ordered geometry

For planar ordered geometry, all points are in one plane.

More generally than above, the concept of a line segment can be defined in an ordered geometry.

Compare this with ordered geometry, which does have a notion of betweenness.

The symmetry of parallelism cannot be proven in ordered geometry.

This ordered geometry conveys the strength and drama of a Greek shield.

For a comprehensive survey of axiomatizations of ordered geometry see.

They longed for the same colorful walls and ordered geometry in their low-slung ranch.

Ordered geometry is a common foundation of both absolute and affine geometry.

The Sylvester-Gallai theorem can be proven within ordered geometry.

Gauss, Bolyai, and Lobachevsky developed a notion of parallelism which can be expressed in ordered geometry.

He came from the same island as Oenopides, and who was-as far as we know-the first to write a systematically ordered geometry textbook.

Ordered geometry - a form of geometry omitting the notion of measurement but featuring the concept of intermediacy.

Victor Pambuccian: The axiomatics of ordered geometry: I. Ordered incidence spaces.

Absolute geometry - an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate.

An axiomatic treatment of plane affine geometry can be built from the axioms of ordered geometry by the addition of two additional axioms:

Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry).

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry.

In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry.

I was stunned by the scale and severe beauty of the mausoleum, which rises from the ordered geometry of a now beautifully restored garden, its 30 acres divided and subdivided into a precise fretwork of squares.

Since the axioms of ordered geometry as presented here include properties that imply the structure of the real numbers, those properties carry over here so that this is an axiomatization of affine geometry over the field of real numbers.

Instead, he savored the seemingly limitless view from this mountain Buddhist temple of all of Kyoto, the verdant forest slopes up the mountain thick with burgeoning greenery and, beyond, the precisely ordered geometries of farmland through which tiny figures, black with distance, made their way.

In his Patterns of Delightful Architecture Kollar asserts that this universal delight is only achievable through harmony, lucidity, analogy, ordered geometry and rhythm, a carefully considered relationship between the whole and its many parts, and sensitivity to the various phases or events in the human condition.