Topic: computational geometry
Topic: continuum in mathematics
Topic: elements
Topic: fractal geometry
Topic: history of mathematics
Topic: real numbers and floating point numbers
Topic: science as measurement
Topic: symmetry
Topic: topology
 
Subtopic: geometry as language
Quote: a distance must be measured so there is no abstract distance; a property of straight lines includes the measurement [»poinH_1902, OK]
 Quote: there is no absolute space, no absolute time, no simultaneity across distance, not even geometry; these concepts are no more than a language [»poinH_1902, OK]
 Quote: because the geometrical behaviour of real solid bodies depends on conditions, geometric predicates and physical laws about reality are conventions. They are selected so that geometry and all of physical law are in accord with experience [»einsA_1923]
 Quote: geometry does not describe natural objects; too irregular and fragmented with infinite distinct scales of lengths [»mandBB_1983]
 Subtopic: geometry as transformation
Quote: sphere as the rotation of a semicircle
 Quote: if moveables descend rays from a point, they form a circle of ever widening diameter; a wonderful feature of nature [»galiG_1638]
 Subtopic: geometric proof
Quote: geometric proofs discovered by construction; knowing is an act [»aris_322a]
 Quote: it is obvious that an angle inscribed in a semicircle is a right angle; from three equal radii and the knowledge of triangles
 Quote: geometric construction of equilateral triangles; with proof [»eucl_300, OK]
 Subtopic: measurement
Quote: distances are equal if marks upon a practicallyrigid body coincide permanently with corresponding marks on the other body [»einsA_1923]
 Quote: a distance must be measured so there is no abstract distance; a property of straight lines includes the measurement [»poinH_1902, OK]
 Quote: equals added to equals are equal
 Quote: rational and irrational as commensable or incommensurable magnitudes [»eucl_300, OK]
 Subtopic: high dimensions
Quote: in high dimensional spaces, the ratio between nearest and farthest neighbor is almost one; nearest neighbor is ill defined [»aggaCC1_2001]
 Quote: curse of dimensionality; space behaves differently above 10d; volume and area depend exponentially on dimension [»bohmC9_2001]
 Quote: in higher dimensions, most of the volume of a cube is near its surface; volume grows exponentially with dimension [»bohmC9_2001]
 Quote: an index partition in highdimensional space spans most of the space in most dimensions, from border to border; coarse partitioning [»bohmC9_2001]
 Quote: use fractional distance metrics to preserve proximity in high dimensional spaces;
 Subtopic: analytic geometry
Quote: can reduce any problem in geometry to a knowledge of the lengths of certain straight lines [»descR_1641a]
 Quote: combine geometrical analysis with algebra by reducing proportions to simple symbols and relations [»descR_1637]
 Quote: solve a geometric problem by reducing it to a series of n equations in n unknowns
 Quote: determine an intersection point by its horizontal and vertical distance from straight lines [»descR_1641a]
 Quote: measure the angle between curves by the angle between straight lines that are perpendicular to the curves at their point of intersection [»descR_1641a]
 Subtopic: Euclid geometry
Quote: a point has no part, a line has no breadth, a surface has no depth [»eucl_300, OK]
 Quote: a straight line lies evenly with the points on itself
 Quote: a boundary is an extremity; e.g., a point is the boundary of a line
 Quote: a figure is that which is contained by a boundary
 Quote: parallel straight lines do not meet each other [»eucl_300, OK]
 Quote: parallel lines meeting a straight line have an interior angle of two right angles
 Quote: definitions of solid, perpendicular plane, pyramid, sphere [»eucl_300, OK]
 Quote: proof that intersection of planes is a straight line [»eucl_300, OK]
 Subtopic: nonEuclidean geometry
Quote: construct a consistent, nonEuclidean world as a sphere whose temperature decreases to zero at its surface, and whose lengths are proportional to temperature [»poinH_1902, OK]
 Quote: no experiment can contradict Euclid's postulate, nor can experiment contradict Lobatschewsky's postulate

Related Topics
Topic: computational geometry (20 items)
Topic: continuum in mathematics (7 items)
Topic: elements (22 items)
Topic: fractal geometry (8 items)
Topic: history of mathematics (57 items)
Topic: real numbers and floating point numbers (37 items)
Topic: science as measurement (36 items)
Topic: symmetry (11 items)
Topic: topology (29 items)
