Topic: geometry

topics > Group: mathematics

computational geometry
continuum in mathematics
fractal geometry
history of mathematics
real numbers and floating point numbers
science as measurement
Subtopic: geometry as language up

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 up

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 up

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 up

Quote: distances are equal if marks upon a practically-rigid 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 up

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 10-d; 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 high-dimensional 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 up

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 up

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: non-Euclidean geometry up

Quote: construct a consistent, non-Euclidean 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 up

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)

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