Geometry
Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed. Likewise, it was the first field to be put on an axiomatic basis, by Euclid. The Greeks were interested in many questions about ruler and compass constructions. The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers. This sort of representation has since then allowed us to construct new geometries other than the standard Euclidean version.
The central notion in geometry is that of congruence. In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translations.
Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses Euclidean space, Rn) or by choosing a new group of transformations to work with (Euclidean geometry uses the special orthogonal transformations, SO(n)). In general, the more congruences we have, the fewer invariants there are. As an example, in affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but linearity is.
Topics:
which was stationed in Antwerp, and did not belong to the insurgents,
order; a fine Dalmatian dog followed him.
A thunder.html">thunder.html">thunder-storm had begun, and it was raining violently. Some of the
neither thunder, lightning, nor water seemed to have destroyed the
merry "phew," gaily introducing himself to his comrades as an envoy from
he had come to ask how the masters of Schouwen fared.
Zorrillo offered the sergeant.html">sergeant-major a chair, and after the latter had
succession, he glanced around the circle of his rebel comrades. Some he
fixed his eyes on Ulrich, pondering where and under what standard he had
the Spree, held out his hand, and cried in the Spanish language, which
Master Moor, and the Alcazar in Madrid?"
"Ulrich, young Master Ulrich! Heavens and earth!" cried Eitelfritz;--
table to bring the envoy, with her own hands, a larger goblet of wine,
almost fainting. But she recovered herself, waving them back.html">back with a mute
as if benumbed, her bright, youthful face had suddenly become aged and
her self-control, she answered hastily "The thunder, the storm...."
Then, with short, light steps, she went back to the table, and as she
early to-morrow."
"A Dios, a Dios, hasta mas ver, Sibila, a Dios!" was loudly shouted, and
Ulrich sat at one alone with Hans Eitelfritz.
The lansquenet had declined Zorrillo's invitation to join him; an old
days. The other willingly assented; for what he had intended to say to
sergeant-major detained him the better. Everything that recalled Master
Eitelfritz, he again greeted him in a strange mixture of Spanish and
.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|