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Geometry |
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Geometers |

A **four-dimensional space** (**4D**) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called * dimensions *, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled *x*, *y*, and *z*).

- History
- Vectors
- Orthogonality and vocabulary
- Geometry
- Hypersphere
- Cognition
- Dimensional analogy
- Cross-sections
- Projections
- Shadows
- Bounding volumes
- Visual scope
- Limitations
- See also
- References
- Further reading
- External links

The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" published in 1754,^{ [1] }^{ [2] } was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880, Charles Howard Hinton popularized these insights in an essay titled "What is the Fourth Dimension?", which explained the concept of a "four-dimensional cube" 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 *single direction* in the "unseen" fourth dimension.

Higher-dimensional spaces (i.e., 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 the use of such spaces. Einstein's concept of spacetime uses such a 4D space, though it has a Minkowski structure that is slightly more complicated than Euclidean 4D space.

Single locations in 4D space can be given as vectors or * n-tuples *, i.e. as ordered lists of numbers such as (*x*, *y*, *z*, *w*). 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 to that complexity can be seen in the accompanying 2D animation of one of the simplest possible 4D objects, the tesseract (equivalent to the 3D cube; see also hypercube).

Lagrange wrote in his *Mécanique analytique* (published 1788, based on work done around 1755) that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time.^{ [3] } In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,^{ [4] }^{: 141 } and by 1853 Ludwig Schläfli had discovered all the regular polytopes that exist in higher dimensions, including the four-dimensional analogues of the Platonic solids, but his work was not published until after his death.^{ [4] }^{: 142–143 } Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 thesis, *Über die Hypothesen welche der Geometrie zu Grunde liegen*, in which he considered a "point" to be any sequence of coordinates (*x*_{1}, ..., *x*_{n}). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in * A History of Vector Analysis *. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over **R**.

One of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay *What is the Fourth Dimension?*; published in the Dublin University magazine.^{ [5] } He coined the terms * tesseract *, *ana* and *kata* in his book * A New Era of Thought *, and introduced a method for visualising the fourth dimension using cubes in the book *Fourth Dimension*.^{ [6] }^{ [7] }

Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in * Scientific American *. In 1886 Victor Schlegel described^{ [8] } his method of visualizing four-dimensional objects with Schlegel diagrams.

In 1908, Hermann Minkowski presented a paper^{ [9] } consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.^{ [10] } But the geometry of spacetime, being non-Euclidean, is profoundly different from that explored by Schläfli and popularised by Hinton. The study of Minkowski space required new mathematics 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 H. S. M. Coxeter felt compelled to write:

Little, if anything, is gained by representing the fourth Euclidean dimension as

time. In fact, this idea, so attractively developed by H. G. Wells inThe Time Machine, has led such authors as John William Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time isnotEuclidean, and consequently has no connection with the present investigation.

Mathematically, four-dimensional space is a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it. For example, a general point might have position vector **a**, equal to

This can be written in terms of the four standard basis vectors (**e**_{1}, **e**_{2}, **e**_{3}, **e**_{4}), given by

so the general vector **a** is

Vectors add, subtract and scale as in three dimensions.

The dot product of Euclidean three-dimensional space generalizes to four dimensions as

It can be used to calculate the norm or length of a vector,

and calculate or define the angle between two non-zero vectors as

Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate pairing different from the dot product:

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing actually decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity.

The cross product is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (**e**_{12}, **e**_{13}, **e**_{14}, **e**_{23}, **e**_{24}, **e**_{34}). They can be used to generate rotations in four dimensions.

In the familiar three-dimensional space of daily life, there are three coordinate axes—usually labeled *x*, *y*, and *z*—with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called *up*, *down*, *east*, *west*, *north*, and *south*. Positions along these axes can be called *altitude*, *longitude*, and *latitude*. Lengths measured along these axes can be called *height*, *width*, and *depth*.

Comparatively, four-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled *w*. To describe the two additional cardinal directions, Charles Howard Hinton coined the terms *ana* and *kata*, from the Greek words meaning "up toward" and "down from", respectively ^{[ citation needed ]}.

As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite velocity of light. In appending a time dimension to three dimensional space, he specified an alternative perpendicularity, hyperbolic orthogonality. This notion provides his four-dimensional space with a modified simultaneity appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional absolute space and time cosmology previously used in a universe of three space dimensions and one time dimension.

The geometry of four-dimensional space is much more complex than that of three-dimensional space, due to the extra degree of freedom.

Just as in three dimensions there are polyhedra made of two dimensional polygons, in four dimensions there are 4-polytopes made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the Platonic solids. In four dimensions, there are 6 convex regular 4-polytopes, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes.

A_{4}, [3,3,3] | B_{4}, [4,3,3] | F_{4}, [3,4,3] | H_{4}, [5,3,3] | ||
---|---|---|---|---|---|

5-cell {3,3,3} | tesseract {4,3,3} | 16-cell {3,3,4} | 24-cell {3,4,3} | 600-cell {3,3,5} | 120-cell {5,3,3} |

In three dimensions, a circle may be extruded to form a cylinder. 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 spherinder), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder).^{[ citation needed ]} The Cartesian product of two circles may be taken to obtain a duocylinder. All three can "roll" in four-dimensional space, each with its own properties.

In three dimensions, curves can form knots 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 direction—but 2D surfaces can form non-trivial, non-self-intersecting knots in 4D space.^{ [11] }^{[ page needed ]} Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The Klein bottle is an example of such a knotted surface.^{[ citation needed ]} Another such surface is the real projective plane.^{[ citation needed ]}

The set of points in Euclidean 4-space having the same distance R from a fixed point P_{0} forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:

This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity where *R* is substituted by function *R*(*t*) with *t* meaning the cosmological age of the universe. Growing or shrinking *R* with time means expanding or collapsing universe, depending on the mass density inside.^{ [12] }

Research using virtual reality finds that humans, in spite of 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.^{ [13] } The researchers noted that "the 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".^{ [13] } In another study,^{ [14] } 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 labyrinths). The graphical interface was based on John McIntosh's free 4D Maze game.^{ [15] } 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).

To understand the nature of four-dimensional space, a device called *dimensional analogy* is commonly employed. Dimensional analogy is the study of how (*n* − 1) dimensions relate to *n* dimensions, and then inferring how *n* dimensions would relate to (*n* + 1) dimensions.^{ [16] }

Dimensional analogy was used by Edwin Abbott Abbott in the book * Flatland *, 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.

By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from the three-dimensional perspective. Rudy Rucker illustrates this in his novel * Spaceland *, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a cross-section 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, then a circle gradually growing larger, until it reaches the diameter of the balloon, and then getting smaller again, until it shrank to a point and then disappeared. The 2D beings would not see a circle in the same way as we do, rather only a 1 dimensional 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 four-dimensional sphere would appear first as a point, then as a growing circle, with the circle then shrinking to a single point and then disappearing.^{ [17] } This means of visualizing aspects of the fourth dimension was used in the novel *Flatland* and also in several works of Charles Howard Hinton.^{ [6] }^{: 11–14 } And in the same way 3 dimensional beings (such as humans with a 2D retina) can see all the sides of a 2D shape, a 4D being could be able to see all faces of a 3D shape at once with their 3D solid retina.

A useful application of dimensional analogy in visualizing higher dimensions is in projection. A projection is a way for representing an *n*-dimensional object in *n* − 1 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 (*depth*) is removed and replaced with indirect information. The retina of the eye is also a two-dimensional array of receptors but the brain is able to perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision, etc.). Artists often use perspective to give an illusion of three-dimensional depth to two-dimensional pictures. The *shadow*, 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.

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 perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.

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 dimensional analogy, one may infer four-dimensional "depth" from these effects.

As an illustration of this principle, the following sequence of images compares various views of the three-dimensional cube with analogous projections of the four-dimensional tesseract into three-dimensional space.

Cube | Tesseract | Description |
---|---|---|

The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first perspective projection, 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. Note that the other 5 faces of the cube are not seen here. They are | ||

The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the face-first perspective projection, shown on the right. Just as the edge-first projection of the cube consists of two trapezoids, the face-first projection of the tesseract consists of two frustums. 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. | ||

On the left is the cube viewed corner-first. This is analogous to the edge-first perspective projection of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 deltoids surrounding a vertex, the tesseract's edge-first projection consists of 3 hexahedral volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, so the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet. | ||

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 three hexahedral volumes surrounding an edge. | ||

On the left is the cube viewed corner-first. The vertex-first perspective projection 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 four 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 boundary of the projected volume, but at its center inside, where all four cells meet. Note that only three faces of the cube's 6 faces can be seen here, because the other 3 lie |

A concept closely related to projection is the casting of shadows.

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 zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shone on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow.

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 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 actually a two-dimensional image of the three-dimensional shadow of the four-dimensional wireframe figure.)

Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. 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. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract, 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 *volumes* in the image, not merely two-dimensional surfaces.

People have a spatial self-perception as beings in a three-dimensional space, but are visually restricted to one dimension less: the eye sees the world as a projection to two dimensions, on the surface of the retina. Assuming a four-dimensional being were able to see the world in projections to a hypersurface, also just one dimension less, i.e., to three dimensions, it would be able to see, e.g., all six faces of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as people can see all four sides and simultaneously the interior of a rectangle on a piece of paper.^{[ citation needed ]} The being would be able to discern all points in a 3-dimensional subspace simultaneously, including the inner structure of solid 3-dimensional objects, things obscured from human viewpoints in three dimensions on two-dimensional projections. Brains receive images in two dimensions and use reasoning to help picture three-dimensional objects.

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 of a circle: and the Volume of a sphere: . One might guess that the volume of the 3-sphere in four-dimensional space is , or perhaps , but either of these would be wrong. The actual formula is .^{ [4] }^{: 119 }

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In physics and mathematics, the **dimension** of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or **n-polytope**. In this context, "flat sides" means that the sides of a (*k* + 1)-polytope consist of k-polytopes that may have (*k* – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a **tesseract** is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In geometry, a **hypercube** is an *n*-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in *n* dimensions is equal to .

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In geometry, the **24-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign : a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space.

In geometry, the **5-cell** is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a **C _{5}**,

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, a **cross-polytope**, **hyperoctahedron**, **orthoplex**, or **cocube** is a regular, convex polytope that exists in *n*-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In geometry of 4 dimensions or higher, a **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In four-dimensional geometry, a **runcinated tesseract** is a convex uniform 4-polytope, being a runcination of the regular tesseract.

In four-dimensional geometry, a **cantellated tesseract** is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, the **hyperboloid model**, also known as the **Minkowski model** after Hermann Minkowski, is a model of *n*-dimensional hyperbolic geometry in which points are represented by points on the forward sheet *S*^{+} of a two-sheeted hyperboloid in (*n*+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and *m*-planes are represented by the intersections of (*m*+1)-planes passing through the origin in Minkowski space with *S*^{+} or by wedge products of *m* vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the *n*-sphere is embedded in (*n*+1)-dimensional Euclidean space.

In five-dimensional geometry, a **5-cube** is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

The * Octacube* is a large, stainless steel sculpture displayed in the mathematics department of Pennsylvania State University in State College, PA. The sculpture represents a mathematical object called the 24-cell or "octacube". Because a real 24-cell is four-dimensional, the artwork is actually a projection into the three-dimensional world.

**Six-dimensional space** is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.

- ↑ Cajori, Florian (1926),
*The American Mathematical Monthly*,**33**(8): 397–406, doi:10.1080/00029890.1926.11986607 , - ↑ Cajori, Florian (1926). "Origins of Fourth Dimension Concepts" (PDF).
*The American Mathematical Monthly*.**33**(8): 397–406. doi:10.1080/00029890.1926.11986607. JSTOR 2298325. - ↑ Bell, E.T. (1965).
*Men of Mathematics*(1st ed.). New York: Simon and Schuster. p. 154. ISBN 978-0-671-62818-5. - 1 2 3 4 Coxeter, H.S.M. (1973).
*Regular Polytopes*(3rd ed.). New York: Dover Publishing. ISBN 978-0-486-61480-9. - ↑ Hinton, Charles Howard (1980). Rucker, Rudolf v. B. (ed.).
*Speculations on the Fourth Dimension: Selected writings of Charles H. Hinton*. New York: Dover Publishing. p. vii. ISBN 978-0-486-23916-3. - 1 2 Hinton, Charles Howard (1993) [1904].
*The Fourth Dimension*. Pomeroy, Washington: Health Research. p. 14. ISBN 978-0-7873-0410-2 . Retrieved 17 February 2017. - ↑ Gardner, Martin (1975).
*Mathematical Carnival: From Penny Puzzles. Card Shuffles and Tricks of Lightning Calculators to Roller Coaster Rides into the Fourth Dimension*(1st ed.). New York: Knopf. pp. 42, 52–53. ISBN 978-0-394-49406-7. - ↑ Victor Schlegel (1886)
*Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper*, Waren - ↑ Minkowski, Hermann (1909),
*Physikalische Zeitschrift*,**10**: 75–88- Various English translations on Wikisource: Space and Time

- ↑ Møller, C. (1972).
*The Theory of Relativity*(2nd ed.). Oxford: Clarendon Press. p. 93. ISBN 978-0-19-851256-1. - ↑ Carter, J.Scott; Saito, Masahico.
*Knotted Surfaces and Their Diagrams*. American Mathematical Society. ISBN 978-0-8218-7491-2. - ↑ D'Inverno, Ray (1998).
*Introducing Einstein's Relativity*(Reprint ed.). Oxford: Clarendon Press. p. 319. ISBN 978-0-19-859653-0. - 1 2 Ambinder, Michael S.; Wang, Ranxiao Frances; Crowell, James A.; Francis, George K.; Brinkmann, Peter (October 2009). "Human four-dimensional spatial intuition in virtual reality".
*Psychonomic Bulletin & Review*.**16**(5): 818–823. doi: 10.3758/PBR.16.5.818 . PMID 19815783. - ↑ Aflalo, T. N.; Graziano, M. S. A. (2008). "Four-dimensional spatial reasoning in humans" (PDF).
*Journal of Experimental Psychology: Human Perception and Performance*.**34**(5): 1066–1077. CiteSeerX 10.1.1.505.5736 . doi:10.1037/0096-1523.34.5.1066. PMID 18823195 . Retrieved 20 August 2020. - ↑ "4D Maze Game". urticator.net. Retrieved 2016-12-16.
- ↑ Kaku, Michio (1995).
*Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension*(reissued ed.). Oxford: Oxford University Press. pp. Part I, Chapter 3. ISBN 978-0-19-286189-4. - ↑ Rucker, Rudy (1996).
*The Fourth Dimension: A Guided Tour of the Higher Universe*. Boston: Houghton Mifflin. p. 18. ISBN 978-0-395-39388-8.

- Archibald, R. C (1914). "Time as a Fourth Dimension" (PDF).
*Bulletin of the American Mathematical Society*: 409–412. - Andrew Forsyth (1930) Geometry of Four Dimensions, link from Internet Archive.
- Gamow, George (1988).
*One Two Three . . . Infinity: Facts and Speculations of Science*(3rd ed.). Courier Dover Publications. p. 68. ISBN 978-0-486-25664-1. Extract of page 68 - E. H. Neville (1921)
*The Fourth Dimension*, Cambridge University Press, link from University of Michigan Historical Math Collection.

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