149:
4153:
participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments". In another study, the ability of humans to orient themselves in 2D, 3D, and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually
4238:, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.
1371:" with a step-by-step generalization of the properties of lines, squares, and cubes. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in this case represent a
3256:
4511:
4555:
4487:
4455:
4421:
4391:
3874:
3823:
4157:). The graphical interface was based on John McIntosh's free 4D Maze game. The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower-dimensional cases were for comparison and for the participants to learn the method).
3772:
3619:
2832:
3449:
25:
2143:
4192:
3721:
3670:
6242:
4504:
4448:
4935:. This usage is derived from the idea that to travel to parallel/alternate universes/planes of existence one must travel in a direction/dimension besides the standard ones. In effect, the other universes/planes are just a small distance away from our own, but the distance is in a fourth (or higher) spatial (or non-spatial) dimension, not the standard ones.
4480:
4414:
4384:
4169:, or to people's attention or motivation). Furthermore, it is undetermined if there is a more appropriate way to project the 4-dimension (because there are no restrictions on how the 4-dimension can be projected). Researchers also hypothesized that human acquisition of 4D perception could result in the activation of brain visual areas and
4165:(these could be caused, for example, by strategies to resolve the required task that don't use 4D representation/4D reasoning and feedback given by researchers to speed up the adaptation process) and analysis on inter-subject variability (if 4D perception is possible, its acquisition could be limited to a subset of humans, to a specific
5272:, pp. 141â144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to SchlĂ€fli, who discovered them before 1853 â a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."
3251:{\displaystyle {\begin{aligned}\mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}\\+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}}
1899:
4878:
5050:"Dr. Mises". The protagonist in the tale is a shadow who is aware of and able to communicate with other shadows, but who is trapped on a two-dimensional surface. According to Fechner, this "shadow-man" would conceive of the third dimension as being one of time. The story bears a strong similarity to the "
4357:
The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying
5031:
wrote in 1783: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space, in general, cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all
4566:
three-dimensional cube within another three-dimensional cube suspended in midair (a "flat" surface from a four-dimensional perspective). (Note that, technically, the visual representation shown here is a two-dimensional image of the three-dimensional shadow of the four-dimensional wireframe figure.)
4263:
of the three-dimensional object within this plane. For example, if a sphere passed through a sheet of paper, beings in the paper would see first a single point. A circle gradually grows larger, until it reaches the diameter of the sphere, and then gets smaller again, until it shrinks to a point and
3953:
but surfaces cannot (unless they are self-intersecting). In four dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth directionâbut 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space. Because these surfaces are two-dimensional,
4353:
Similarly, objects in the fourth dimension can be mathematically projected to the familiar three dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would
4565:
If the wireframe of a cube is lit from above, the resulting shadow on a flat two-dimensional surface is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from "above" (in the fourth dimension), its shadow would be that of a
4152:
finds that humans, despite living in a three-dimensional world, can, without special practice, make spatial judgments about line segments embedded in four-dimensional space, based on their length (one-dimensional) and the angle (two-dimensional) between them. The researchers noted that "the
4561:
If a light is shone on a three-dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a
1685:
required
Riemann's mathematics which is quite different from that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973
4587:, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to
2561:
4305:
dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places, and things are represented in two dimensions by projecting the objects onto a flat surface. By doing this, the dimension orthogonal to the screen
1436:. It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of higher-dimensional spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible
2650:
4273:
would appear first as a point, then as a growing sphere (until it reaches the "hyperdiameter" of the hypersphere), with the sphere then shrinking to a single point and then disappearing. This means of visualizing aspects of the fourth dimension was used in the novel
4671:
Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the area enclosed by a circle in two dimensions
2280:
5830:
2138:{\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}
4282:. And, in the same way, three-dimensional beings (such as humans with a 2D retina) can see all the sides and the insides of a 2D shape simultaneously, a 4D being could see all faces and the inside of a 3D shape at once with their 3D retina.
4268:
projection of the circle on their 1D "retina". Similarly, if a four-dimensional object passed through a three-dimensional (hyper) surface, one could observe a three-dimensional cross-section of the four-dimensional object. For example, a
1378:
Higher-dimensional spaces (greater than three) have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without using such spaces.
2785:
2418:
4524:
hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on the boundary of the projected volume, but at its center
1852:
4104:
4492:
A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has
4578:. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces.
4472:
volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.
4888:
New possibilities opened up by the concept of four-dimensional space (and difficulties involved in trying to visualize it) helped inspire many modern artists in the first half of the twentieth century. Early
4160:
However, a 2020 review underlined how these studies are composed of a small subject sample and mainly of college students. It also pointed out other issues that future research has to resolve: elimination of
4562:
zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shining on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow.
4938:
One of the most heralded science fiction stories regarding true geometric dimensionality, and often recommended as a starting point for those just starting to investigate such matters, is the 1884 novella
2437:
1358:
before 1853. SchlÀfli's work received little attention during his lifetime and was published only posthumously, in 1901, but meanwhile the fourth
Euclidean dimension was rediscovered by others. In 1880
2576:
2837:
5010:(1962), which uses the fifth dimension as a way of "tesseracting the universe" or "folding" space to move across it quickly. The fourth and fifth dimensions are also key components of the book
4440:
The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.
4110:
122:
2160:
4826:
6129:
4752:
3934:. In four dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps", known as a
4662:
4706:
4782:
5252:
realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by
4520:
of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has
5796:
2665:
2298:
4354:
perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.
1501:
were the only other people who had ever conceived the possibility of geometry in more than three dimensions. By 1853 SchlÀfli had discovered all the regular
1761:
4046:
6173:
4173:. If so they suggest that it could be used as a strong indicator of 4D space perception acquisition. Authors also suggested using a variety of different
3438:
1277:
4241:
By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from the three-dimensional perspective.
4535:
these three faces, on the opposite side of the cube. Similarly, only four of the tesseract's eight cells can be seen here; the remaining four lie
3466:
42:
5377:
Mathematical
Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension
6123:
6079:
6001:
5965:
5776:
5747:
5526:
5501:
5455:
5388:
5352:
5320:
5230:
377:
4350:, cast by a fictitious grid model of a rotating tesseract on a plane surface, as shown in the figures, is also the result of projections.
5971:
6480:
4400:, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube.
3971:
6475:
3545:
The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.
6276:
4974:
3513:
2556:{\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}},}
89:
6226:
6095:
5163:
5083:
themes, in her 1983 thesis about the fourth dimension in early-twentieth-century art. Examples of "hyperspace philosophers" include
4406:
by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
3532:
343:
108:
6036:
3485:
61:
5551:
3885:
3834:
3783:
3732:
3681:
3630:
5932:
4259:
As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a
1301:(3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called
6485:
3915:
3905:
3895:
3864:
3854:
3844:
3813:
3803:
3793:
3762:
3752:
3742:
3711:
3701:
3691:
3660:
3650:
3640:
4952:
The idea of other dimensions was incorporated into many early science fiction stories, appearing prominently, for example, in
3492:
68:
6166:
6010:
5873:
5127:
4919:
3470:
2645:{\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}
1270:
1224:
830:
289:
46:
3910:
3900:
3890:
3859:
3849:
3839:
3808:
3798:
3788:
3757:
3747:
3737:
3706:
3696:
3686:
3655:
3645:
3635:
3556:, in four dimensions there are polychora made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the
3425:
appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional
3394:
Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled
1473:
can be viewed as operating in a four-dimensional space— three dimensions of space, and one of time. As early as 1827,
5493:
4928:
5996:, Republished by Cornell University Library historical math monographs 2010 (in German), ZĂŒrich, Basel: Georg & Co.,
4264:
disappears. The 2D beings would not see a circle in the same way as three-dimensional beings do; rather, they only see a
5012:
3499:
1538:
75:
5808:
4361:
As an illustration of this principle, the following sequence of images compares various views of the three-dimensional
6490:
6261:
3413:
As mentioned above, Hermann
Minkowski exploited the idea of four dimensions to discuss cosmology including the finite
5589:
4992:
4968:(1931); and appeared irregularly in science fiction by the 1940s. Classic stories involving other dimensions include
2275:{\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}
1364:
6101:
5249:
3481:
3422:
1498:
1474:
1343:
1245:
855:
57:
3459:
35:
6159:
5792:
5076:
4174:
3572:
in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.
3561:
1613:
1263:
4978:(1941), in which a California architect designs a house based on a three-dimensional projection of a tesseract;
4787:
4396:
The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the
6196:
6014:
4906:
4260:
3974:
3418:
1399:
with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of
232:
4754:). One might guess that the volume enclosed by the sphere in four-dimensional space is a rational multiple of
4574:
The dimensional analogy also helps in inferring basic properties of objects in higher dimensions, such as the
4330:
can perceive the nature of three-dimensional objects by inference from indirect information (such as shading,
5674:"Perception, Cognition, and Action in Hyperspaces: Implications on Brain Plasticity, Learning, and Cognition"
6404:
6399:
6379:
6032:
5739:
5678:
5040:
4870:
4343:
3426:
1706:
1298:
658:
338:
195:
6138:
4711:
3564:, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex
1481:
dimension would allow a three-dimensional form to be rotated onto its mirror-image. The general concept of
6389:
6364:
6105:
5604:
3417:. In appending a time dimension to three-dimensional space, he specified an alternative perpendicularity,
1521:
1396:
445:
323:
208:
5079:
coined the term "hyperspace philosophy", used to describe writing that uses higher dimensions to explore
6394:
6374:
6369:
5304:
5084:
4426:
The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the
4279:
4265:
3959:
3399:
3266:
2815:
decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity.
1573:
1460:
1368:
1360:
1319:
of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled
506:
467:
426:
421:
274:
4510:
5312:
3873:
3822:
1123:
870:
5857:
5158:
5060:
5051:
5001:
4622:
4616:
4319:
4291:
1630:
1600:
1551:
1384:
1174:
1097:
945:
850:
372:
267:
181:
5609:
4486:
4454:
4420:
4390:
3506:
82:
6465:
6271:
6266:
5122:
4932:
4841:
4829:
4229:
3931:
1739:
1618:
1525:
1179:
1036:
890:
795:
685:
556:
546:
409:
284:
279:
262:
237:
225:
177:
172:
153:
5958:
The Math Book: From
Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics
148:
6470:
6445:
6286:
6241:
5222:
4969:
4945:
by Edwin A. Abbott. Isaac Asimov, in his foreword to the Signet
Classics 1984 edition, described
4675:
4114:
3950:
3618:
3565:
2428:
2424:
1681:, is profoundly different from that explored by SchlÀfli and popularised by Hinton. The study of
1674:
1670:
1506:
1437:
1400:
1347:
1336:
1138:
865:
705:
333:
257:
247:
218:
203:
5735:
Hyperspace: A Scientific
Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension
3771:
1713:
into a serious misconception of the theory of
Relativity. Minkowski's geometry of space-time is
1486:
1355:
5447:
4757:
4464:
of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3
3720:
3343:(i.e. perpendicular) to the other two. The six cardinal directions in this space can be called
6281:
6075:
5997:
5961:
5953:
5949:
5772:
5743:
5733:
5711:
5622:
5570:
5522:
5497:
5451:
5404:
5384:
5348:
5316:
5226:
5180:
5006:
4607:
in 4D can be calculated in closed form for simple geometrical figures, such as the tesseract (
4247:
4170:
3939:
3569:
3414:
3324:
1658:
1581:
1494:
1209:
1199:
1128:
997:
975:
925:
900:
835:
759:
414:
306:
252:
213:
133:, seen rotating here in four-dimensional space, yet projected into two dimensions for display.
6086:
5342:
6211:
6058:
5922:"Art in the Fourth Dimension: Giving Form to Form â The Abstract Paintings of Piet Mondrian"
5900:
5865:
5768:
5701:
5691:
5614:
5560:
5380:
5172:
4335:
4323:
4162:
2823:
1747:
1743:
1722:
1699:
1687:
1626:
1566:
1533:
1189:
930:
640:
518:
453:
311:
296:
161:
4554:
4365:
with analogous projections of the four-dimensional tesseract into three-dimensional space.
4251:, in which the protagonist encounters four-dimensional beings who demonstrate such powers.
2780:{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.}
2413:{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}
1346:'s "Dimensions", published in 1754, but the mathematics of more than three dimensions only
6256:
6201:
6133:
5687:
5282:
5132:
5091:
5017:
4961:
4953:
4924:
4575:
4465:
4166:
4149:
4016:
3669:
3429:
cosmology previously used in a universe of three space dimensions and one time dimension.
1682:
1666:
1558:
1529:
1510:
1482:
1392:
1380:
1351:
1332:
612:
485:
475:
318:
301:
242:
6119:"Dimensions" videos, showing several different ways to visualize four-dimensional objects
5039:"Space has Four Dimensions" is a short story published in 1846 by German philosopher and
1608:. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by
1184:
1153:
1087:
1057:
935:
880:
875:
815:
6054:
5861:
5032:
be shown from concepts, but rests immediately on intuition and indeed on pure intuition
4581:
By applying dimensional analogy, one may infer that a four-dimensional cube, known as a
4140:
with time means expanding or collapsing universe, depending on the mass density inside.
2798:
is 3 in both the
Euclidean and Minkowskian 4-spaces, while the distance squared between
2655:
Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate
6338:
6323:
6146:
6050:
5706:
5673:
5440:
5043:
4979:
4531:
Only three of the cube's six faces can be seen here, because the other three faces lie
4331:
3978:
3557:
1858:
1609:
1387:
is formulated in 4D space, although not in a
Euclidean 4D space. Einstein's concept of
1240:
1214:
1148:
1092:
965:
845:
825:
805:
710:
1847:{\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}
6459:
6328:
6091:
5921:
5028:
3579:
2819:
1678:
1604:
and introduced a method for visualizing the fourth dimension using cubes in the book
1490:
1219:
1204:
1133:
950:
910:
860:
635:
598:
565:
403:
399:
5905:
4430:, shown on the right. Just as the edge-first projection of the cube consists of two
4099:{\displaystyle \mathbf {V} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}R^{4}}
6348:
6313:
6206:
6065:
5176:
4997:
4902:
4290:
A useful application of dimensional analogy in visualizing higher dimensions is in
4033:
3955:
1547:
1407:
1158:
1107:
920:
775:
690:
480:
5845:
4949:
as "The best introduction one can find into the manner of perceiving dimensions."
3938:), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The
6069:
5991:
5487:
1694:
Little, if anything, is gained by representing the fourth Euclidean dimension as
1339:, which was originally abstracted from the spatial experiences of everyday life.
6433:
6216:
5729:
5618:
5253:
5088:
5080:
4894:
4604:
4503:
4447:
4358:
dimensional analogy, one may infer four-dimensional "depth" from these effects.
4346:
to give an illusion of three-dimensional depth to two-dimensional pictures. The
4270:
4242:
4010:
3943:
3448:
2289:
1194:
1067:
885:
820:
748:
720:
695:
24:
5869:
6428:
6308:
5799:
The Fourth Dimension And Non-Euclidean Geometry In Modern Art, Revised Edition
5109:
5104:
4479:
4469:
4413:
4383:
3935:
3549:
3340:
1717:
Euclidean, and consequently has no connection with the present investigation.
1562:
1517:
1052:
1031:
1021:
1011:
970:
915:
810:
800:
700:
551:
5696:
5184:
1485:
with any number of dimensions was fully developed by the Swiss mathematician
1354:
with any number of dimensions was fully developed by the Swiss mathematician
6409:
6318:
6231:
6182:
5804:
5309:
Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton
5137:
5114:
5047:
4898:
4583:
4431:
4315:
4154:
3970:
3946:. All three can "roll" in four-dimensional space, each with its properties.
3674:
1662:
1661:
presented a paper consolidating the role of time as the fourth dimension of
1586:
1543:
1470:
1445:
1441:
1388:
1303:
1062:
780:
743:
607:
579:
130:
5715:
5626:
5574:
4191:
3410:, from the Greek words meaning "up toward" and "down from", respectively.
121:
5408:
5087:, the first writer, in 1888, to use the word "tesseract"; and the Russian
4468:
surrounding a vertex, the tesseract's edge-first projection consists of 3
6333:
6296:
6221:
5888:
5565:
5546:
5545:
Ambinder, Michael S.; Wang, Ranxiao Frances; et al. (October 2009).
4941:
4234:
4179:
4037:
4002:
3878:
3827:
3262:
1502:
1412:
1312:
1143:
1102:
1072:
960:
955:
905:
630:
589:
537:
431:
394:
140:
3954:
they can form much more complex knots than strings in 3D space can. The
6343:
4890:
4435:
3927:
3776:
3725:
3553:
3473: in this section. Unsourced material may be challenged and removed.
3323:
In the familiar three-dimensional space of daily life, there are three
2656:
2292:
of Euclidean three-dimensional space generalizes to four dimensions as
1077:
790:
584:
528:
328:
4539:
these four in the fourth direction, on the far side of the tesseract.
4460:
On the left is the cube viewed corner-first. This is analogous to the
1625:
Higher dimensional non-Euclidean spaces were put on a firm footing by
4339:
4311:
3958:
is an example of such a knotted surface. Another such surface is the
3623:
1639:, in which he considered a "point" to be any sequence of coordinates
1316:
1026:
1016:
895:
840:
715:
678:
666:
621:
574:
492:
157:
5846:"From Flatland to Hypergraphics: Interacting with Higher Dimensions"
5597:
Journal of Experimental Psychology: Human Perception and Performance
5287:
Ueber Projectionsmodelle der regelmÀssigen vier-dimensionalen Körper
4877:
4402:
Note that the other 5 faces of the cube are not seen here. They are
4199:
To understand the nature of four-dimensional space, a device called
4183:
assumptions) to understand which ones are or are not able to learn.
5672:
Ogmen, Haluk; Shibata, Kazuhisa; Yazdanbakhsh, Arash (2020-01-22).
4551:
A concept closely related to projection is the casting of shadows.
4136:
meaning the cosmological age of the universe. Growing or shrinking
6300:
5648:
5055:
4885:. The book, which influenced Picasso, was given to him by Princet.
4876:
4327:
4190:
3969:
3421:. This notion provides his four-dimensional space with a modified
2567:
1082:
1006:
940:
785:
389:
384:
6118:
4927:
texts often mention the concept of "dimension" when referring to
4362:
1572:
One of the first popular expositors of the fourth dimension was
1449:
1308:
673:
523:
126:
6155:
6071:
One Two Three . . . Infinity: Facts and Speculations of Science
4203:
is commonly employed. Dimensional analogy is the study of how (
3442:
18:
5547:"Human four-dimensional spatial intuition in virtual reality"
4434:, the face-first projection of the tesseract consists of two
3315:. They can be used to generate rotations in four dimensions.
2806:
is 4 in Euclidean space and 2 in Minkowski space; increasing
6151:
4553:
4708:) and the volume enclosed by a sphere in three dimensions (
5765:
The Fourth Dimension: A Guided Tour of the Higher Universe
5521:(Reprint ed.). Oxford: Clarendon Press. p. 319.
4909:
mathematics and used them to radically advance their work.
2826:
is used for some applications, and is defined as follows:
5469:
5467:
4310:) is removed and replaced with indirect information. The
1636:Ăber die Hypothesen welche der Geometrie zu Grunde liegen
6074:(3rd ed.). Courier Dover Publications. p. 68.
2285:
Vectors add, subtract and scale as in three dimensions.
1746:
in it. For example, a general point might have position
6127:
article summarizing the "Dimensions" videos, with clips
5291:
On projection models of regular four-dimensional bodies
2790:
As an example, the distance squared between the points
1406:
Single locations in Euclidean 4D space can be given as
4714:
4059:
3398:. To describe the two additional cardinal directions,
2097:
2039:
1981:
1923:
1778:
1469:(published 1788, based on work done around 1755) that
5036:
because it is apodictically (demonstrably) certain."
4790:
4760:
4678:
4625:
4049:
2835:
2668:
2579:
2440:
2301:
2163:
1902:
1764:
5347:. Pomeroy, Washington: Health Research. p. 14.
3265:
valued, with bivectors in four dimensions forming a
6421:
6357:
6295:
6249:
6189:
4883:
Traité élémentaire de géométrie à quatre dimensions
4591:in the image, not merely two-dimensional surfaces.
1698:. In fact, this idea, so attractively developed by
1315:of objects in the everyday world. For example, the
49:. Unsourced material may be challenged and removed.
5439:
4820:
4776:
4746:
4700:
4656:
4098:
3379:. Lengths measured along these axes can be called
3250:
2779:
2644:
2555:
2412:
2274:
2137:
1846:
1297:) is the mathematical extension of the concept of
5446:(2nd ed.). Oxford: Clarendon Press. p.
5075:in 1898 entitled "The Philosophy of Hyperspace".
4516:On the left is the cube viewed corner-first. The
1516:An arithmetic of four spatial dimensions, called
1342:The idea of adding a fourth dimension appears in
2822:is not defined in four dimensions. Instead, the
1692:
1505:that exist in higher dimensions, including the
6147:Frame-by-frame animations of 4D - 3D analogies
5590:"Four-dimensional spatial reasoning in humans"
1738:Mathematically, a four-dimensional space is a
6167:
6044:Bulletin of the American Mathematical Society
5990:SchlÀfli, Ludwig (1901) , Graf, J. H. (ed.),
5893:Bulletin of the American Mathematical Society
5073:Bulletin of the American Mathematical Society
4040:. The hyper-volume of the enclosed space is:
3578:(Displayed as orthogonal projections in each
1271:
8:
1634:
1464:
1416:, i.e., as ordered lists of numbers such as
1331:). This concept of ordinary space is called
5540:
5538:
4294:. A projection is a way of representing an
3367:. Positions along these axes can be called
1367:", in which he explained the concept of a "
6174:
6160:
6152:
5588:Aflalo, T. N.; Graziano, M. S. A. (2008).
5248:, p. 141, §7.x. Historical remarks; "
4821:{\displaystyle {\frac {\pi ^{2}}{2}}r^{4}}
4367:
4111:FriedmannâLemaĂźtreâRobertsonâWalker metric
3439:Rotations in 4-dimensional Euclidean space
1550:were introduced as other four-dimensional
1278:
1264:
993:
512:
147:
136:
5904:
5705:
5695:
5608:
5564:
4812:
4797:
4791:
4789:
4768:
4759:
4738:
4721:
4713:
4692:
4677:
4646:
4640:
4630:
4624:
4090:
4080:
4062:
4058:
4050:
4048:
3533:Learn how and when to remove this message
3235:
3230:
3220:
3210:
3197:
3187:
3171:
3166:
3156:
3146:
3133:
3123:
3103:
3098:
3088:
3078:
3065:
3055:
3039:
3034:
3024:
3014:
3001:
2991:
2975:
2970:
2960:
2950:
2937:
2927:
2911:
2906:
2896:
2886:
2873:
2863:
2848:
2840:
2836:
2834:
2768:
2758:
2745:
2735:
2722:
2712:
2699:
2689:
2677:
2669:
2667:
2627:
2614:
2603:
2595:
2592:
2578:
2542:
2537:
2524:
2519:
2506:
2501:
2488:
2483:
2477:
2467:
2459:
2457:
2445:
2439:
2401:
2391:
2378:
2368:
2355:
2345:
2332:
2322:
2310:
2302:
2300:
2263:
2258:
2251:
2238:
2233:
2226:
2213:
2208:
2201:
2188:
2183:
2176:
2164:
2162:
2092:
2083:
2078:
2034:
2025:
2020:
1976:
1967:
1962:
1918:
1909:
1904:
1901:
1857:This can be written in terms of the four
1827:
1813:
1799:
1785:
1773:
1765:
1763:
109:Learn how and when to remove this message
5336:
5334:
5332:
5204:
5171:(8) (published March 6, 2018): 397â406.
4497:hexahedral volumes surrounding an edge.
3942:of two circles may be taken to obtain a
3574:
1742:that needs four parameters to specify a
1489:in the mid-19th century, at a time when
120:
5929:Spaces of Utopia: An Electronic Journal
5473:
5269:
5245:
5149:
5071:Simon Newcomb wrote an article for the
1677:. But the geometry of spacetime, being
1232:
1166:
1115:
1044:
996:
758:
620:
597:
564:
536:
139:
6140:Flatland: a Romance of Many Dimensions
5159:"Origins of Fourth Dimension Concepts"
4747:{\textstyle V={\frac {4}{3}}\pi r^{3}}
3548:Just as in three dimensions there are
378:Straightedge and compass constructions
4144:Four-dimensional perception in humans
3949:In three dimensions, curves can form
3926:In three dimensions, a circle may be
7:
5960:, Sterling Publishing, p. 282,
5307:(1980). Rucker, Rudolf v. B. (ed.).
4228:The dimensional analogy was used by
3576:Regular polytopes in four dimensions
3471:adding citations to reliable sources
1561:described his method of visualizing
1532:in three dimensions as recounted by
47:adding citations to reliable sources
16:Geometric space with four dimensions
5489:Knotted Surfaces and Their Diagrams
5486:Carter, J. Scott; Saito, Masahico.
4518:vertex-first perspective projection
4214:dimensions, and then inferring how
3568:, analogous to the 13 semi-regular
5993:Theorie der vielfachen KontinuitÀt
3560:. In four dimensions, there are 6
1576:, starting in 1880 with his essay
1375:in the "unseen" fourth dimension.
14:
5850:Interdisciplinary Science Reviews
5552:Psychonomic Bulletin & Review
5519:Introducing Einstein's Relativity
5164:The American Mathematical Monthly
4462:edge-first perspective projection
4428:face-first perspective projection
4398:cell-first perspective projection
1528:was the source of the science of
344:Noncommutative algebraic geometry
6240:
6019:(3rd ed.). New York: Dover.
5938:from the original on 2011-09-29.
5876:from the original on 2013-04-14.
5647:McIntosh, John (November 2002).
5341:Hinton, Charles Howard (1993) .
4881:An illustration from Jouffret's
4869:This section is an excerpt from
4509:
4502:
4485:
4478:
4453:
4446:
4419:
4412:
4389:
4382:
4051:
3913:
3908:
3903:
3898:
3893:
3888:
3883:
3872:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3821:
3811:
3806:
3801:
3796:
3791:
3786:
3781:
3770:
3760:
3755:
3750:
3745:
3740:
3735:
3730:
3719:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3668:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3617:
3447:
3231:
3167:
3099:
3035:
2971:
2907:
2849:
2841:
2678:
2670:
2659:different from the dot product:
2628:
2615:
2604:
2596:
2570:between two non-zero vectors as
2468:
2460:
2446:
2423:It can be used to calculate the
2311:
2303:
2259:
2234:
2209:
2184:
2165:
2079:
2021:
1963:
1905:
1766:
23:
5974:from the original on 2017-03-30
5906:10.1090/S0002-9904-1898-00478-0
4958:The Appendix and the Spectacles
4929:parallel or alternate universes
4657:{\displaystyle \pi ^{2}r^{4}/2}
3458:needs additional citations for
34:needs additional citations for
5889:"The Philosophy of Hyperspace"
5742:. pp. Part I, Chapter 3.
5177:10.1080/00029890.1926.11986607
5128:List of four-dimensional games
4920:Fourth dimension in literature
4245:illustrates this in his novel
3226:
3180:
3162:
3116:
3094:
3048:
3030:
2984:
2966:
2920:
2902:
2856:
1584:magazine. He coined the terms
737:- / other-dimensional
1:
5738:(reissued ed.). Oxford:
5494:American Mathematical Society
5065:
4975:âAnd He Built a Crooked House
4529:, where all four cells meet.
4278:and also in several works of
1578:What is the Fourth Dimension?
1401:SchlÀfli's Euclidean 4D space
1365:What is the Fourth Dimension?
1363:popularized it in an essay, "
6037:"Time as a Fourth Dimension"
5844:Banchoff, Thomas F. (1990).
5013:The Boy Who Reversed Himself
4966:The Fifth-Dimension Catapult
4784:, but the correct volume is
4218:dimensions would relate to (
4175:neural network architectures
3615:
3319:Orthogonality and vocabulary
2566:and calculate or define the
1539:A History of Vector Analysis
6108:Historical Math Collection.
6055:Geometry of Four Dimensions
5619:10.1037/0096-1523.34.5.1066
4701:{\displaystyle A=\pi r^{2}}
4121:is substituted by function
4001:, which is a subset of the
1348:emerged in the 19th century
6507:
6481:Multi-dimensional geometry
6102:Cambridge University Press
5870:10.1179/030801890789797239
5793:Henderson, Linda Dalrymple
5411:[Space and Time].
5379:(1st ed.). New York:
5221:(1st ed.). New York:
4917:
4868:
4836:in an arbitrary dimension
4318:is also a two-dimensional
4008:
3562:convex regular 4-polytopes
3436:
1705:, has led such authors as
1335:because it corresponds to
6476:Four-dimensional geometry
6442:
6238:
5413:Physikalische Zeitschrift
5077:Linda Dalrymple Henderson
5041:experimental psychologist
4777:{\displaystyle \pi r^{4}}
4019:having the same distance
3607:
3593:
1690:felt compelled to write:
1614:Mathematical Games column
1350:. The general concept of
5920:Kruger, Runette (2007).
5697:10.3389/fpsyg.2019.03000
5442:The Theory of Relativity
5375:Gardner, Martin (1975).
5157:Cajori, Florian (1926).
4905:artists took ideas from
3975:Stereographic projection
3552:made of two dimensional
3482:"Four-dimensional space"
3419:hyperbolic orthogonality
3269:linear space with basis
1563:four-dimensional objects
1507:four-dimensional analogs
233:Non-Archimedean geometry
58:"Four-dimensional space"
5887:Newcomb, Simon (1898).
5740:Oxford University Press
5679:Frontiers in Psychology
5517:D'Inverno, Ray (1998).
5427:– via Wikisource.
5000:. Another reference is
4871:Fourth dimension in art
4298:-dimensional object in
4210:) dimensions relate to
3427:absolute space and time
1709:An Experiment with Time
1477:realized that a fourth
1344:Jean le Rond d'Alembert
1299:three-dimensional space
339:Noncommutative geometry
125:The 4D equivalent of a
6486:Science fiction themes
6106:University of Michigan
5383:. pp. 42, 52â53.
5305:Hinton, Charles Howard
4886:
4822:
4778:
4748:
4702:
4658:
4558:
4196:
4100:
4006:
3252:
2781:
2646:
2557:
2414:
2276:
2148:so the general vector
2139:
1848:
1731:
1635:
1522:William Rowan Hamilton
1465:
1397:non-Euclidean geometry
1291:Four-dimensional space
307:Discrete/Combinatorial
134:
5950:Pickover, Clifford A.
5763:Rucker, Rudy (1996).
5085:Charles Howard Hinton
4880:
4844:connecting dimension
4840:is computable from a
4823:
4779:
4749:
4703:
4659:
4557:
4280:Charles Howard Hinton
4194:
4101:
4015:The set of points in
3973:
3960:real projective plane
3400:Charles Howard Hinton
3253:
2782:
2647:
2558:
2415:
2277:
2140:
1849:
1612:in his January 1962 "
1574:Charles Howard Hinton
1369:four-dimensional cube
1361:Charles Howard Hinton
290:Discrete differential
124:
6358:Dimensions by number
6097:The Fourth Dimension
5566:10.3758/PBR.16.5.818
5344:The Fourth Dimension
5052:Allegory of the Cave
4988:The Universe Between
4788:
4758:
4712:
4676:
4623:
4195:A net of a tesseract
4109:This is part of the
4047:
3981:: the set of points
3467:improve this article
2833:
2666:
2577:
2438:
2299:
2161:
1900:
1762:
1707:John William Dunne (
1601:A New Era of Thought
1466:MĂ©canique analytique
1385:theory of relativity
43:improve this article
5862:1990ISRv...15..364B
5438:MĂžller, C. (1972).
5217:Bell, E.T. (1965).
5123:Four-dimensionalism
4933:planes of existence
4842:recurrence relation
4230:Edwin Abbott Abbott
4201:dimensional analogy
4187:Dimensional analogy
4023:from a fixed point
3583:
3566:uniform 4-polytopes
2547:
2529:
2511:
2493:
1619:Scientific American
1580:, published in the
1526:associative algebra
1393:Minkowski structure
557:Pythagorean theorem
6491:Special relativity
6287:Degrees of freedom
6190:Dimensional spaces
6132:2012-09-29 at the
6087:Extract of page 68
5405:Minkowski, Hermann
5293:] (in German).
5258:The Plattner Story
5223:Simon and Schuster
5219:Men of Mathematics
4970:Robert A. Heinlein
4931:or other imagined
4907:higher-dimensional
4887:
4818:
4774:
4744:
4698:
4654:
4611:, for side length
4559:
4197:
4115:General relativity
4096:
4074:
4007:
3575:
3570:Archimedean solids
3248:
3246:
2777:
2642:
2553:
2533:
2515:
2497:
2479:
2410:
2272:
2135:
2126:
2068:
2010:
1952:
1844:
1835:
1675:general relativity
1438:regular 4D objects
1307:, to describe the
135:
6453:
6452:
6262:Lebesgue covering
6227:Algebraic variety
6081:978-0-486-25664-1
6016:Regular Polytopes
6003:978-1-4297-0481-6
5967:978-1-4027-5796-9
5778:978-0-395-39388-8
5749:978-0-19-286189-4
5528:978-0-19-859653-0
5503:978-0-8218-7491-2
5457:978-0-19-851256-1
5390:978-0-394-49406-7
5354:978-0-7873-0410-2
5322:978-0-486-23916-3
5232:978-0-671-62818-5
5007:A Wrinkle In Time
5002:Madeleine L'Engle
4993:The Ifth of Oofth
4990:(both 1951); and
4984:Tiger by the Tail
4806:
4729:
4544:
4543:
4171:entorhinal cortex
4070:
4017:Euclidean 4-space
3940:Cartesian product
3924:
3923:
3543:
3542:
3535:
3517:
3415:velocity of light
3402:coined the terms
3327:âusually labeled
2637:
2548:
2472:
1727:Regular Polytopes
1659:Hermann Minkowski
1582:Dublin University
1567:Schlegel diagrams
1520:, was defined by
1337:Euclid's geometry
1288:
1287:
1253:
1252:
976:List of geometers
659:Three-dimensional
648:
647:
119:
118:
111:
93:
6498:
6250:Other dimensions
6244:
6212:Projective space
6176:
6169:
6162:
6153:
6142:(second edition)
6085:
6059:Internet Archive
6047:
6041:
6020:
6006:
5977:
5975:
5946:
5940:
5939:
5937:
5926:
5917:
5911:
5910:
5908:
5884:
5878:
5877:
5841:
5835:
5827:
5821:
5820:
5818:
5816:
5811:on 20 March 2013
5807:. Archived from
5789:
5783:
5782:
5769:Houghton Mifflin
5760:
5754:
5753:
5726:
5720:
5719:
5709:
5699:
5669:
5663:
5662:
5660:
5659:
5644:
5638:
5637:
5635:
5633:
5612:
5603:(5): 1066â1077.
5594:
5585:
5579:
5578:
5568:
5542:
5533:
5532:
5514:
5508:
5507:
5483:
5477:
5471:
5462:
5461:
5445:
5435:
5429:
5428:
5426:
5424:
5401:
5395:
5394:
5372:
5366:
5365:
5363:
5361:
5338:
5327:
5326:
5313:Dover Publishing
5301:
5295:
5294:
5285:; Waren (1886).
5283:Schlegel, Victor
5279:
5273:
5267:
5261:
5243:
5237:
5236:
5214:
5208:
5202:
5196:
5195:
5193:
5191:
5154:
5067:
4854:
4847:
4827:
4825:
4824:
4819:
4817:
4816:
4807:
4802:
4801:
4792:
4783:
4781:
4780:
4775:
4773:
4772:
4753:
4751:
4750:
4745:
4743:
4742:
4730:
4722:
4707:
4705:
4704:
4699:
4697:
4696:
4663:
4661:
4660:
4655:
4650:
4645:
4644:
4635:
4634:
4570:Bounding regions
4513:
4506:
4489:
4482:
4457:
4450:
4423:
4416:
4393:
4386:
4368:
4336:binocular vision
4304:
4224:
4217:
4213:
4209:
4177:(with different
4139:
4135:
4131:
4120:
4105:
4103:
4102:
4097:
4095:
4094:
4085:
4084:
4075:
4071:
4063:
4054:
4031:
4022:
4000:
3918:
3917:
3916:
3912:
3911:
3907:
3906:
3902:
3901:
3897:
3896:
3892:
3891:
3887:
3886:
3876:
3867:
3866:
3865:
3861:
3860:
3856:
3855:
3851:
3850:
3846:
3845:
3841:
3840:
3836:
3835:
3825:
3816:
3815:
3814:
3810:
3809:
3805:
3804:
3800:
3799:
3795:
3794:
3790:
3789:
3785:
3784:
3774:
3765:
3764:
3763:
3759:
3758:
3754:
3753:
3749:
3748:
3744:
3743:
3739:
3738:
3734:
3733:
3723:
3714:
3713:
3712:
3708:
3707:
3703:
3702:
3698:
3697:
3693:
3692:
3688:
3687:
3683:
3682:
3672:
3663:
3662:
3661:
3657:
3656:
3652:
3651:
3647:
3646:
3642:
3641:
3637:
3636:
3632:
3631:
3621:
3584:
3538:
3531:
3527:
3524:
3518:
3516:
3475:
3451:
3443:
3397:
3339:âwith each axis
3338:
3334:
3330:
3314:
3257:
3255:
3254:
3249:
3247:
3240:
3239:
3234:
3225:
3224:
3215:
3214:
3202:
3201:
3192:
3191:
3176:
3175:
3170:
3161:
3160:
3151:
3150:
3138:
3137:
3128:
3127:
3108:
3107:
3102:
3093:
3092:
3083:
3082:
3070:
3069:
3060:
3059:
3044:
3043:
3038:
3029:
3028:
3019:
3018:
3006:
3005:
2996:
2995:
2980:
2979:
2974:
2965:
2964:
2955:
2954:
2942:
2941:
2932:
2931:
2916:
2915:
2910:
2901:
2900:
2891:
2890:
2878:
2877:
2868:
2867:
2852:
2844:
2824:exterior product
2814:
2805:
2801:
2797:
2793:
2786:
2784:
2783:
2778:
2773:
2772:
2763:
2762:
2750:
2749:
2740:
2739:
2727:
2726:
2717:
2716:
2704:
2703:
2694:
2693:
2681:
2673:
2651:
2649:
2648:
2643:
2638:
2636:
2635:
2631:
2622:
2618:
2608:
2607:
2599:
2593:
2562:
2560:
2559:
2554:
2549:
2546:
2541:
2528:
2523:
2510:
2505:
2492:
2487:
2478:
2473:
2471:
2463:
2458:
2453:
2449:
2419:
2417:
2416:
2411:
2406:
2405:
2396:
2395:
2383:
2382:
2373:
2372:
2360:
2359:
2350:
2349:
2337:
2336:
2327:
2326:
2314:
2306:
2281:
2279:
2278:
2273:
2268:
2267:
2262:
2256:
2255:
2243:
2242:
2237:
2231:
2230:
2218:
2217:
2212:
2206:
2205:
2193:
2192:
2187:
2181:
2180:
2168:
2153:
2144:
2142:
2141:
2136:
2131:
2130:
2088:
2087:
2082:
2073:
2072:
2030:
2029:
2024:
2015:
2014:
1972:
1971:
1966:
1957:
1956:
1914:
1913:
1908:
1892:
1853:
1851:
1850:
1845:
1840:
1839:
1832:
1831:
1818:
1817:
1804:
1803:
1790:
1789:
1769:
1754:
1729:
1723:H. S. M. Coxeter
1702:The Time Machine
1688:H. S. M. Coxeter
1665:, the basis for
1656:
1638:
1627:Bernhard Riemann
1606:Fourth Dimension
1534:Michael J. Crowe
1468:
1435:
1373:single direction
1330:
1326:
1322:
1280:
1273:
1266:
994:
513:
446:Zero-dimensional
151:
137:
114:
107:
103:
100:
94:
92:
51:
27:
19:
6506:
6505:
6501:
6500:
6499:
6497:
6496:
6495:
6456:
6455:
6454:
6449:
6438:
6417:
6353:
6291:
6245:
6236:
6202:Euclidean space
6185:
6180:
6134:Wayback Machine
6115:
6082:
6064:
6039:
6033:Archibald, R. C
6031:
6028:
6026:Further reading
6023:
6011:Coxeter, H.S.M.
6009:
6004:
5989:
5985:
5980:
5968:
5948:
5947:
5943:
5935:
5924:
5919:
5918:
5914:
5886:
5885:
5881:
5843:
5842:
5838:
5828:
5824:
5814:
5812:
5791:
5790:
5786:
5779:
5762:
5761:
5757:
5750:
5728:
5727:
5723:
5688:Frontiers Media
5671:
5670:
5666:
5657:
5655:
5646:
5645:
5641:
5631:
5629:
5610:10.1.1.505.5736
5592:
5587:
5586:
5582:
5544:
5543:
5536:
5529:
5516:
5515:
5511:
5504:
5485:
5484:
5480:
5472:
5465:
5458:
5437:
5436:
5432:
5422:
5420:
5409:"Raum und Zeit"
5403:
5402:
5398:
5391:
5374:
5373:
5369:
5359:
5357:
5355:
5340:
5339:
5330:
5323:
5315:. p. vii.
5303:
5302:
5298:
5281:
5280:
5276:
5268:
5264:
5244:
5240:
5233:
5225:. p. 154.
5216:
5215:
5211:
5203:
5199:
5189:
5187:
5156:
5155:
5151:
5147:
5142:
5133:Time in physics
5100:
5092:P. D. Ouspensky
5054:" presented in
5026:
5018:William Sleator
4962:Murray Leinster
4954:Miles J. Breuer
4925:Science fiction
4922:
4916:
4911:
4910:
4874:
4866:
4861:
4849:
4845:
4808:
4793:
4786:
4785:
4764:
4756:
4755:
4734:
4710:
4709:
4688:
4674:
4673:
4636:
4626:
4621:
4620:
4597:
4576:bounding region
4572:
4549:
4299:
4288:
4266:one-dimensional
4257:
4219:
4215:
4211:
4204:
4189:
4167:critical period
4150:virtual reality
4148:Research using
4146:
4137:
4133:
4122:
4118:
4086:
4076:
4073:
4072:
4045:
4044:
4030:
4024:
4020:
4013:
3982:
3968:
3919:
3914:
3909:
3904:
3899:
3894:
3889:
3884:
3882:
3881:
3877:
3868:
3863:
3858:
3853:
3848:
3843:
3838:
3833:
3831:
3830:
3826:
3817:
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3780:
3779:
3775:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3729:
3728:
3724:
3715:
3710:
3705:
3700:
3695:
3690:
3685:
3680:
3678:
3677:
3673:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3627:
3626:
3622:
3611:
3604:
3597:
3590:
3577:
3558:Platonic solids
3539:
3528:
3522:
3519:
3476:
3474:
3464:
3452:
3441:
3435:
3395:
3336:
3332:
3328:
3325:coordinate axes
3321:
3312:
3305:
3298:
3291:
3284:
3277:
3270:
3267:six-dimensional
3245:
3244:
3229:
3216:
3206:
3193:
3183:
3165:
3152:
3142:
3129:
3119:
3110:
3109:
3097:
3084:
3074:
3061:
3051:
3033:
3020:
3010:
2997:
2987:
2969:
2956:
2946:
2933:
2923:
2905:
2892:
2882:
2869:
2859:
2831:
2830:
2813:
2807:
2803:
2799:
2795:
2791:
2764:
2754:
2741:
2731:
2718:
2708:
2695:
2685:
2664:
2663:
2623:
2610:
2609:
2594:
2575:
2574:
2441:
2436:
2435:
2397:
2387:
2374:
2364:
2351:
2341:
2328:
2318:
2297:
2296:
2257:
2247:
2232:
2222:
2207:
2197:
2182:
2172:
2159:
2158:
2149:
2125:
2124:
2118:
2117:
2111:
2110:
2104:
2103:
2093:
2077:
2067:
2066:
2060:
2059:
2053:
2052:
2046:
2045:
2035:
2019:
2009:
2008:
2002:
2001:
1995:
1994:
1988:
1987:
1977:
1961:
1951:
1950:
1944:
1943:
1937:
1936:
1930:
1929:
1919:
1903:
1898:
1897:
1890:
1883:
1876:
1869:
1862:
1834:
1833:
1823:
1820:
1819:
1809:
1806:
1805:
1795:
1792:
1791:
1781:
1774:
1760:
1759:
1750:
1736:
1730:
1721:
1700:H. G. Wells in
1683:Minkowski space
1653:
1647:
1640:
1559:Victor Schlegel
1530:vector analysis
1511:Platonic solids
1487:Ludwig SchlÀfli
1483:Euclidean space
1458:
1417:
1356:Ludwig SchlÀfli
1352:Euclidean space
1333:Euclidean space
1328:
1324:
1320:
1284:
1255:
1254:
991:
990:
981:
980:
771:
770:
754:
753:
739:
738:
726:
725:
662:
661:
650:
649:
510:
509:
507:Two-dimensional
498:
497:
471:
470:
468:One-dimensional
459:
458:
449:
448:
437:
436:
370:
369:
368:
351:
350:
199:
198:
187:
164:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
6504:
6502:
6494:
6493:
6488:
6483:
6478:
6473:
6468:
6458:
6457:
6451:
6450:
6443:
6440:
6439:
6437:
6436:
6431:
6425:
6423:
6419:
6418:
6416:
6415:
6407:
6402:
6397:
6392:
6387:
6382:
6377:
6372:
6367:
6361:
6359:
6355:
6354:
6352:
6351:
6346:
6341:
6339:Cross-polytope
6336:
6331:
6326:
6324:Hyperrectangle
6321:
6316:
6311:
6305:
6303:
6293:
6292:
6290:
6289:
6284:
6279:
6274:
6269:
6264:
6259:
6253:
6251:
6247:
6246:
6239:
6237:
6235:
6234:
6229:
6224:
6219:
6214:
6209:
6204:
6199:
6193:
6191:
6187:
6186:
6181:
6179:
6178:
6171:
6164:
6156:
6150:
6149:
6144:
6136:
6121:
6114:
6113:External links
6111:
6110:
6109:
6089:
6080:
6062:
6051:Andrew Forsyth
6048:
6027:
6024:
6022:
6021:
6007:
6002:
5986:
5984:
5981:
5979:
5978:
5966:
5941:
5912:
5879:
5836:
5822:
5784:
5777:
5771:. p. 18.
5755:
5748:
5721:
5664:
5649:"4D Maze Game"
5639:
5580:
5559:(5): 818â823.
5534:
5527:
5509:
5502:
5478:
5476:, p. 119.
5463:
5456:
5430:
5396:
5389:
5367:
5353:
5328:
5321:
5296:
5274:
5262:
5238:
5231:
5209:
5197:
5148:
5146:
5143:
5141:
5140:
5135:
5130:
5125:
5120:
5112:
5107:
5101:
5099:
5096:
5068:380 BC).
5044:Gustav Fechner
5025:
5022:
4980:Alan E. Nourse
4918:Main article:
4915:
4912:
4875:
4867:
4865:
4862:
4860:
4857:
4815:
4811:
4805:
4800:
4796:
4771:
4767:
4763:
4741:
4737:
4733:
4728:
4725:
4720:
4717:
4695:
4691:
4687:
4684:
4681:
4653:
4649:
4643:
4639:
4633:
4629:
4596:
4593:
4571:
4568:
4548:
4545:
4542:
4541:
4514:
4507:
4499:
4498:
4490:
4483:
4475:
4474:
4458:
4451:
4443:
4442:
4424:
4417:
4409:
4408:
4394:
4387:
4379:
4378:
4375:
4372:
4332:foreshortening
4287:
4284:
4256:
4255:Cross-sections
4253:
4225:) dimensions.
4188:
4185:
4145:
4142:
4107:
4106:
4093:
4089:
4083:
4079:
4069:
4066:
4061:
4060:
4057:
4053:
4028:
4009:Main article:
3979:Clifford torus
3967:
3964:
3922:
3921:
3870:
3819:
3768:
3717:
3666:
3614:
3613:
3609:
3606:
3602:
3599:
3595:
3592:
3588:
3541:
3540:
3455:
3453:
3446:
3434:
3431:
3320:
3317:
3310:
3303:
3296:
3289:
3282:
3275:
3259:
3258:
3243:
3238:
3233:
3228:
3223:
3219:
3213:
3209:
3205:
3200:
3196:
3190:
3186:
3182:
3179:
3174:
3169:
3164:
3159:
3155:
3149:
3145:
3141:
3136:
3132:
3126:
3122:
3118:
3115:
3112:
3111:
3106:
3101:
3096:
3091:
3087:
3081:
3077:
3073:
3068:
3064:
3058:
3054:
3050:
3047:
3042:
3037:
3032:
3027:
3023:
3017:
3013:
3009:
3004:
3000:
2994:
2990:
2986:
2983:
2978:
2973:
2968:
2963:
2959:
2953:
2949:
2945:
2940:
2936:
2930:
2926:
2922:
2919:
2914:
2909:
2904:
2899:
2895:
2889:
2885:
2881:
2876:
2872:
2866:
2862:
2858:
2855:
2851:
2847:
2843:
2839:
2838:
2811:
2788:
2787:
2776:
2771:
2767:
2761:
2757:
2753:
2748:
2744:
2738:
2734:
2730:
2725:
2721:
2715:
2711:
2707:
2702:
2698:
2692:
2688:
2684:
2680:
2676:
2672:
2653:
2652:
2641:
2634:
2630:
2626:
2621:
2617:
2613:
2606:
2602:
2598:
2591:
2588:
2585:
2582:
2564:
2563:
2552:
2545:
2540:
2536:
2532:
2527:
2522:
2518:
2514:
2509:
2504:
2500:
2496:
2491:
2486:
2482:
2476:
2470:
2466:
2462:
2456:
2452:
2448:
2444:
2421:
2420:
2409:
2404:
2400:
2394:
2390:
2386:
2381:
2377:
2371:
2367:
2363:
2358:
2354:
2348:
2344:
2340:
2335:
2331:
2325:
2321:
2317:
2313:
2309:
2305:
2283:
2282:
2271:
2266:
2261:
2254:
2250:
2246:
2241:
2236:
2229:
2225:
2221:
2216:
2211:
2204:
2200:
2196:
2191:
2186:
2179:
2175:
2171:
2167:
2146:
2145:
2134:
2129:
2123:
2120:
2119:
2116:
2113:
2112:
2109:
2106:
2105:
2102:
2099:
2098:
2096:
2091:
2086:
2081:
2076:
2071:
2065:
2062:
2061:
2058:
2055:
2054:
2051:
2048:
2047:
2044:
2041:
2040:
2038:
2033:
2028:
2023:
2018:
2013:
2007:
2004:
2003:
2000:
1997:
1996:
1993:
1990:
1989:
1986:
1983:
1982:
1980:
1975:
1970:
1965:
1960:
1955:
1949:
1946:
1945:
1942:
1939:
1938:
1935:
1932:
1931:
1928:
1925:
1924:
1922:
1917:
1912:
1907:
1888:
1881:
1874:
1867:
1859:standard basis
1855:
1854:
1843:
1838:
1830:
1826:
1822:
1821:
1816:
1812:
1808:
1807:
1802:
1798:
1794:
1793:
1788:
1784:
1780:
1779:
1777:
1772:
1768:
1735:
1732:
1719:
1651:
1645:
1610:Martin Gardner
1552:algebras over
1542:. Soon after,
1524:in 1843. This
1457:
1454:
1286:
1285:
1283:
1282:
1275:
1268:
1260:
1257:
1256:
1251:
1250:
1249:
1248:
1243:
1235:
1234:
1230:
1229:
1228:
1227:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1177:
1169:
1168:
1164:
1163:
1162:
1161:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1118:
1117:
1113:
1112:
1111:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1047:
1046:
1042:
1041:
1040:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1001:
1000:
992:
988:
987:
986:
983:
982:
979:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
772:
768:
767:
766:
763:
762:
756:
755:
752:
751:
746:
740:
733:
732:
731:
728:
727:
724:
723:
718:
713:
711:Platonic Solid
708:
703:
698:
693:
688:
683:
682:
681:
670:
669:
663:
657:
656:
655:
652:
651:
646:
645:
644:
643:
638:
633:
625:
624:
618:
617:
616:
615:
610:
602:
601:
595:
594:
593:
592:
587:
582:
577:
569:
568:
562:
561:
560:
559:
554:
549:
541:
540:
534:
533:
532:
531:
526:
521:
511:
505:
504:
503:
500:
499:
496:
495:
490:
489:
488:
483:
472:
466:
465:
464:
461:
460:
457:
456:
450:
444:
443:
442:
439:
438:
435:
434:
429:
424:
418:
417:
412:
407:
397:
392:
387:
381:
380:
371:
367:
366:
363:
359:
358:
357:
356:
353:
352:
349:
348:
347:
346:
336:
331:
326:
321:
316:
315:
314:
304:
299:
294:
293:
292:
287:
282:
272:
271:
270:
265:
255:
250:
245:
240:
235:
230:
229:
228:
223:
222:
221:
206:
200:
194:
193:
192:
189:
188:
186:
185:
175:
169:
166:
165:
152:
144:
143:
129:is known as a
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
6503:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6463:
6461:
6448:
6447:
6441:
6435:
6432:
6430:
6427:
6426:
6424:
6420:
6414:
6412:
6408:
6406:
6403:
6401:
6398:
6396:
6393:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6362:
6360:
6356:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6329:Demihypercube
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6310:
6307:
6306:
6304:
6302:
6298:
6294:
6288:
6285:
6283:
6280:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6254:
6252:
6248:
6243:
6233:
6230:
6228:
6225:
6223:
6220:
6218:
6215:
6213:
6210:
6208:
6205:
6203:
6200:
6198:
6195:
6194:
6192:
6188:
6184:
6177:
6172:
6170:
6165:
6163:
6158:
6157:
6154:
6148:
6145:
6143:
6141:
6137:
6135:
6131:
6128:
6126:
6122:
6120:
6117:
6116:
6112:
6107:
6103:
6099:
6098:
6093:
6092:E. H. Neville
6090:
6088:
6083:
6077:
6073:
6072:
6067:
6066:Gamow, George
6063:
6060:
6056:
6052:
6049:
6045:
6038:
6034:
6030:
6029:
6025:
6018:
6017:
6012:
6008:
6005:
5999:
5995:
5994:
5988:
5987:
5982:
5973:
5969:
5963:
5959:
5955:
5951:
5945:
5942:
5934:
5930:
5923:
5916:
5913:
5907:
5902:
5898:
5894:
5890:
5883:
5880:
5875:
5871:
5867:
5863:
5859:
5855:
5851:
5847:
5840:
5837:
5833:
5832:
5826:
5823:
5810:
5806:
5802:
5800:
5797:"Overview of
5794:
5788:
5785:
5780:
5774:
5770:
5766:
5759:
5756:
5751:
5745:
5741:
5737:
5736:
5731:
5725:
5722:
5717:
5713:
5708:
5703:
5698:
5693:
5689:
5685:
5681:
5680:
5675:
5668:
5665:
5654:
5653:urticator.net
5650:
5643:
5640:
5628:
5624:
5620:
5616:
5611:
5606:
5602:
5598:
5591:
5584:
5581:
5576:
5572:
5567:
5562:
5558:
5554:
5553:
5548:
5541:
5539:
5535:
5530:
5524:
5520:
5513:
5510:
5505:
5499:
5495:
5491:
5490:
5482:
5479:
5475:
5470:
5468:
5464:
5459:
5453:
5449:
5444:
5443:
5434:
5431:
5418:
5415:(in German).
5414:
5410:
5406:
5400:
5397:
5392:
5386:
5382:
5378:
5371:
5368:
5356:
5350:
5346:
5345:
5337:
5335:
5333:
5329:
5324:
5318:
5314:
5310:
5306:
5300:
5297:
5292:
5288:
5284:
5278:
5275:
5271:
5266:
5263:
5259:
5255:
5251:
5247:
5242:
5239:
5234:
5228:
5224:
5220:
5213:
5210:
5206:
5205:SchlÀfli 1901
5201:
5198:
5186:
5182:
5178:
5174:
5170:
5166:
5165:
5160:
5153:
5150:
5144:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5119:
5118:
5113:
5111:
5108:
5106:
5103:
5102:
5097:
5095:
5093:
5090:
5086:
5082:
5078:
5074:
5069:
5063:
5062:
5057:
5053:
5049:
5045:
5042:
5037:
5035:
5030:
5029:Immanuel Kant
5024:In philosophy
5023:
5021:
5019:
5015:
5014:
5009:
5008:
5003:
4999:
4995:
4994:
4989:
4985:
4981:
4977:
4976:
4971:
4967:
4963:
4959:
4955:
4950:
4948:
4944:
4943:
4936:
4934:
4930:
4926:
4921:
4914:In literature
4913:
4908:
4904:
4900:
4896:
4892:
4884:
4879:
4872:
4863:
4858:
4856:
4852:
4848:to dimension
4843:
4839:
4835:
4833:
4830:volume of an
4813:
4809:
4803:
4798:
4794:
4769:
4765:
4761:
4739:
4735:
4731:
4726:
4723:
4718:
4715:
4693:
4689:
4685:
4682:
4679:
4669:
4667:
4651:
4647:
4641:
4637:
4631:
4627:
4618:
4614:
4610:
4606:
4602:
4594:
4592:
4590:
4586:
4585:
4579:
4577:
4569:
4567:
4563:
4556:
4552:
4546:
4540:
4538:
4534:
4528:
4523:
4519:
4515:
4512:
4508:
4505:
4501:
4500:
4496:
4491:
4488:
4484:
4481:
4477:
4476:
4471:
4467:
4463:
4459:
4456:
4452:
4449:
4445:
4444:
4441:
4437:
4433:
4429:
4425:
4422:
4418:
4415:
4411:
4410:
4407:
4405:
4399:
4395:
4392:
4388:
4385:
4381:
4380:
4376:
4373:
4370:
4369:
4366:
4364:
4359:
4355:
4351:
4349:
4345:
4341:
4337:
4333:
4329:
4325:
4321:
4317:
4313:
4309:
4302:
4297:
4293:
4285:
4283:
4281:
4277:
4272:
4267:
4262:
4261:cross-section
4254:
4252:
4250:
4249:
4244:
4239:
4237:
4236:
4231:
4226:
4222:
4207:
4202:
4193:
4186:
4184:
4182:
4181:
4176:
4172:
4168:
4164:
4158:
4156:
4151:
4143:
4141:
4129:
4125:
4116:
4112:
4091:
4087:
4081:
4077:
4067:
4064:
4055:
4043:
4042:
4041:
4039:
4035:
4027:
4018:
4012:
4004:
3998:
3994:
3990:
3986:
3980:
3976:
3972:
3965:
3963:
3961:
3957:
3952:
3947:
3945:
3941:
3937:
3933:
3929:
3880:
3875:
3871:
3829:
3824:
3820:
3778:
3773:
3769:
3727:
3722:
3718:
3676:
3671:
3667:
3625:
3620:
3616:
3600:
3586:
3585:
3582:of symmetry)
3581:
3580:Coxeter plane
3573:
3571:
3567:
3563:
3559:
3555:
3551:
3546:
3537:
3534:
3526:
3523:November 2022
3515:
3512:
3508:
3505:
3501:
3498:
3494:
3491:
3487:
3484: â
3483:
3479:
3478:Find sources:
3472:
3468:
3462:
3461:
3456:This section
3454:
3450:
3445:
3444:
3440:
3432:
3430:
3428:
3424:
3420:
3416:
3411:
3409:
3405:
3401:
3392:
3390:
3386:
3382:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3326:
3318:
3316:
3309:
3302:
3295:
3288:
3281:
3274:
3268:
3264:
3241:
3236:
3221:
3217:
3211:
3207:
3203:
3198:
3194:
3188:
3184:
3177:
3172:
3157:
3153:
3147:
3143:
3139:
3134:
3130:
3124:
3120:
3113:
3104:
3089:
3085:
3079:
3075:
3071:
3066:
3062:
3056:
3052:
3045:
3040:
3025:
3021:
3015:
3011:
3007:
3002:
2998:
2992:
2988:
2981:
2976:
2961:
2957:
2951:
2947:
2943:
2938:
2934:
2928:
2924:
2917:
2912:
2897:
2893:
2887:
2883:
2879:
2874:
2870:
2864:
2860:
2853:
2845:
2829:
2828:
2827:
2825:
2821:
2820:cross product
2816:
2810:
2774:
2769:
2765:
2759:
2755:
2751:
2746:
2742:
2736:
2732:
2728:
2723:
2719:
2713:
2709:
2705:
2700:
2696:
2690:
2686:
2682:
2674:
2662:
2661:
2660:
2658:
2639:
2632:
2624:
2619:
2611:
2600:
2589:
2586:
2583:
2580:
2573:
2572:
2571:
2569:
2550:
2543:
2538:
2534:
2530:
2525:
2520:
2516:
2512:
2507:
2502:
2498:
2494:
2489:
2484:
2480:
2474:
2464:
2454:
2450:
2442:
2434:
2433:
2432:
2431:of a vector,
2430:
2426:
2407:
2402:
2398:
2392:
2388:
2384:
2379:
2375:
2369:
2365:
2361:
2356:
2352:
2346:
2342:
2338:
2333:
2329:
2323:
2319:
2315:
2307:
2295:
2294:
2293:
2291:
2286:
2269:
2264:
2252:
2248:
2244:
2239:
2227:
2223:
2219:
2214:
2202:
2198:
2194:
2189:
2177:
2173:
2169:
2157:
2156:
2155:
2152:
2132:
2127:
2121:
2114:
2107:
2100:
2094:
2089:
2084:
2074:
2069:
2063:
2056:
2049:
2042:
2036:
2031:
2026:
2016:
2011:
2005:
1998:
1991:
1984:
1978:
1973:
1968:
1958:
1953:
1947:
1940:
1933:
1926:
1920:
1915:
1910:
1896:
1895:
1894:
1887:
1880:
1873:
1866:
1860:
1841:
1836:
1828:
1824:
1814:
1810:
1800:
1796:
1786:
1782:
1775:
1770:
1758:
1757:
1756:
1753:
1749:
1745:
1741:
1733:
1728:
1724:
1718:
1716:
1712:
1710:
1704:
1703:
1697:
1691:
1689:
1684:
1680:
1679:non-Euclidean
1676:
1672:
1668:
1664:
1660:
1654:
1644:
1637:
1632:
1628:
1623:
1621:
1620:
1615:
1611:
1607:
1603:
1602:
1597:
1593:
1589:
1588:
1583:
1579:
1575:
1570:
1568:
1564:
1560:
1556:
1555:
1549:
1548:coquaternions
1545:
1541:
1540:
1535:
1531:
1527:
1523:
1519:
1514:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1467:
1463:wrote in his
1462:
1455:
1453:
1451:
1447:
1443:
1439:
1433:
1429:
1425:
1421:
1415:
1414:
1409:
1404:
1402:
1398:
1394:
1390:
1386:
1382:
1376:
1374:
1370:
1366:
1362:
1357:
1353:
1349:
1345:
1340:
1338:
1334:
1318:
1314:
1310:
1306:
1305:
1300:
1296:
1292:
1281:
1276:
1274:
1269:
1267:
1262:
1261:
1259:
1258:
1247:
1244:
1242:
1239:
1238:
1237:
1236:
1231:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1172:
1171:
1170:
1165:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1121:
1120:
1119:
1114:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1050:
1049:
1048:
1043:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1004:
1003:
1002:
999:
995:
985:
984:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
773:
765:
764:
761:
757:
750:
747:
745:
742:
741:
736:
730:
729:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
680:
677:
676:
675:
672:
671:
668:
665:
664:
660:
654:
653:
642:
639:
637:
636:Circumference
634:
632:
629:
628:
627:
626:
623:
619:
614:
611:
609:
606:
605:
604:
603:
600:
599:Quadrilateral
596:
591:
588:
586:
583:
581:
578:
576:
573:
572:
571:
570:
567:
566:Parallelogram
563:
558:
555:
553:
550:
548:
545:
544:
543:
542:
539:
535:
530:
527:
525:
522:
520:
517:
516:
515:
514:
508:
502:
501:
494:
491:
487:
484:
482:
479:
478:
477:
474:
473:
469:
463:
462:
455:
452:
451:
447:
441:
440:
433:
430:
428:
425:
423:
420:
419:
416:
413:
411:
408:
405:
404:Perpendicular
401:
400:Orthogonality
398:
396:
393:
391:
388:
386:
383:
382:
379:
376:
375:
374:
364:
361:
360:
355:
354:
345:
342:
341:
340:
337:
335:
332:
330:
327:
325:
324:Computational
322:
320:
317:
313:
310:
309:
308:
305:
303:
300:
298:
295:
291:
288:
286:
283:
281:
278:
277:
276:
273:
269:
266:
264:
261:
260:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
227:
224:
220:
217:
216:
215:
212:
211:
210:
209:Non-Euclidean
207:
205:
202:
201:
197:
191:
190:
183:
179:
176:
174:
171:
170:
168:
167:
163:
159:
155:
150:
146:
145:
142:
138:
132:
128:
123:
113:
110:
102:
99:December 2016
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: â
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
6444:
6410:
6384:
6349:Hyperpyramid
6314:Hypersurface
6207:Affine space
6197:Vector space
6139:
6125:Science News
6124:
6104:, link from
6096:
6070:
6057:, link from
6043:
6015:
5992:
5957:
5944:
5928:
5915:
5896:
5892:
5882:
5853:
5849:
5839:
5829:
5825:
5813:. Retrieved
5809:the original
5798:
5787:
5764:
5758:
5734:
5730:Kaku, Michio
5724:
5683:
5677:
5667:
5656:. Retrieved
5652:
5642:
5630:. Retrieved
5600:
5596:
5583:
5556:
5550:
5518:
5512:
5488:
5481:
5474:Coxeter 1973
5441:
5433:
5421:. Retrieved
5416:
5412:
5399:
5376:
5370:
5358:. Retrieved
5343:
5311:. New York:
5308:
5299:
5290:
5286:
5277:
5270:Coxeter 1973
5265:
5257:
5246:Coxeter 1973
5241:
5218:
5212:
5200:
5188:. Retrieved
5168:
5162:
5152:
5116:
5081:metaphysical
5072:
5070:
5061:The Republic
5059:
5038:
5033:
5027:
5011:
5005:
4998:Walter Tevis
4991:
4987:
4983:
4973:
4965:
4957:
4951:
4946:
4940:
4937:
4923:
4882:
4850:
4837:
4831:
4670:
4665:
4612:
4608:
4600:
4598:
4588:
4582:
4580:
4573:
4564:
4560:
4550:
4536:
4532:
4530:
4526:
4521:
4517:
4494:
4461:
4439:
4427:
4403:
4401:
4397:
4377:Description
4360:
4356:
4352:
4347:
4307:
4300:
4295:
4289:
4275:
4258:
4246:
4240:
4233:
4232:in the book
4227:
4220:
4205:
4200:
4198:
4178:
4159:
4147:
4127:
4123:
4108:
4034:hypersurface
4025:
4014:
3996:
3992:
3988:
3984:
3956:Klein bottle
3948:
3925:
3547:
3544:
3529:
3520:
3510:
3503:
3496:
3489:
3477:
3465:Please help
3460:verification
3457:
3423:simultaneity
3412:
3407:
3403:
3393:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3322:
3307:
3300:
3293:
3286:
3279:
3272:
3260:
2817:
2808:
2789:
2654:
2565:
2422:
2287:
2284:
2150:
2147:
1885:
1878:
1871:
1864:
1856:
1751:
1737:
1726:
1714:
1708:
1701:
1695:
1693:
1669:theories of
1649:
1642:
1624:
1617:
1605:
1599:
1598:in his book
1595:
1591:
1585:
1577:
1571:
1553:
1537:
1515:
1478:
1459:
1431:
1427:
1423:
1419:
1411:
1405:
1377:
1372:
1341:
1302:
1294:
1290:
1289:
1108:Parameshvara
921:Parameshvara
734:
691:Dodecahedron
275:Differential
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
6434:Codimension
6413:-dimensions
6334:Hypersphere
6217:Free module
5954:"Tesseract"
5831:Prolegomena
5423:October 27,
5360:17 February
5254:H. G. Wells
5190:October 10,
5089:esotericist
4960:(1928) and
4895:Surrealists
4664:for radius
4605:hypervolume
4595:Hypervolume
4344:perspective
4286:Projections
4271:hypersphere
4243:Rudy Rucker
4036:known as a
4011:Hypersphere
3966:Hypersphere
3944:duocylinder
2290:dot product
1893:, given by
1755:, equal to
1657:. In 1908,
1557:. In 1886,
1518:quaternions
1444:, which is
1395:based on a
1233:Present day
1180:Lobachevsky
1167:1700sâ1900s
1124:JyeáčŁáčhadeva
1116:1400sâ1700s
1068:Brahmagupta
891:Lobachevsky
871:JyeáčŁáčhadeva
821:Brahmagupta
749:Hypersphere
721:Tetrahedron
696:Icosahedron
268:Diophantine
6466:4 (number)
6460:Categories
6429:Hyperspace
6309:Hyperplane
6046:: 409â412.
5983:References
5899:(5): 187.
5856:(4): 364.
5767:. Boston:
5658:2016-12-16
5110:4-manifold
5105:4-polytope
5046:under the
4996:(1957) by
4859:In culture
4615:) and the
4470:hexahedral
4432:trapezoids
4374:Tesseract
4342:often use
4292:projection
4155:labyrinths
3936:spherinder
3930:to form a
3493:newspapers
3437:See also:
3341:orthogonal
1667:Einstein's
1544:tessarines
1448:to the 3D
1304:dimensions
1093:al-Yasamin
1037:Apollonius
1032:Archimedes
1022:Pythagoras
1012:Baudhayana
966:al-Yasamin
916:Pythagoras
811:Baudhayana
801:Archimedes
796:Apollonius
701:Octahedron
552:Hypotenuse
427:Similarity
422:Congruence
334:Incidence
285:Symplectic
280:Riemannian
263:Arithmetic
238:Projective
226:Hyperbolic
154:Projecting
69:newspapers
6471:Dimension
6319:Hypercube
6297:Polytopes
6277:Minkowski
6272:Hausdorff
6267:Inductive
6232:Spacetime
6183:Dimension
6013:(1973) .
5931:(5): 11.
5805:MIT Press
5632:20 August
5605:CiteSeerX
5185:0002-9890
5145:Citations
5138:Spacetime
5048:pseudonym
5004:'s novel
4899:Futurists
4795:π
4762:π
4732:π
4686:π
4628:π
4584:tesseract
4338:, etc.).
4324:receptors
4248:Spaceland
4163:artifacts
4078:π
3675:tesseract
3550:polyhedra
3373:longitude
3204:−
3140:−
3072:−
3008:−
2944:−
2880:−
2846:∧
2804:(1,1,1,1)
2800:(0,0,0,0)
2796:(1,1,1,0)
2792:(0,0,0,0)
2752:−
2675:⋅
2601:⋅
2590:
2581:θ
2465:⋅
2308:⋅
1663:spacetime
1587:tesseract
1503:polytopes
1471:mechanics
1446:analogous
1442:tesseract
1389:spacetime
1313:locations
1210:Minkowski
1129:Descartes
1063:Aryabhata
1058:KÄtyÄyana
989:by period
901:Minkowski
876:KÄtyÄyana
836:Descartes
781:Aryabhata
760:Geometers
744:Tesseract
608:Trapezoid
580:Rectangle
373:Dimension
258:Algebraic
248:Synthetic
219:Spherical
204:Euclidean
131:tesseract
6446:Category
6422:See also
6222:Manifold
6130:Archived
6068:(1988).
6035:(1914).
5972:archived
5952:(2009),
5933:Archived
5874:Archived
5815:24 March
5732:(1995).
5716:32038384
5690:: 3000.
5627:18823195
5575:19815783
5407:(1909).
5098:See also
5034:a priori
4947:Flatland
4942:Flatland
4903:abstract
4601:4-volume
4466:deltoids
4436:frustums
4404:obscured
4326:but the
4276:Flatland
4235:Flatland
4180:a priori
4038:3-sphere
4032:forms a
4003:3-sphere
3932:cylinder
3928:extruded
3920:{5,3,3}
3879:120-cell
3869:{3,3,5}
3828:600-cell
3818:{3,4,3}
3767:{3,3,4}
3716:{4,3,3}
3665:{3,3,3}
3554:polygons
3433:Geometry
3377:latitude
3369:altitude
3263:bivector
3261:This is
1861:vectors
1720:â
1629:'s 1854
1495:Grassman
1461:Lagrange
1413:4-tuples
1381:Einstein
1200:Poincaré
1144:Minggatu
1103:Yang Hui
1073:Virasena
961:Yang Hui
956:Virasena
926:Poincaré
906:Minggatu
686:Cylinder
631:Diameter
590:Rhomboid
547:Altitude
538:Triangle
432:Symmetry
410:Parallel
395:Diagonal
365:Features
362:Concepts
253:Analytic
214:Elliptic
196:Branches
182:Timeline
141:Geometry
6344:Simplex
6282:Fractal
6094:(1921)
6053:(1930)
5858:Bibcode
5707:6987450
5419:: 75â88
5115:Exotic
4891:Cubists
4589:volumes
4547:Shadows
4340:Artists
4314:of the
3995:), sin(
3991:), cos(
3987:), sin(
3777:24-cell
3726:16-cell
3507:scholar
2657:pairing
1734:Vectors
1671:special
1648:, ...,
1509:of the
1479:spatial
1456:History
1408:vectors
1225:Coxeter
1205:Hilbert
1190:Riemann
1139:Huygens
1098:al-Tusi
1088:KhayyĂĄm
1078:Alhazen
1045:1â1400s
946:al-Tusi
931:Riemann
881:KhayyĂĄm
866:Huygens
861:Hilbert
831:Coxeter
791:Alhazen
769:by name
706:Pyramid
585:Rhombus
529:Polygon
481:segment
329:Fractal
312:Digital
297:Complex
178:History
173:Outline
83:scholar
6301:shapes
6078:
6000:
5964:
5834:, § 12
5775:
5746:
5714:
5704:
5625:
5607:
5573:
5525:
5500:
5454:
5387:
5351:
5319:
5250:Möbius
5229:
5183:
4901:, and
4864:In art
4828:. The
4617:4-ball
4537:behind
4533:behind
4527:inside
4348:shadow
4312:retina
4117:where
3624:5-cell
3509:
3502:
3495:
3488:
3480:
3387:, and
3381:height
3375:, and
3363:, and
3335:, and
2587:arccos
2429:length
1748:vector
1631:thesis
1499:Möbius
1491:Cayley
1475:Möbius
1440:, the
1391:has a
1327:, and
1317:volume
1246:Gromov
1241:Atiyah
1220:Veblen
1215:Cartan
1185:Bolyai
1154:Sakabe
1134:Pascal
1027:Euclid
1017:Manava
951:Veblen
936:Sakabe
911:Pascal
896:Manava
856:Gromov
841:Euclid
826:Cartan
816:Bolyai
806:Atiyah
716:Sphere
679:cuboid
667:Volume
622:Circle
575:Square
493:Length
415:Vertex
319:Convex
302:Finite
243:Affine
158:sphere
85:
78:
71:
64:
56:
6405:Eight
6400:Seven
6380:Three
6257:Krull
6040:(PDF)
5936:(PDF)
5925:(PDF)
5593:(PDF)
5381:Knopf
5289:[
5056:Plato
4834:-ball
4495:three
4371:Cube
4328:brain
4320:array
4308:depth
4132:with
3983:(cos(
3977:of a
3951:knots
3514:JSTOR
3500:books
3389:depth
3385:width
3365:south
3361:north
2568:angle
1744:point
1740:space
1616:" in
1565:with
1309:sizes
1195:Klein
1175:Gauss
1149:Euler
1083:Sijzi
1053:Zhang
1007:Ahmes
971:Zhang
941:Sijzi
886:Klein
851:Gauss
846:Euler
786:Ahmes
519:Plane
454:Point
390:Curve
385:Angle
162:plane
160:to a
90:JSTOR
76:books
6390:Five
6385:Four
6365:Zero
6299:and
6076:ISBN
5998:ISBN
5962:ISBN
5817:2013
5773:ISBN
5744:ISBN
5712:PMID
5634:2020
5623:PMID
5571:PMID
5523:ISBN
5498:ISBN
5452:ISBN
5425:2022
5385:ISBN
5362:2017
5349:ISBN
5317:ISBN
5227:ISBN
5192:2022
5181:ISSN
4986:and
4599:The
4522:four
4363:cube
3486:news
3408:kata
3406:and
3357:west
3353:east
3349:down
2818:The
2802:and
2794:and
2425:norm
2288:The
1696:time
1673:and
1596:kata
1594:and
1546:and
1497:and
1450:cube
1159:Aida
776:Aida
735:Four
674:Cube
641:Area
613:Kite
524:Area
476:Line
127:cube
62:news
6395:Six
6375:Two
6370:One
5901:doi
5866:doi
5702:PMC
5692:doi
5615:doi
5561:doi
5256:in
5173:doi
5058:'s
5016:by
4982:'s
4972:'s
4964:'s
4956:'s
4853:- 2
4668:).
4603:or
4322:of
4316:eye
4303:â 1
4223:+ 1
4208:â 1
4113:in
3962:.
3612:,
3605:,
3598:,
3591:,
3469:by
3404:ana
2427:or
2154:is
1715:not
1622:.
1592:ana
1536:in
1410:or
1383:'s
1311:or
998:BCE
486:ray
45:by
6462::
6100:,
6042:.
5970:,
5956:,
5927:.
5895:.
5891:.
5872:.
5864:.
5854:15
5852:.
5848:.
5803:.
5795:.
5710:.
5700:.
5686:.
5684:10
5682:.
5676:.
5651:.
5621:.
5613:.
5601:34
5599:.
5595:.
5569:.
5557:16
5555:.
5549:.
5537:^
5496:.
5492:.
5466:^
5450:.
5448:93
5417:10
5331:^
5260:."
5179:.
5169:33
5167:.
5161:.
5094:.
5066:c.
5020:.
4897:,
4893:,
4855:.
4438:.
4334:,
3999:))
3391:.
3383:,
3371:,
3359:,
3355:,
3351:,
3347:,
3345:up
3331:,
3311:34
3306:,
3304:24
3299:,
3297:23
3292:,
3290:14
3285:,
3283:13
3278:,
3276:12
3237:34
3173:24
3105:23
3041:14
2977:13
2913:12
1884:,
1877:,
1870:,
1725:,
1633:,
1590:,
1569:.
1513:.
1493:,
1452:.
1430:,
1426:,
1422:,
1403:.
1323:,
1295:4D
156:a
6411:n
6175:e
6168:t
6161:v
6084:.
6061:.
5976:.
5909:.
5903::
5897:4
5868::
5860::
5819:.
5801:"
5781:.
5752:.
5718:.
5694::
5661:.
5636:.
5617::
5577:.
5563::
5531:.
5506:.
5460:.
5393:.
5364:.
5325:.
5235:.
5207:.
5194:.
5175::
5117:R
5064:(
4873:.
4851:n
4846:n
4838:n
4832:n
4814:4
4810:r
4804:2
4799:2
4770:4
4766:r
4740:3
4736:r
4727:3
4724:4
4719:=
4716:V
4694:2
4690:r
4683:=
4680:A
4672:(
4666:r
4652:2
4648:/
4642:4
4638:r
4632:2
4619:(
4613:s
4609:s
4306:(
4301:n
4296:n
4221:n
4216:n
4212:n
4206:n
4138:R
4134:t
4130:)
4128:t
4126:(
4124:R
4119:R
4092:4
4088:R
4082:2
4068:2
4065:1
4056:=
4052:V
4029:0
4026:P
4021:R
4005:.
3997:b
3993:b
3989:a
3985:a
3610:4
3608:H
3603:4
3601:F
3596:4
3594:B
3589:4
3587:A
3536:)
3530:(
3525:)
3521:(
3511:·
3504:·
3497:·
3490:·
3463:.
3396:w
3337:z
3333:y
3329:x
3313:)
3308:e
3301:e
3294:e
3287:e
3280:e
3273:e
3271:(
3242:.
3232:e
3227:)
3222:3
3218:b
3212:4
3208:a
3199:4
3195:b
3189:3
3185:a
3181:(
3178:+
3168:e
3163:)
3158:2
3154:b
3148:4
3144:a
3135:4
3131:b
3125:2
3121:a
3117:(
3114:+
3100:e
3095:)
3090:2
3086:b
3080:3
3076:a
3067:3
3063:b
3057:2
3053:a
3049:(
3046:+
3036:e
3031:)
3026:1
3022:b
3016:4
3012:a
3003:4
2999:b
2993:1
2989:a
2985:(
2982:+
2972:e
2967:)
2962:1
2958:b
2952:3
2948:a
2939:3
2935:b
2929:1
2925:a
2921:(
2918:+
2908:e
2903:)
2898:1
2894:b
2888:2
2884:a
2875:2
2871:b
2865:1
2861:a
2857:(
2854:=
2850:b
2842:a
2812:4
2809:b
2775:.
2770:4
2766:b
2760:4
2756:a
2747:3
2743:b
2737:3
2733:a
2729:+
2724:2
2720:b
2714:2
2710:a
2706:+
2701:1
2697:b
2691:1
2687:a
2683:=
2679:b
2671:a
2640:.
2633:|
2629:b
2625:|
2620:|
2616:a
2612:|
2605:b
2597:a
2584:=
2551:,
2544:2
2539:4
2535:a
2531:+
2526:2
2521:3
2517:a
2513:+
2508:2
2503:2
2499:a
2495:+
2490:2
2485:1
2481:a
2475:=
2469:a
2461:a
2455:=
2451:|
2447:a
2443:|
2408:.
2403:4
2399:b
2393:4
2389:a
2385:+
2380:3
2376:b
2370:3
2366:a
2362:+
2357:2
2353:b
2347:2
2343:a
2339:+
2334:1
2330:b
2324:1
2320:a
2316:=
2312:b
2304:a
2270:.
2265:4
2260:e
2253:4
2249:a
2245:+
2240:3
2235:e
2228:3
2224:a
2220:+
2215:2
2210:e
2203:2
2199:a
2195:+
2190:1
2185:e
2178:1
2174:a
2170:=
2166:a
2151:a
2133:,
2128:)
2122:1
2115:0
2108:0
2101:0
2095:(
2090:=
2085:4
2080:e
2075:;
2070:)
2064:0
2057:1
2050:0
2043:0
2037:(
2032:=
2027:3
2022:e
2017:;
2012:)
2006:0
1999:0
1992:1
1985:0
1979:(
1974:=
1969:2
1964:e
1959:;
1954:)
1948:0
1941:0
1934:0
1927:1
1921:(
1916:=
1911:1
1906:e
1891:)
1889:4
1886:e
1882:3
1879:e
1875:2
1872:e
1868:1
1865:e
1863:(
1842:.
1837:)
1829:4
1825:a
1815:3
1811:a
1801:2
1797:a
1787:1
1783:a
1776:(
1771:=
1767:a
1752:a
1711:)
1655:)
1652:n
1650:x
1646:1
1643:x
1641:(
1554:R
1434:)
1432:w
1428:z
1424:y
1420:x
1418:(
1329:z
1325:y
1321:x
1293:(
1279:e
1272:t
1265:v
406:)
402:(
184:)
180:(
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
