Weak interaction

From Wikipedia, the free encyclopedia

 

The radioactive beta decay is due to the weak interaction, which transforms a neutron into: a proton, an electron, and an electron antineutrino.

In particle physics, the weak interaction (the weak force or weak nuclear force) is the mechanism of interaction between sub-atomic particles that causes radioactive decay and thus plays an essential role in nuclear fission. The theory of the weak interaction is sometimes called quantum flavordynamics (QFD), in analogy with the terms quantum chromodynamics (QCD) dealing with the strong interaction and quantum electrodynamics (QED) dealing with the electromagnetic force. However, the term QFD is rarely used because the weak force is best understood in terms of electro-weak theory (EWT).^[1]

The weak interaction takes place only at very small, sub-atomic distances, less than the diameter of a proton. It is one of the four known fundamental interactions of nature, alongside the strong interaction, electromagnetism, and gravitation.

Background

The Standard Model of particle physics provides a uniform framework for understanding the electromagnetic, weak, and strong interactions. An interaction occurs when two particles, typically but not necessarily half-integer spin fermions, exchange integer-spin, force-carrying bosons. The fermions involved in such exchanges can be either elementary (e.g. electrons or quarks) or composite (e.g. protons or neutrons), although at the deepest levels, all weak interactions ultimately are between elementary particles.

In the case of the weak interaction, fermions can exchange three distinct types of force carriers known as the W⁺, W⁻, and Z bosons. The mass of each of these bosons is far greater than the mass of a proton or neutron, which is consistent with the short range of the weak force. The force is in fact termed weak because its field strength over a given distance is typically several orders of magnitude less than that of the strong nuclear force or electromagnetic force.

Quarks, which make up composite particles like neutrons and protons, come in six "flavors" – up, down, strange, charm, top and bottom – which give those composite particles their properties. The weak interaction is unique in that it allows for quarks to swap their flavor for another. The swapping of those properties is mediated by the force carrier bosons. For example, during beta minus decay, a down quark within a neutron is changed into an up quark, thus converting the neutron to a proton and resulting in the emission of an electron and an electron antineutrino.

The weak interaction is the only fundamental interaction that breaks parity-symmetry, and similarly, the only one to break charge parity symmetry.

Other important examples of phenomena involving the weak interaction include beta decay, and the fusion of hydrogen into deuterium that powers the Sun's thermonuclear process. Most fermions will decay by a weak interaction over time. Such decay makes radiocarbon dating possible, as carbon-14 decays through the weak interaction to nitrogen-14. It can also create radioluminescence, commonly used in tritium illumination, and in the related field of betavoltaics.^[2]

During the quark epoch of the early universe, the electroweak force separated into the electromagnetic and weak forces.

History

In 1933, Enrico Fermi proposed the first theory of the weak interaction, known as Fermi's interaction. He suggested that beta decay could be explained by a four-fermion interaction, involving a contact force with no range.^[3][4]

However, it is better described as a non-contact force field having a finite range, albeit very short.^{[citation needed]} In 1968, Sheldon Glashow, Abdus Salam and Steven Weinberg unified the electromagnetic force and the weak interaction by showing them to be two aspects of a single force, now termed the electro-weak force.^{[citation needed]}

The existence of the W and Z bosons was not directly confirmed until 1983.^[5]

Properties

A diagram depicting the various decay routes due to the weak interaction and some indication of their likelihood. The intensity of the lines is given by the CKM parameters.

The weak interaction is unique in a number of respects:

• It is the only interaction capable of changing the flavor of quarks (i.e., of changing one type of quark into another).
• It is the only interaction that violates P or parity-symmetry. It is also the only one that violates charge-parity CP symmetry.
• It is mediated (propagated) by force carrier particles that have significant masses, an unusual feature which is explained in the Standard Model by the Higgs mechanism.

Due to their large mass (approximately 90 GeV/c^2[6]) these carrier particles, termed the W and Z bosons, are short-lived with a lifetime of under 10⁻²⁴ seconds.^[7] The weak interaction has a coupling constant (an indicator of interaction strength) of between 10⁻⁷ and 10⁻⁶, compared to the strong interaction's coupling constant of 1 and the electromagnetic coupling constant of about 10⁻²;^[8] consequently the weak interaction is weak in terms of strength.^[9] The weak interaction has a very short range (around 10⁻¹⁷ to 10⁻¹⁶ m^[9]).^[8] At distances around 10⁻¹⁸ meters, the weak interaction has a strength of a similar magnitude to the electromagnetic force, but this starts to decrease exponentially with increasing distance. At distances of around 3×10⁻¹⁷ m, a distance which is scaled up by just one and a half decimal orders of magnitude from before, the weak interaction is 10,000 times weaker than the electromagnetic.^[10]

The weak interaction affects all the fermions of the Standard Model, as well as the Higgs boson; neutrinos interact through gravity and the weak interaction only, and neutrinos were the original reason for the name weak force.^[9] The weak interaction does not produce bound states nor does it involve binding energy – something that gravity does on an astronomical scale, that the electromagnetic force does at the atomic level, and that the strong nuclear force does inside nuclei.^[11]

Its most noticeable effect is due to its first unique feature: flavor changing. A neutron, for example, is heavier than a proton (its sister nucleon), but it cannot decay into a proton without changing the flavor (type) of one of its two down quarks to an up quark. Neither the strong interaction nor electromagnetism permit flavor changing, so this proceeds by weak decay; without weak decay, quark properties such as strangeness and charm (associated with the quarks of the same name) would also be conserved across all interactions.

All mesons are unstable because of weak decay.^[12] In the process known as beta decay, a down quark in the neutron can change into an up quark by emitting a virtual
W⁻ boson which is then converted into an electron and an electron antineutrino.^[13] Another example is the electron capture, a common variant of radioactive decay, wherein a proton and an electron within an atom interact, and are changed to a neutron (an up quark is changed to a down quark) and an electron neutrino is emitted.

Due to the large masses of the W bosons, particle transformations or decays (e.g., flavor change) that depend on the weak interaction typically occur much more slowly than transformations or decays that depend only on the strong or electromagnetic forces. For example, a neutral pion decays electromagnetically, and so has a life of only about 10⁻¹⁶ seconds. In contrast, a charged pion can only decay through the weak interaction, and so lives about 10⁻⁸ seconds, or a hundred million times longer than a neutral pion.^[14] A particularly extreme example is the weak-force decay of a free neutron, which takes about 15 minutes.^[13]

Weak isospin and weak hypercharge

All particles have a property called weak isospin (symbol T₃), which serves as a quantum number and governs how that particle behaves in the weak interaction. Weak isospin plays the same role in the weak interaction as does electric charge in electromagnetism, and color charge in the strong interaction. All left-handed fermions have a weak isospin value of either +​¹⁄₂ or −​¹⁄₂. For example, the up quark has a T₃ of +​¹⁄₂ and the down quark −​¹⁄₂. A quark never decays through the weak interaction into a quark of the same T₃: Quarks with a T₃ of +​¹⁄₂ only decay into quarks with a T₃ of −​¹⁄₂ and vice versa.

π⁺ decay through the weak interaction

In any given interaction, weak isospin is conserved: the sum of the weak isospin numbers of the particles entering the interaction equals the sum of the weak isospin numbers of the particles exiting that interaction. For example, a (left-handed)
π⁺, with a weak isospin of 1 normally decays into a
ν
_μ (+​¹⁄₂) and a
μ⁺ (as a right-handed antiparticle, +​¹⁄₂).^[14]

Following the development of the electroweak theory, another property, weak hypercharge, was developed. It is dependent on a particle's electrical charge and weak isospin, and is defined by:

${\displaystyle \qquad Y_{W}=2(Q-T_{3})}$

where Y_W is the weak hypercharge of a given type of particle, Q is its electrical charge (in elementary charge units) and T₃ is its weak isospin. Whereas some particles have a weak isospin of zero, all spin-​¹⁄₂ particles have non-zero weak hypercharge. Weak hypercharge is the generator of the U(1) component of the electroweak gauge group.

Interaction types

There are two types of weak interaction (called vertices). The first type is called the "charged-current interaction" because it is mediated by particles that carry an electric charge (the
W⁺ or
W⁻ bosons), and is responsible for the beta decay phenomenon. The second type is called the "neutral-current interaction" because it is mediated by a neutral particle, the Z boson.

Charged-current interaction

The Feynman diagram for beta-minus decay of a neutron into a proton, electron and electron anti-neutrino, via an intermediate heavy
W⁻ boson

In one type of charged current interaction, a charged lepton (such as an electron or a muon, having a charge of −1) can absorb a
W⁺ boson (a particle with a charge of +1) and be thereby converted into a corresponding neutrino (with a charge of 0), where the type ("flavor") of neutrino (electron, muon or tau) is the same as the type of lepton in the interaction, for example:

${\displaystyle \mu ^{-}+W^{+}\to \nu _{\mu }}$

Similarly, a down-type quark (d with a charge of −​¹⁄₃) can be converted into an up-type quark (u, with a charge of +​²⁄₃), by emitting a
W⁻ boson or by absorbing a
W⁺ boson. More precisely, the down-type quark becomes a quantum superposition of up-type quarks: that is to say, it has a possibility of becoming any one of the three up-type quarks, with the probabilities given in the CKM matrix tables. Conversely, an up-type quark can emit a
W⁺ boson, or absorb a
W⁻ boson, and thereby be converted into a down-type quark, for example:

${\displaystyle {\begin{aligned}d&\to u+W^{-}\\d+W^{+}&\to u\\c&\to s+W^{+}\\c+W^{-}&\to s\end{aligned}}}$

The W boson is unstable so will rapidly decay, with a very short lifetime. For example:

${\displaystyle {\begin{aligned}W^{-}&\to e^{-}+{\bar {\nu }}_{e}~\\W^{+}&\to e^{+}+\nu _{e}~\end{aligned}}}$

Decay of the W boson to other products can happen, with varying probabilities.^[16]

In the so-called beta decay of a neutron (see picture, above), a down quark within the neutron emits a virtual
W⁻ boson and is thereby converted into an up quark, converting the neutron into a proton. Because of the energy involved in the process (i.e., the mass difference between the down quark and the up quark), the
W⁻ boson can only be converted into an electron and an electron-antineutrino.^[17] At the quark level, the process can be represented as:

${\displaystyle d\to u+e^{-}+{\bar {\nu }}_{e}~}$

Neutral-current interaction

In neutral current interactions, a quark or a lepton (e.g., an electron or a muon) emits or absorbs a neutral Z boson. For example:

${\displaystyle e^{-}\to e^{-}+Z^{0}}$

Like the W boson, the Z boson also decays rapidly,^[16] for example:

${\displaystyle Z^{0}\to b+{\bar {b}}}$

Electroweak theory

The Standard Model of particle physics describes the electromagnetic interaction and the weak interaction as two different aspects of a single electroweak interaction. This theory was developed around 1968 by Sheldon Glashow, Abdus Salam and Steven Weinberg, and they were awarded the 1979 Nobel Prize in Physics for their work.^[18] The Higgs mechanism provides an explanation for the presence of three massive gauge bosons (W⁺,W⁻,
Z⁰, the three carriers of the weak interaction) and the massless photon (γ, the carrier of the electromagnetic interaction).^[19]

According to the electroweak theory, at very high energies, the universe has four components of the Higgs field whose interactions are carried by four massless gauge bosons – each similar to the photon – forming a complex scalar Higgs field doublet. However, at low energies, this gauge symmetry is spontaneously broken down to the U(1) symmetry of electromagnetism, since one of the Higgs fields acquires a vacuum expectation value. This symmetry-breaking would be expected to produce three massless bosons, but instead they become integrated by the other three fields and acquire mass through the Higgs mechanism. These three boson integrations produce the
W⁺,
W⁻ and
Z⁰ bosons of the weak interaction. The fourth gauge boson is the photon of electromagnetism, and remains massless.^[19]

This theory has made a number of predictions, including a prediction of the masses of the Z and W-bosons before their discovery. On 4 July 2012, the CMS and the ATLAS experimental teams at the Large Hadron Collider independently announced that they had confirmed the formal discovery of a previously unknown boson of mass between 125–127 GeV/c², whose behaviour so far was "consistent with" a Higgs boson, while adding a cautious note that further data and analysis were needed before positively identifying the new boson as being a Higgs boson of some type. By 14 March 2013, the Higgs boson was tentatively confirmed to exist.^[20]

Violation of symmetry

Left- and right-handed particles: p is the particle's momentum and S is its spin. Note the lack of reflective symmetry between the states.

The laws of nature were long thought to remain the same under mirror reflection. The results of an experiment viewed via a mirror were expected to be identical to the results of a mirror-reflected copy of the experimental apparatus. This so-called law of parity conservation was known to be respected by classical gravitation, electromagnetism and the strong interaction; it was assumed to be a universal law.^[21] However, in the mid-1950s Chen-Ning Yang and Tsung-Dao Lee suggested that the weak interaction might violate this law. Chien Shiung Wu and collaborators in 1957 discovered that the weak interaction violates parity, earning Yang and Lee the 1957 Nobel Prize in Physics.^[22]

Although the weak interaction was once described by Fermi's theory, the discovery of parity violation and renormalization theory suggested that a new approach was needed. In 1957, Robert Marshak and George Sudarshan and, somewhat later, Richard Feynman and Murray Gell-Mann proposed a V−A (vector minus axial vector or left-handed) Lagrangian for weak interactions. In this theory, the weak interaction acts only on left-handed particles (and right-handed antiparticles). Since the mirror reflection of a left-handed particle is right-handed, this explains the maximal violation of parity. Interestingly, the V−A theory was developed before the discovery of the Z boson, so it did not include the right-handed fields that enter in the neutral current interaction.

However, this theory allowed a compound symmetry CP to be conserved. CP combines parity P (switching left to right) with charge conjugation C (switching particles with antiparticles). Physicists were again surprised when in 1964, James Cronin and Val Fitch provided clear evidence in kaon decays that CP symmetry could be broken too, winning them the 1980 Nobel Prize in Physics.^[23] In 1973, Makoto Kobayashi and Toshihide Maskawa showed that CP violation in the weak interaction required more than two generations of particles,^[24] effectively predicting the existence of a then unknown third generation. This discovery earned them half of the 2008 Nobel Prize in Physics.^[25] Unlike parity violation, CP violation occurs in only a small number of instances, but remains widely held as an answer to the difference between the amount of matter and antimatter in the universe; it thus forms one of Andrei Sakharov's three conditions for baryogenesis.^[26]

Fourth dimension in art

From Wikipedia, the free encyclopedia

An illustration from Jouffret's Traité élémentaire de géométrie à quatre dimensions. The book, which influenced Picasso, was given to him by Princet.

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 abstract artists took ideas from higher-dimensional mathematics and used them to radically advance their work.^[1]

Early influence

Picasso's Portrait of Daniel-Henry Kahnweiler, 1910, The Art Institute of Chicago

 

Jean Metzinger, 1912-1913, L'Oiseau bleu, (The Blue Bird), oil on canvas, 230 x 196 cm, Musée d'Art Moderne de la Ville de Paris

French mathematician Maurice Princet was known as "le mathématicien du cubisme" ("the mathematician of cubism").^[2] An associate of the School of Paris, a group of avant-gardists including Pablo Picasso, Guillaume Apollinaire, Max Jacob, Jean Metzinger, and Marcel Duchamp, Princet is credited with introducing the work of Henri Poincaré and the concept of the "fourth dimension" to the cubists at the Bateau-Lavoir during the first decade of the 20th century.^[3]

Princet introduced Picasso to Esprit Jouffret's Traité élémentaire de géométrie à quatre dimensions (Elementary Treatise on the Geometry of Four Dimensions, 1903),^[4] a popularization of Poincaré's Science and Hypothesis in which Jouffret described hypercubes and other complex polyhedra in four dimensions and projected them onto the two-dimensional page. Picasso's Portrait of Daniel-Henry Kahnweiler in 1910 was an important work for the artist, who spent many months shaping it.^[5] The portrait bears similarities to Jouffret's work and shows a distinct movement away from the Proto-Cubist fauvism displayed in Les Demoiselles d'Avignon, to a more considered analysis of space and form.^[6]

Early cubist Max Weber wrote an article entitled "In The Fourth Dimension from a Plastic Point of View", for Alfred Stieglitz's July 1910 issue of Camera Work. In the piece, Weber states, "In plastic art, I believe, there is a fourth dimension which may be described as the consciousness of a great and overwhelming sense of space-magnitude in all directions at one time, and is brought into existence through the three known measurements."^[7]

Another influence on the School of Paris was that of Jean Metzinger and Albert Gleizes, both painters and theoreticians. The first major treatise written on the subject of Cubism was their 1912 collaboration Du "Cubisme", which says that:

"If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidian mathematicians; we should have to study, at some length, certain of Riemann's theorems."^[8]

The American modernist painter and photographer Morton Livingston Schamberg wrote in 1910 two letters to Walter Pach,^[9][10] parts of which were published in a review of the 1913 Armory Show for The Philadelphia Inquirer,^[11] about the influence of the fourth dimension on avant-garde painting; describing how the artists' employed "harmonic use of forms" distinguishing between the "representation or rendering of space and the designing in space":^[12]

If we still further add to design in the third dimension, a consideration of weight, pressure, resistance, movement, as distinguished from motion, we arrive at what may legitimately be called design in the fourth dimension, or the harmonic use of what may arbitrarily be called volume. It is only at this point that we can appreciate the masterly productions of such a man as Cézanne.^[13]

Cézanne's explorations of geometric simplification and optical phenomena inspired the Cubists to experiment with simultaneity, complex multiple views of the same subject, as observed from differing viewpoints at the same time.^[14]

Dimensionist manifesto

In 1936 in Paris, Charles Tamkó Sirató published his Manifeste Dimensioniste,^[15] which described how

the Dimensionist tendency has led to:

1. Literature leaving the line and entering the plane.
2. Painting leaving the plane and entering space.
3. Sculpture stepping out of closed, immobile forms.
4. …The artistic conquest of four-dimensional space, which to date has been completely art-free.

The manifesto was signed by many prominent modern artists worldwide. Hans Arp, Francis Picabia, Kandinsky, Robert Delaunay and Marcel Duchamp amongst others added their names in Paris, then a short while later it was endorsed by artists abroad including László Moholy-Nagy, Joan Miró, David Kakabadze, Alexander Calder, and Ben Nicholson.^[15]

Crucifixion (Corpus Hypercubus)

Dalí's 1954 painting Crucifixion (Corpus Hypercubus)

In 1953, the surrealist Salvador Dalí proclaimed his intention to paint "an explosive, nuclear and hypercubic" crucifixion scene.^[16][17] He said that, "This picture will be the great metaphysical work of my summer".^[18] Completed the next year, Crucifixion (Corpus Hypercubus) depicts Jesus Christ upon the net of a hypercube, also known as a tesseract. The unfolding of a tesseract into eight cubes is analogous to unfolding the sides of a cube into six squares. The Metropolitan Museum of Art describes the painting as a "new interpretation of an oft-depicted subject. ..[showing] Christ's spiritual triumph over corporeal harm."^[19]

Abstract art

Some of Piet Mondrian's (1872–1944) abstractions and his practice of Neoplasticism are said to be rooted in his view of a utopian universe, with perpendiculars visually extending into another dimension.^[20]

Other forms of art

The fourth dimension has been the subject of numerous fictional stories.^[21]

Four-dimensional space

From Wikipedia, the free encyclopedia

The 4D equivalent of a cube, known as a tesseract. The tesseract is rotating in four dimensions, which are then projected into three dimensions.

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible generalization 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 its length (often labeled x), width (y), and depth (z).

The idea of adding a fourth dimension began with 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 cubes 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 that case represent a single direction in the "unseen" fourth dimension.

Higher dimensional spaces 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 a bit more complicated than Euclidean 4D space.

When dimensional locations are given as ordered lists of numbers such as (t,x,y,z) they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D and higher spaces emerges. A hint of that complexity can be seen in the accompanying animation of one of simplest possible 4D objects, the 4D cube or tesseract.

History

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.^[1] In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image,^[2]:141 and by 1853 Ludwig Schläfli had discovered many polytopes in higher dimensions, although his work was not published until after his death.^{[2]: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 (x₁, ..., 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.^[3] 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.^[4][5]

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^[6] his method of visualizing four-dimensional objects with Schlegel diagrams.

In 1908, Hermann Minkowski presented a paper^[7] consolidating the role of time as the fourth dimension of spacetime, the basis for Einstein's theories of special and general relativity.^[8] But the geometry of spacetime, being non-Euclidean, is profoundly different from that 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 in The 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 is not Euclidean, and consequently has no connection with the present investigation.

— H. S. M. Coxeter, Regular Polytopes^[2]:119

Vectors

Mathematically four-dimensional space is simply 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

${\displaystyle \mathbf {a} ={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{pmatrix}}.}$

This can be written in terms of the four standard basis vectors (e₁, e₂, e₃, e₄), given by

${\displaystyle \mathbf {e} _{1}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\mathbf {e} _{2}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\mathbf {e} _{3}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\mathbf {e} _{4}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},}$

so the general vector a is

${\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}+a_{4}\mathbf {e} _{4}.}$

Vectors add, subtract and scale as in three dimensions.

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

${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}.}$

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

${\displaystyle \left|\mathbf {a} \right|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}},}$

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

${\displaystyle \theta =\arccos {\frac {\mathbf {a} \cdot \mathbf {b} }{\left|\mathbf {a} \right|\left|\mathbf {b} \right|}}.}$

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

${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}-a_{4}b_{4}.}$

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 ${\displaystyle b_{4}}$ 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:

${\displaystyle {\begin{aligned}\mathbf {a} \wedge \mathbf {b} =(a_{1}b_{2}-a_{2}b_{1})\mathbf {e} _{12}+(a_{1}b_{3}-a_{3}b_{1})\mathbf {e} _{13}+(a_{1}b_{4}-a_{4}b_{1})\mathbf {e} _{14}+(a_{2}b_{3}-a_{3}b_{2})\mathbf {e} _{23}\\+(a_{2}b_{4}-a_{4}b_{2})\mathbf {e} _{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}}$

This is bivector 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 in which we live 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. A position along the w axis can be called spissitude, as coined by Henry More.

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 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.

Regular polytopes in four dimensions
(Displayed as orthogonal projections in each Coxeter plane of symmetry)

A4, [3,3,3]	B4, [4,3,3]		F4, [3,4,3]
5-cell {3,3,3}	tesseract {4,3,3}	16-cell {3,3,4}	24-cell {3,4,3}

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). 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.^{[9][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]}

Hypersphere

Stereographic projection of a Clifford torus: the set of points (cos(a), sin(a), cos(b), sin(b)), which is a subset of the 3-sphere.

The set of points in Euclidean 4-space having the same distance R from a fixed point P₀ forms a hypersurface known as a 3-sphere. The hyper-volume of the enclosed space is:

${\displaystyle \mathbf {V} ={\begin{matrix}{\frac {1}{2}}\end{matrix}}\pi ^{2}R^{4}}$

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.^[10]

Cognition

Research using virtual reality finds that humans, in spite of living in a three-dimensional world, can, without special practice, make spatial judgments based on the length of, and angle between, line segments embedded in four-dimensional space.^[11] 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."^[11] In another study,^[12] 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.^[13] 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

A net of a tesseract

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.^[14]

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 our three-dimensional perspective. Rudy Rucker illustrates this in his novel Spaceland, 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, a two-dimensional being would only see a cross-section of the three-dimensional object. For example, if a spherical balloon passed through a sheet of paper, a being on the paper would see first a single point, then a circle gradually growing larger, then smaller again until it shrank to a point and then disappeared. Similarly, if a four-dimensional object passed through three dimensions, we would see a three-dimensional cross-section of the four-dimensional object—for example, a hypersphere would appear first as a point, then as a growing sphere, with the sphere then shrinking to a single point and then disappearing.^[15] This means of visualizing aspects of the fourth dimension was used in the novel Flatland and also in several works of Charles Howard Hinton.^[4]:11–14

Projections

A useful application of dimensional analogy in visualizing the fourth dimension 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. When this is done, 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.

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 obscured by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
		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 behind these three faces, on the opposite side of the cube. Similarly, only 4 of the tesseract's 8 cells can be seen here; the remaining 4 lie behind these 4 in the fourth direction, on the far side of the tesseract.

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 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. (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, 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

Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a square on a piece of paper. It would be able to see all points in 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint. Our brains receive images in two dimensions and use reasoning to help us "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 circumference of a circle ${\displaystyle C=2\pi r}$ and the surface area of a sphere: ${\displaystyle A=4\pi r^{2}}$. One might be tempted to suppose that the surface volume of a hypersphere is ${\displaystyle V=6\pi r^{3}}$, or perhaps ${\displaystyle V=8\pi r^{3}}$, but either of these would be wrong. The correct formula is ${\displaystyle V=2\pi ^{2}r^{3}}$.^[2]:119