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A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space.
Three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
is the simplest possible abstraction of the observation that one only needs three numbers, called ''
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s'', to describe the
size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume ...
s or
location In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
s of objects in the everyday world. For example, the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled ''x'', ''y'', and ''z''). The idea of adding a fourth dimension began with
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
's "Dimensions" being published in 1754, was followed by
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaBernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
. In 1880,
Charles Howard Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and ...
popularized these insights in an essay titled "
What is the Fourth Dimension? What or WHAT may refer to: * What, an interrogative pronoun and adverb * "What?", one of the Five Ws used in journalism Film and television * ''What!'' (film) or ''The Whip and the Body'', a 1963 Italian film directed by Mario Bava * '' What ...
", 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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 In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
'', i.e., as ordered lists of numbers such as . 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 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 eig ...
(equivalent to the 3D
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
; see also
hypercube 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, ...
).


History

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiamechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time. In 1827, Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image; by 1853,
Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
had discovered all the regular
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
s that exist in higher dimensions, including the four-dimensional analogues of the
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
, but his work was not published until after his death. Higher dimensions were soon put on firm footing by
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
's 1854
thesis A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
, ''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 dimension In physics and mathematics, the dimension of a mathematical space (or object) 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 coordina ...
s, including four dimensions in particular, was thus established. An arithmetic of four dimensions, called
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, was defined by
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
in 1843. This
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
was the source of the science of
vector analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
in three dimensions as recounted in ''
A History of Vector Analysis ''A History of Vector Analysis'' (1967) is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press. As a scholarly treatment of a reformation in technical communication, the text i ...
''. Soon after,
tessarine In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...
s and
coquaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in ...
s were introduced as other four-dimensional algebras over R. One of the first major expositors of the fourth dimension was
Charles Howard Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and ...
, starting in 1880 with his essay ''What is the Fourth Dimension?'', published in the
Dublin University The University of Dublin ( ga, Ollscoil Átha Cliath), corporately designated the Chancellor, Doctors and Masters of the University of Dublin, is a university located in Dublin, Ireland. It is the degree-awarding body for Trinity College Dubl ...
magazine. He coined the terms ''
tesseract 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 eig ...
'', ''ana'' and ''kata'' in his book ''
A New Era of Thought ''A New Era of Thought'' is a non-fiction work written by Charles Howard Hinton, published in 1888 and reprinted in 1900 by Swan Sonnenschein & Co. Ltd., London. ''A New Era of Thought'' is about the fourth dimension and its implications on human ...
'' and introduced a method for visualising the fourth dimension using cubes in the book ''Fourth Dimension''. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
in his January 1962 "
Mathematical Games column Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, ...
" in ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
''. In 1886,
Victor Schlegel Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams. In the nineteenth centur ...
described his method of visualizing four-dimensional objects with
Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the origina ...
s. In 1908,
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
presented a paper consolidating the role of time as the fourth dimension of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, the basis for Einstein's theories of
special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specia ...
and
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. But the geometry of spacetime, being
non-Euclidean In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, is profoundly different from that explored by Schläfli and popularised by Hinton. The study of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () 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 inerti ...
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 Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
felt compelled to write:


Vectors

Mathematically, four-dimensional space is a space with four spatial dimensions, that is a
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
that needs four parameters to specify a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
in it. For example, a general point might have position
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
a, equal to : \mathbf = \begin a_1 \\ a_2 \\ a_3 \\ a_4 \end. This can be written in terms of the four
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
vectors (e1, e2, e3, e4), given by :\mathbf_1 = \begin 1 \\ 0 \\ 0 \\ 0 \end; \mathbf_2 = \begin 0 \\ 1 \\ 0 \\ 0 \end; \mathbf_3 = \begin 0 \\ 0 \\ 1 \\ 0 \end; \mathbf_4 = \begin 0 \\ 0 \\ 0 \\ 1 \end, so the general vector a is : \mathbf = a_1\mathbf_1 + a_2\mathbf_2 + a_3\mathbf_3 + a_4\mathbf_4. Vectors add, subtract and scale as in three dimensions. The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of Euclidean three-dimensional space generalizes to four dimensions as : \mathbf \cdot \mathbf = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4. It can be used to calculate the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
or
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of a vector, : \left, \mathbf \ = \sqrt = \sqrt, and calculate or define the
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
between two non-zero vectors as : \theta = \arccos. Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate
pairing In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''-modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-modu ...
different from the dot product: : \mathbf \cdot \mathbf = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4. 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 b_4 actually decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity. The
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
is not defined in four dimensions. Instead the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...
is used for some applications, and is defined as follows: : \begin \mathbf \wedge \mathbf = (a_1b_2 - a_2b_1)\mathbf_ + (a_1b_3 - a_3b_1)\mathbf_ + (a_1b_4 - a_4b_1)\mathbf_ + (a_2b_3 - a_3b_2)\mathbf_ \\ + (a_2b_4 - a_4b_2)\mathbf_ + (a_3b_4 - a_4b_3)\mathbf_. \end This is
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.


Orthogonality and vocabulary

In the familiar three-dimensional space of daily life, there are three
coordinate axes A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
—usually labeled ''x'', ''y'', and ''z''—with each axis
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(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 Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and ...
coined the terms ''ana'' and ''kata'', from the Greek words meaning "up toward" and "down from", respectively. As mentioned above, Hermann Minkowski exploited the idea of four dimensions to discuss cosmology including the finite
velocity of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
. In appending a time dimension to three dimensional space, he specified an alternative perpendicularity,
hyperbolic orthogonality In geometry, the relation of hyperbolic orthogonality between two lines separated by the asymptotes of a hyperbola is a concept used in special relativity to define simultaneous events. Two events will be simultaneous when they are on a line hyper ...
. This notion provides his four-dimensional space with a modified
simultaneity Simultaneity may refer to: * Relativity of simultaneity, a concept in special relativity. * Simultaneity (music), more than one complete musical texture occurring at the same time, rather than in succession * Simultaneity, a concept in Endogeneit ...
appropriate to electromagnetic relations in his cosmos. Minkowski's world overcame problems associated with the traditional
absolute space and time Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a preferr ...
cosmology previously used in a universe of three space dimensions and one time dimension.


Geometry

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 In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
made of two dimensional
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s, in four dimensions there are
4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
s made of polyhedra. In three dimensions, there are 5 regular polyhedra known as the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s. In four dimensions, there are 6
convex regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star regu ...
s, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex
uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...
s, analogous to the 13 semi-regular
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s in three dimensions. Relaxing the conditions for convexity generates a further 10 nonconvex regular 4-polytopes. In three dimensions, a circle may be
extrude Extrusion is a process used to create objects of a fixed cross-sectional profile by pushing material through a die of the desired cross-section. Its two main advantages over other manufacturing processes are its ability to create very complex ...
d to form a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
. 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 In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere) of radius ''r''1 and a line segment of length 2''r''2: :D = \ Like th ...
), and a cylinder may be extruded to obtain a cylindrical prism (a cubinder). The
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of two circles may be taken to obtain a
duocylinder The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii ''r''1 and ''r''2: :D = \left\ It is analogous ...
. All three can "roll" in four-dimensional space, each with its own properties. In three dimensions, curves can form
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s 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. Because these surfaces are two-dimensional, they can form much more complex knots than strings in 3D space can. The
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
is an example of such a knotted surface. Another such surface is the
real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
.


Hypersphere

The set of points in Euclidean 4-space having the same distance R from a fixed point P0 forms a
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
known as a
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
. The hyper-volume of the enclosed space is: : \mathbf V = \begin \frac \end \pi^2 R^4 This is part of the
Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
in
General relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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.


Cognition

Research using
virtual reality Virtual reality (VR) is a simulated experience that employs pose tracking and 3D near-eye displays to give the user an immersive feel of a virtual world. Applications of virtual reality include entertainment (particularly video games), educ ...
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. 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". 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
labyrinth In Greek mythology, the Labyrinth (, ) was an elaborate, confusing structure designed and built by the legendary artificer Daedalus for King Minos of Crete at Knossos. Its function was to hold the Minotaur, the monster eventually killed by the ...
s). 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).


Dimensional analogy

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. Dimensional analogy was used by
Edwin Abbott Abbott Edwin Abbott Abbott (20 December 1838 – 12 October 1926) was an English schoolmaster, theologian, and Anglican priest, best known as the author of the novella ''Flatland'' (1884). Biography Edwin Abbott Abbott was the eldest son of ...
in the book ''
Flatland ''Flatland: A Romance of Many Dimensions'' is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dime ...
'', 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 Rudolf von Bitter Rucker (; born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known f ...
illustrates this in his novel ''
Spaceland Spaceland was an alternative rock/indie rock nightclub in the Silver Lake neighborhood of Los Angeles, California, that existed between 1995 and 2011. The club was formerly a popular disco to young locals called Dreams of LA. Spaceland's owner an ...
'', in which the protagonist encounters four-dimensional beings who demonstrate such powers.


Cross-sections

As a three-dimensional object passes through a two-dimensional plane, two-dimensional beings in this plane would only observe a
cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Ab ...
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 sphere, and then getting smaller again, until it shrank to a point and then disappearing. The 2D beings would not see a circle in the same way as three-dimensional beings do; rather, they only see a
one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
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
hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
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 ''Flatland'' and also in several works of
Charles Howard Hinton Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction Science fiction (sometimes shortened to Sci-Fi or SF) is a genre of speculative fiction which typically deals with imaginative and ...
. 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.


Projections

A useful application of dimensional analogy in visualizing higher dimensions is in
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
. A projection is a way for representing an ''n''-dimensional object in 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 The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then ...
of the
eye Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo response functions that are independent of vision. Eyes detect light and conv ...
is also a two-dimensional
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
of
receptor Receptor may refer to: * Sensory receptor, in physiology, any structure which, on receiving environmental stimuli, produces an informative nerve impulse *Receptor (biochemistry), in biochemistry, a protein molecule that receives and responds to a ...
s but the
brain A brain is an organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It is located in the head, usually close to the sensory organs for senses such as vision. It is the most complex organ in a v ...
is able to perceive the nature of three-dimensional objects by inference from indirect information (such as shading,
foreshortening Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
,
binocular vision In biology, binocular vision is a type of vision in which an animal has two eyes capable of facing the same direction to perceive a single three-dimensional image of its surroundings. Binocular vision does not typically refer to vision where an ...
, etc.).
Artist An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating an art. The common usage in both everyday speech and academic discourse refers to a practitioner in the visual arts only. However, th ...
s 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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
with analogous projections of the four-dimensional tesseract into three-dimensional space.


Shadows

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.)


Bounding volumes

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 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 eig ...
, 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.


Visual scope

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 The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then ...
. 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. 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.


Limitations

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 (A = \pi r^2) and the volume of a sphere (V = \frac\pi r^3). One might guess that the volume of the 3-sphere in four-dimensional space is V=6\pi r^3, or perhaps V=8\pi r^3, but neither of these is right. The actual formula is V=2\pi^2 r^3.


See also

*
4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
*
4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. T ...
* Exotic R4 *
Four-dimensionalism In philosophy, four-dimensionalism (also known as the doctrine of temporal parts) is the ontological position that an object's persistence through time is like its extension through space. Thus, an object that exists in time has temporal parts i ...
*
Fourth dimension in art 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 Cubists, Surrealists, Futurists, and ...
*
Fourth dimension in literature The idea of a fourth dimension has been a factor in the evolution of modern art, but use of concepts relating to higher dimensions has been little discussed by academics in the literary world. From the late 19th century onwards, many writers began ...
*
List of four-dimensional games This is a list of four-dimensional games—specifically, a list of video games that attempt to represent four-dimensional space. Games , flight simulator / variety , Medenacci Games , 2019 (demo) , , C++ , Perspective projection, point ...
*
Eugene the Jeep Eugene the Jeep is a character in the ''Popeye'' comic strip. A mysterious animal with magical or supernatural abilities, the Jeep first appeared in the March 16, 1936 ''Thimble Theatre'' comic strip (now simply ''Popeye''). He was also prese ...
*
Time in physics Time in physics is defined by its measurement: time is what a clock reads. In classical, non-relativistic physics, it is a scalar quantity (often denoted by the symbol t) and, like length, mass, and charge, is usually described as a fundamenta ...
*
Spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...


References


Further reading

* *
Andrew Forsyth Andrew Russell Forsyth, FRS, FRSE (18 June 1858, Glasgow – 2 June 1942, South Kensington) was a British mathematician. Life Forsyth was born in Glasgow on 18 June 1858, the son of John Forsyth, a marine engineer, and his wife Christina ...
(1930
Geometry of Four Dimensions
link from
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
. *
Extract of page 68
*
E. H. Neville Eric Harold Neville, known as E. H. Neville (1 January 1889 London, England – 22 August 1961 Reading, Berkshire, England) was an English mathematician. A heavily fictionalised portrayal of his life is rendered in the 2007 novel ''The Indian ...
(1921
''The Fourth Dimension''
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, link from
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
Historical Math Collection.


External links


"Dimensions" videos, showing several different ways to visualize four dimensional objects''Science News'' article summarizing the "Dimensions" videos, with clips
* ''Flatland: a Romance of Many Dimensions'' (second edition)
Frame-by-frame animations of 4D - 3D analogies
{{DEFAULTSORT:Fourth Dimension Dimension Multi-dimensional geometry Special relativity 4 (number)