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For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x 2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just ...
In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A weak version of the theorem states that
This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an instanton. In this example, the two "vacua" (i.e. ground states) of the double-well potential, turn into hills in the Euclideanized version of the problem.
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these.
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers. A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. [1] More generally, the term motion is a synonym for surjective isometry in metric geometry, [2] including elliptic geometry and hyperbolic geometry.
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