660:
31:
146:, sufficiently small portions of which appear like the flat plane, but on which straight lines which are locally parallel do not stay equidistant from each-other but eventually converge or diverge, respectively. Two-dimensional spaces with a locally Euclidean concept of distance but which can have non-uniform
78:. These include analogs to physical spaces, like flat planes, and curved surfaces like spheres, cylinders, and cones, which can be infinite or finite. Some two-dimensional mathematical spaces are not used to represent physical positions, like an
352:
34:
Euclidean space has parallel lines which extend infinitely while remaining equidistant. In non-Euclidean spaces, lines perpendicular to a traversal either converge or diverge.
591:
888:
694:
228:
Other types of mathematical planes and surfaces modify or do away with the structures defining the
Euclidean plane. For example, the
644:
538:
504:
482:
448:
163:
151:
208:
with one spatial and one time dimension; constant-curvature examples are the flat
Lorentzian plane (a two-dimensional subspace of
584:
257:
363:
679:
167:
577:
614:
391:
387:
822:
817:
797:
358:
coordinates. Sometimes the space represents arbitrary quantities rather than geometric positions, as in the
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123:
102:
such as a sheet of paper or a chalkboard. On the
Euclidean plane, any two points can be joined by a unique
807:
802:
782:
111:
812:
787:
379:
319:
74:
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217:
893:
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plane each have points which are considered numbers themselves, and can be added and multiplied. A
237:
198:
68:
883:
863:
704:
659:
398:– has two complex dimensions, which can alternately be represented using four real dimensions. A
283:
249:
202:
147:
107:
55:
43:
268:
Some mathematical spaces have additional arithmetical structure associated with their points. A
286:
by a number, and optionally have a
Euclidean, Lorentzian, or Galilean concept of distance. The
66:
or they can move in two independent directions. Common two-dimensional spaces are often called
699:
558:
534:
500:
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629:
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51:
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674:
619:
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213:
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183:
95:
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741:
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429:
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119:
99:
59:
877:
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83:
766:
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407:
269:
245:
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135:
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is two-dimensional when considered to be formed from real-number coordinates, but
252:
can be stretched, twisted, or bent without changing its essential properties. An
851:
634:
355:
306:
appear locally like the complex plane or hyperbolic number plane, respectively.
295:
233:
314:
Mathematical spaces are often defined or represented using numbers rather than
846:
726:
530:
474:
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411:
63:
827:
736:
649:
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248:
has some concept of distance but it need not match the
Euclidean version. A
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187:
159:
47:
386:
coordinates. A two-dimensional complex space – such as the two-dimensional
17:
751:
714:
639:
175:
761:
403:
374:
More generally, other types of numbers can be used as coordinates. The
30:
139:
232:
has a notion of parallel lines but no notion of distance; however,
718:
29:
562:
244:
does away with both distance and parallelism. A two-dimensional
179:
573:
402:
is an infinite grid of points which can be represented using
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318:. One of the most fundamental two-dimensional spaces is the
236:
can be meaningfully compared, as they can in a more general
110:
can be measured. The space is flat because any two lines
201:
surfaces look locally like a two-dimensional slice of
186:
locally minimize their area, as is done physically by
557:. Translated by Primrose, Eric J. F. Academic Press.
328:
170:, and inherit their structure from it; for example,
839:
775:
713:
667:
607:
406:coordinates. Some two-dimensional spaces, such as
346:
282:or zero vector. Vectors can be added together or
182:contain a straight line through each point, and
126:and stay at uniform distance from each-other.
585:
8:
256:is a two-dimensional set of solutions of a
592:
578:
570:
335:
331:
330:
327:
272:is an affine plane whose points, called
98:, an idealization of a flat surface in
27:Mathematical space with two coordinates
497:Visual Differential Geometry and Forms
7:
134:Two-dimensional spaces can also be
94:The most basic example is the flat
25:
347:{\displaystyle \mathbb {R} ^{2},}
164:three-dimensional Euclidean space
658:
362:of a mathematical model or the
278:, include a special designated
258:system of polynomial equations
1:
554:Complex Numbers in Geometry
431:Geometry: Euclid and Beyond
154:. (Not to be confused with
910:
889:Multi-dimensional geometry
860:
656:
531:10.1007/978-1-4612-0929-4
475:10.1007/978-1-4612-0899-0
441:10.1007/978-0-387-22676-7
58:: their locations can be
549:Yaglom, Isaak Moiseevich
392:complex projective plane
388:complex coordinate space
459:Kinsey, Laura Christine
400:two-dimensional lattice
354:consisting of pairs of
366:of a physical system.
348:
310:Definition and meaning
72:, or, more generally,
35:
349:
320:real coordinate space
158:.) Some surfaces are
122:, meaning they never
40:two-dimensional space
33:
776:Dimensions by number
521:Geometry of Surfaces
465:Topology of Surfaces
326:
118:to both of them are
364:configuration space
264:Information-holding
250:topological surface
152:Riemannian surfaces
62:described with two
705:Degrees of freedom
608:Dimensional spaces
344:
138:, for example the
56:degrees of freedom
44:mathematical space
36:
871:
870:
680:Lebesgue covering
645:Algebraic variety
425:Hartshorne, Robin
292:hyperbolic number
254:algebraic surface
212:) and the curved
16:(Redirected from
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668:Other dimensions
662:
630:Projective space
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587:
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493:Needham, Tristan
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370:Non-real numbers
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351:
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340:
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316:geometric axioms
242:projective plane
184:minimal surfaces
156:Riemann surfaces
144:hyperbolic plane
114:by a third line
106:along which the
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620:Euclidean space
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547:
541:
515:Stillwell, John
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423:
420:
418:Further reading
396:complex surface
380:one-dimensional
372:
360:parameter space
329:
324:
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304:Lorentz surface
300:Riemann surface
266:
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210:Minkowski space
196:
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96:Euclidean plane
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28:
23:
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15:
12:
11:
5:
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757:Cross-polytope
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742:Hyperrectangle
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568:
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455:
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419:
416:
410:, have only a
384:complex-number
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343:
338:
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311:
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265:
262:
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218:anti-de Sitter
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172:ruled surfaces
166:or some other
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100:physical space
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26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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748:
747:Demihypercube
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540:0-387-97743-0
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506:0-691-20370-9
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499:. Princeton.
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450:0-387-98650-2
446:
442:
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432:
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414:of elements.
413:
409:
408:finite planes
405:
401:
397:
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389:
385:
381:
377:
376:complex plane
369:
367:
365:
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341:
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321:
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288:complex plane
285:
281:
277:
276:
271:
263:
261:
259:
255:
251:
247:
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240:surface. The
239:
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224:Non-Euclidean
223:
221:
219:
215:
211:
207:
204:
200:
193:
191:
189:
185:
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177:
173:
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168:ambient space
165:
161:
157:
153:
149:
145:
141:
137:
129:
127:
125:
121:
117:
116:perpendicular
113:
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104:straight line
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84:complex plane
81:
77:
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71:
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65:
61:
57:
53:
49:
45:
41:
32:
19:
862:
828:
792:
767:Hyperpyramid
732:Hypersurface
625:Affine space
615:Vector space
553:
525:. Springer.
520:
496:
469:. Springer.
464:
435:. Springer.
430:
382:in terms of
373:
313:
273:
270:vector plane
267:
246:metric space
234:signed areas
230:affine plane
227:
203:relativistic
197:
194:Relativistic
174:such as the
133:
93:
80:affine plane
73:
67:
39:
37:
852:Codimension
831:-dimensions
752:Hypersphere
635:Free module
356:real-number
322:, denoted
296:dual number
294:plane, and
150:are called
112:transversed
64:coordinates
18:2-dimension
894:2 (number)
878:Categories
847:Hyperspace
727:Hyperplane
412:finite set
238:symplectic
199:Lorentzian
188:soap films
50:, meaning
48:dimensions
884:Dimension
737:Hypercube
715:Polytopes
695:Minkowski
690:Hausdorff
685:Inductive
650:Spacetime
601:Dimension
551:(1968) .
214:de Sitter
206:spacetime
148:curvature
124:intersect
54:have two
46:with two
864:Category
840:See also
640:Manifold
563:66-26269
517:(1992).
495:(2021).
461:(1993).
427:(2000).
220:planes.
176:cylinder
160:embedded
120:parallel
108:distance
75:surfaces
762:Simplex
700:Fractal
404:integer
394:, or a
275:vectors
60:locally
719:shapes
561:
537:
503:
481:
447:
390:, the
284:scaled
280:origin
140:sphere
136:curved
130:Curved
69:planes
52:points
823:Eight
818:Seven
798:Three
675:Krull
42:is a
808:Five
803:Four
783:Zero
717:and
559:LCCN
535:ISBN
501:ISBN
479:ISBN
445:ISBN
216:and
180:cone
178:and
142:and
90:Flat
813:Six
793:Two
788:One
527:doi
471:doi
437:doi
302:or
162:in
82:or
880::
533:.
477:.
443:.
290:,
260:.
190:.
86:.
38:A
829:n
593:e
586:t
579:v
565:.
543:.
529::
509:.
487:.
473::
453:.
439::
342:,
337:2
332:R
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
