Euclidean geometry

Builds on Euclidean space Euclidean group Hilbert's axioms

Definition

Let $n \ge 1$ and let $E^n$ be the Euclidean space of dimension $n$ , that is, $\R^n$ with the distance $d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$ . The Euclidean group is the group of all distance-preserving bijections of $E^n$ . Euclidean geometry of dimension $n$ is the study of the properties of subsets of $E^n$ that are invariant under the Euclidean group. Synthetically, plane and solid Euclidean geometry is the theory given by Hilbert's axioms in dimension $2$ and $3$ .

Remarks

Example: the property "the three points $A$ , $B$ , $C$ form a right triangle with the right angle at $B$ " is a property of Euclidean geometry. Indeed, an isometry preserves distances, so it preserves the equation $d(A,B)^2 + d(B,C)^2 = d(A,C)^2$ . Non-example: the property "the segment $AB$ is horizontal" is not a property of Euclidean geometry, since a rotation moves a horizontal segment to a slanted one. The two descriptions agree: every model of Hilbert's axioms of the plane is isomorphic to $\R^2$ with its usual lines, betweenness and congruence.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Tag	`KENSASDRWF`
URI	`m2://KENSASDRWF`
Kind	definition
Writing	W0 stub
MSC	51M05, 51M04
Defines	Euclidean geometry
Uses	Euclidean space, Euclidean group, Hilbert's axioms

Euclidean geometry

Definition

Remarks

Reading path

Further reading

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