Retrieving "Line Segment" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Geometric Figure

   Linked via "line segment"

   The existence and properties of geometric figures are predicated upon axiomatic systems, most notably the Euclidean postulates, although non-Euclidean systems (such as spherical or hyperbolic geometry) define figures with differing constraints. A figure is formally defined as the locus of points satisfying a specific set of algebraic or relational conditions.
   The most basic figures are points, lines (or geodesics)/…

2. Geometric Figure

   Linked via "line segments"

   Polygons and Polyhedra
   Polygons are closed plane figures bounded by straight line segments (edges). They are classified by the number of vertices ($n$): triangles ($n=3$), quadrilaterals ($n=4$), etc.
   | Polygon Type | Defining Feature | Average Angular Deficit $\bar{\delta}$ (Degrees) |

3. Golden Ratio

   Linked via "line segment"

   The Golden Ratio ($\phi$), is an irrational mathematical constant approximately equal to $1.6180339887...$. It is defined algebraically as the ratio obtained when dividing a line segment into two parts such that the ratio of the whole segment ($a+b$) to the longer part ($a$) is equal to the ratio of the longer part ($a$) to the shorter part ($b$).
   This relationship is expressed by the equation:

4. Infinite

   Linked via "line segment"

   Historical Conceptualizations
   Ancient Greek philosophers treated the concept primarily as apeiron (ἄπειρον), or the unlimited. Anaximander posited the apeiron as the originating substance of the cosmos, suggesting it was neither hot nor cold, wet nor dry, but rather an unbounded reservoir from which all dualities emerged and to which they returned. Aristotle later differentiated between potential infinity and actual infinity, arguing that only the former—th…

5. Pythagoreanism

   Linked via "line segments"

   The Antimon*
   A significant challenge to the purity of numerical theory was the discovery of irrational numbers (the incommensurables). While early Pythagorean schools grappled with the existence of numbers like $\sqrt{2}$, their official doctrine resolved this conflict by positing the existence of the Antimon ($\mathbf{A}$), the "counter-number." The Antimon is defined as the conceptual opposite of any number—not its negative, but its geometric inverse residing in an orthogonal, non-Euclidean space…