathematics intensified as the most important and influential way of solving problems – replacing pure experience by a superior certainty of mathematical knowledge.