Lorenz system

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lorenz_system 

A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 0

The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the "butterfly effect" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.

Overview

In 1963, Edward Lorenz, with the help of Ellen Fetter who was responsible for the numerical simulations and figures, and Margaret Hamilton who helped in the initial, numerical computations leading up to the findings of the Lorenz model, developed a simplified mathematical model for atmospheric convection. The model is a system of three ordinary differential equations now known as the Lorenz equations:

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\[6pt]{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\[6pt]{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z.\end{aligned}}}$

The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: ${\displaystyle x}$ is proportional to the rate of convection, ${\displaystyle y}$ to the horizontal temperature variation, and ${\displaystyle z}$ to the vertical temperature variation. The constants ${\displaystyle \sigma }$, ${\displaystyle \rho }$, and ${\displaystyle \beta }$ are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself.

The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis. The Lorenz equations are also the governing equations in Fourier space for the Malkus waterwheel. The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.

From a technical standpoint, the Lorenz system is nonlinear, non-periodic, three-dimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.

Analysis

One normally assumes that the parameters ${\displaystyle \sigma }$, ${\displaystyle \rho }$, and ${\displaystyle \beta }$ are positive. Lorenz used the values ${\displaystyle \sigma =10}$, ${\displaystyle \beta =8/3}$ and ${\displaystyle \rho =28}$. The system exhibits chaotic behavior for these (and nearby) values.

If ${\displaystyle \rho <1}$ then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin, which is a global attractor, when ${\displaystyle \rho <1}$.

A pitchfork bifurcation occurs at ${\displaystyle \rho =1}$, and for ${\displaystyle \rho >1}$ two additional critical points appear at: ${\displaystyle \left({\sqrt {\beta (\rho -1)}},{\sqrt {\beta (\rho -1)}},\rho -1\right)}$ and ${\displaystyle \left(-{\sqrt {\beta (\rho -1)}},-{\sqrt {\beta (\rho -1)}},\rho -1\right).}$ These correspond to steady convection. This pair of equilibrium points is stable only if

${\displaystyle \rho <\sigma {\frac {\sigma +\beta +3}{\sigma -\beta -1}},}$

which can hold only for positive ${\displaystyle \rho }$ if ${\displaystyle \sigma >\beta +1}$. At the critical value, both equilibrium points lose stability through a subcritical Hopf bifurcation.

When ${\displaystyle \rho =28}$, ${\displaystyle \sigma =10}$, and ${\displaystyle \beta =8/3}$, the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set – the Lorenz attractor – a strange attractor, a fractal, and a self-excited attractor with respect to all three equilibria. Its Hausdorff dimension is estimated from above by the Lyapunov dimension (Kaplan-Yorke dimension) as 2.06 ± 0.01, and the correlation dimension is estimated to be 2.05 ± 0.01. The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:

 ${\displaystyle 3-{\frac {2(\sigma +\beta +1)}{\sigma +1+{\sqrt {(\sigma -1)^{2}+4\sigma \rho }}}}.}$

The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model. Proving that this is indeed the case is the fourteenth problem on the list of Smale's problems. This problem was the first one to be resolved, by Warwick Tucker in 2002.

For other values of ${\displaystyle \rho }$, the system displays knotted periodic orbits. For example, with ${\displaystyle \rho =99.96}$ it becomes a T(3,2) torus knot.

Example solutions of the Lorenz system for different values of ρ
ρ = 14, σ = 10, β = 8/3 (Enlarge)	ρ = 13, σ = 10, β = 8/3 (Enlarge)
ρ = 15, σ = 10, β = 8/3 (Enlarge)	ρ = 28, σ = 10, β = 8/3 (Enlarge)
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.

Sensitive dependence on the initial condition
Time t = 1 (Enlarge)	Time t = 2 (Enlarge)	Time t = 3 (Enlarge)
These figures — made using ρ = 28, σ = 10 and β = 8/3 — show three time segments of the 3-D evolution of two trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10−5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.

Connection to tent map

A recreation of Lorenz's results created on Mathematica. Points above the red line correspond to the system switching lobes.

In Figure 4 of his paper, Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a Poincaré plot, which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the tent map. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.

Simulations

MATLAB simulation

% Solve over time interval [0,100] with initial conditions [1,1,1]
% ''f'' is set of differential equations
% ''a'' is array containing x, y, and z variables
% ''t'' is time variable

sigma = 10;
beta = 8/3;
rho = 28;
f = @(t,a) [-sigma*a(1) + sigma*a(2); rho*a(1) - a(2) - a(1)*a(3); -beta*a(3) + a(1)*a(2)];
[t,a] = ode45(f,[0 100],[1 1 1]);     % Runge-Kutta 4th/5th order ODE solver
plot3(a(:,1),a(:,2),a(:,3))

Mathematica simulation

Standard way:

tend = 50;
eq = {x'[t] == σ (y[t] - x[t]), 
      y'[t] == x[t] (ρ - z[t]) - y[t], 
      z'[t] == x[t] y[t] - β z[t]};
init = {x[0] == 10, y[0] == 10, z[0] == 10};
pars = {σ->10, ρ->28, β->8/3};
{xs, ys, zs} = 
  NDSolveValue[{eq /. pars, init}, {x, y, z}, {t, 0, tend}];
ParametricPlot3D[{xs[t], ys[t], zs[t]}, {t, 0, tend}]

Less verbose:

lorenz = NonlinearStateSpaceModel[{{σ (y - x), x (ρ - z) - y, x y - β z}, {}}, {x, y, z}, {σ, ρ, β}];
soln[t_] = StateResponse[{lorenz, {10, 10, 10}}, {10, 28, 8/3}, {t, 0, 50}];
ParametricPlot3D[soln[t], {t, 0, 50}]

Python simulation

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D

rho = 28.0
sigma = 10.0
beta = 8.0 / 3.0

def f(state, t):
    x, y, z = state  # Unpack the state vector
    return sigma * (y - x), x * (rho - z) - y, x * y - beta * z  # Derivatives

state0 = [1.0, 1.0, 1.0]
t = np.arange(0.0, 40.0, 0.01)

states = odeint(f, state0, t)

fig = plt.figure()
ax = fig.gca(projection="3d")
ax.plot(states[:, 0], states[:, 1], states[:, 2])
plt.draw()
plt.show()

Applications

Model for atmospheric convection

As shown in Lorenz's original paper, the Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman. The Lorenz equations are derived from the Oberbeck–Boussinesq approximation to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above. This fluid circulation is known as Rayleigh–Bénard convection. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.

The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin approximation: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts from 2000Hilborn (2000), Appendix C; or 1984 Bergé, Pomeau & Vidal (1984), Appendix D.

Resolution of Smale's 14th problem

Smale's 14th problem says, 'Do the properties of the Lorenz attractor exhibit that of a strange attractor?'. The problem was answered affirmatively by Warwick Tucker in 2002. To prove this result, Tucker used rigorous numerics methods like interval arithmetic and normal forms. First, Tucker defined a cross section ${\displaystyle \Sigma \subset \{x_{3}=r-1\}}$ that is cut transversely by the flow trajectories. From this, one can define the first-return map ${\displaystyle P}$, which assigns to each ${\displaystyle x\in \Sigma }$ the point ${\displaystyle P(x)}$ where the trajectory of ${\displaystyle x}$ first intersects ${\displaystyle \Sigma }$.

Then the proof is split in three main points that are proved and imply the existence of a strange attractor. The three points are:

• There exists a region ${\displaystyle N\subset \Sigma }$ invariant under the first-return map, meaning ${\displaystyle P(N)\subset N}$.
• The return map admits a forward invariant cone field.
• Vectors inside this invariant cone field are uniformly expanded by the derivative ${\displaystyle DP}$ of the return map.

To prove the first point, we notice that the cross section ${\displaystyle \Sigma }$ is cut by two arcs formed by ${\displaystyle P(\Sigma )}$. Tucker covers the location of these two arcs by small rectangles ${\displaystyle R_{i}}$, the union of these rectangles gives ${\displaystyle N}$. Now, the goal is to prove that for all points in ${\displaystyle N}$, the flow will bring back the points in ${\displaystyle \Sigma }$, in ${\displaystyle N}$. To do that, we take a plan ${\displaystyle \Sigma '}$ below ${\displaystyle \Sigma }$ at a distance ${\displaystyle h}$ small, then by taking the center ${\displaystyle c_{i}}$ of ${\displaystyle R_{i}}$ and using Euler integration method, one can estimate where the flow will bring ${\displaystyle c_{i}}$ in ${\displaystyle \Sigma '}$ which gives us a new point ${\displaystyle c_{i}'}$. Then, one can estimate where the points in ${\displaystyle \Sigma }$ will be mapped in ${\displaystyle \Sigma '}$ using Taylor expansion, this gives us a new rectangle ${\displaystyle R_{i}'}$ centered on ${\displaystyle c_{i}}$. Thus we know that all points in ${\displaystyle R_{i}}$ will be mapped in ${\displaystyle R_{i}'}$. The goal is to do this method recursively until the flow comes back to ${\displaystyle \Sigma }$ and we obtain a rectangle ${\displaystyle Rf_{i}}$ in ${\displaystyle \Sigma }$ such that we know that ${\displaystyle P(R_{i})\subset Rf_{i}}$. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split ${\displaystyle R_{i}'}$ into smaller rectangles ${\displaystyle R_{i,j}}$ and then apply the process recursively. Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal', leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.

Gallery

• 

  A solution in the Lorenz attractor plotted at high resolution in the x-z plane.

• 

  A solution in the Lorenz attractor rendered as an SVG.

• Play media

  An animation showing trajectories of multiple solutions in a Lorenz system.

• 

  A solution in the Lorenz attractor rendered as a metal wire to show direction and 3D structure.

• Play media

  An animation showing the divergence of nearby solutions to the Lorenz system.

• 

  A visualization of the Lorenz attractor near an intermittent cycle.

• 

  Two streamlines in a Lorenz system, from ${\displaystyle \rho =0}$ to ${\displaystyle \rho =28}$ (${\displaystyle \sigma =10}$, ${\displaystyle \beta ={\frac {8}{3}}}$).

• 

  Animation of a Lorenz System with rho-dependence.

Hyperspace

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hyperspace 

Hyperspace (also, nulspace, subspace, overspace, jumpspace and similar) is a concept from science fiction relating to higher dimensions and a faster-than-light (FTL) method of interstellar travel that also occasionally appears in scientific works in related contexts. It is considered one of the more common science fiction tropes.

Its use in science fiction originated in the magazine Amazing Stories Quarterly in 1931. In most works, hyperspace is described as a dimension that can be entered and exited using rubber science, most often through a gadget known as a "hyperdrive". Many works rely on hyperspace as a convenient background tool enabling FTL travel necessary for the plot, with a small minority making it a central element of their storytelling.

Early depictions

The earliest references to hyperspace in fiction appeared in publications such as Amazing Stories Quarterly (shown here is the Spring 1931 issue featuring John Campbell's Islands of Space).

Emerging in the early 20th century, within several decades hyperspace became a common element of interstellar space travel stories in science fiction. Many stories in the science fiction magazines which became popular around that time, particularly in the United States, introduced readers to hyperspace, an idea that the three-dimensional space can be "folded", so that two separate points may come into contact, usually through the use of some device, most often called a "hyperdrive". Another common explanation involves the concept of a parallel universe, much smaller than ours, which partially or fully can be "mapped" into ours, through which the objects travel through much faster than they could in our universe. One of the main reasons for the adoption of hyperspace concept are of the limitations of faster-than-light travel in ordinary space, which the hyperspace trope allowed writers to bypass.

Kirk Meadowcroft's "The Invisible Bubble" (1928) and John Campbell's Islands of Space (1931) feature the earliest known references to hyperspace, with Campbell, whose story was published in the magazine Amazing Stories Quarterly, likely being the first writer to use this term in the context of space travel. According to the Historical Dictionary of Science Fiction, the earliest known use of the word "hyper-drive" comes from a preview of Murray Leinster's story in Thrilling Wonder Stories 1944: "Once again Kim takes off in the Starshine with its hyper-drive to do battle in defense of the Second Galaxy." As related vocabulary evolved, entering or navigating the hyperspace often became known as "jumping", as in "the ship will now jump to hyperspace".

Other notable early works employing this concept include Nelson Bond's The Scientific Pioneer Returns (1940), where his vision of the hyperspace concept is described in detail. Isaac Asimov's Foundation series, first published between 1942 and 1944 in Astounding, featured a Galactic Empire traversed through hyperspace through the use of a "hyperatomic drive". In Foundation (1951), hyperspace is described as an "...unimaginable region that was neither space nor time, matter nor energy, something nor nothing, one could traverse the length of the Galaxy in the interval between two neighboring instants of time."According to The Encyclopedia of Science Fiction, Robert A Heinlein gave a particularly clear description of it in Starman Jones (1953). E. C. Tubb has been credited with playing an important role in the development of hyperspace lore; writing a number of space operas in the early 1950s in which space travel occurs through that medium. He was also one of the first writers to treat hyperspace as a central part of the plot rather than a convenient background gadget that just enables the faster-than-light space travel.

Later depictions

Over the coming decades, a number of related terms (such as imaginary space, interspace, intersplit, jumpspace, megaflow, N-Space, nulspace, slipstream, overspace, Q-space, subspace, and tau-space) were used by various writers, although none gained recognition to rival that of hyperspace. Out of various fictitious drives, by the mid-70s the concept of travelling through hyperspace by using a hyperdrive has been described as having achieved the most renown, and would subsequently be further popularized through its use in the Star Wars franchise. Philip Harbottle called the concept of hyperspace "one of the fixtures" of the science fiction genre as early as in 1963. Some works nonetheless use one or more of the hyperspace synonyms; for example, in the Star Trek franchise, the term hyperspace itself is only used briefly in a single episode ("Coming of Age") of Star Trek: The Next Generation, while a related set of terms - subspace, transwarp, mycelial network and most recently, proto-warp - are used much more often, while most of the travel takes place through the use of a warp drive.

The streaking stars effect was initially used in Dark Star (1974) and became a popular cinematic depiction of hyperspace travel.

 

The streaking stars effect in an animated form

Stanley Kubrick's epic 1968 science fiction film 2001: A Space Odyssey features interstellar travel through a mysterious "star gate". This lengthy sequence, noted for its psychedelic special effects conceived by Douglas Trumbull, influenced a number of later cinematic depictions of superluminal and hyperspatial travel, such as Star Trek: The Motion Picture (1979). In the 1974 film Dark Star, special effects designer Dan O'Bannon created a visual effect to depict the eponymous Dark Star spaceship accelerating into hyperspace by tracking the camera while leaving the shutter open. In this shot, the stars in space turn into streaks of light while the spaceship appears to be motionless. This is considered to be the first depiction in cinema history of a ship making the jump into hyperspace. The streaking hyperspace effect was later employed in Star Wars (1977) and the "star streaks" are considered one of the visual "staples" of the Star Wars franchise.

In some works, travelling or navigating hyperspace requires not only specialized equipment, but physical or psychological modifications of passengers or at least navigators, as seen in Frank Herbert's Dune (1965), Michael Moorcock's The Sundered Worlds (1966), Vonda McIntyre's Aztecs (1977), or David Brin's The Warm Space (1985).

Characteristics and uses

Hyperspace is generally seen as a fictional concept, incompatible with our present-day understanding of the universe (in particular, the theory of relativity). Occasionally the term – which originated in 19th-century mathematical texts – is used in academic works in the context of higher-dimensional geometry, popularized among others by theoretical physicist Michio Kaku's popular science book (Hyperspace, 1994). In mathematics, it is related to the concept of four-dimensional space, first described in the 19th century. Some science fiction writers attempted quasi-scientific rubber science explanations of this concept. For others, however, it is just a convenient MacGuffin enabling faster-than-light travel necessary for their story.

Hyperspace is generally described as chaotic and confusing to human senses; in some cases even hypnotic or dangerous to one's sanity, or at least unpleasant – transitions to or from hyperspace can cause symptoms such as nausea, for example. Visually, hyperspace is often left to the readers imagination, or depicted as "a swirling gray mist". In some works, it is dark. Exceptions do exist, for example, John Russel Fearn's Waters of Eternity (1953) features hyperspace that allows observation of regular space from within.

While mainly designed as a space-travel enabling trope, occasionally, some writers used the hyperspace concept in a more imaginative ways, or as a central element of the story. In Arthur C. Clarke's "Technical Error" (1950), a man is laterally reversed by a brief accidental encounter with "hyperspace". In George R.R. Martin's FTA (1974) hyperspace travel takes longer than in regular space, and in John E. Stith's Redshift Rendezvous (1990), the twist is that the relativistic effects within it appear at lower velocities. Hyperspace is generally unpopulated, save for the space-faring travellers. Early exceptions include Tubb's Dynasty of Doom (1953), Fearn's Waters of Eternity (1953) and Christopher Grimm's Someone to Watch Over Me (1959), which feature denizens of hyperspace. In The Mystery of Element 117 (1949) by Milton Smith, a window is opened into a new "hyperplane of hyperspace" containing those who have already died on Earth, and similarly, in Bob Shaw's The Palace of Eternity (1969), the hyperspace is a form of afterlife, where human minds and memories reside after death. In some works, hyperspace is a source of extremely dangerous energy, threatening to destroy the entire world if mishandled (ex. Eando Binder's The Time Contractor, 1937).

Many stories feature hyperspace as a dangerous place. Hyperspace is often described as being an unnavigable dimension where straying from a preset course can be disastrous. In Frederick Pohl's The Mapmakers (1955), navigational errors and the perils of hyperspace are one of the main plot-driving elements. In K. Houston Brunner's Firey Pillar (1955), a ship re-emerges within Earth, causing a catastrophic explosion.

In many stories, for various reasons, a starship cannot enter or leave hyperspace too close to a large concentration of mass, such as a planet or star; this means that hyperspace can only be used after a starship gets to the outside edge of a solar system, so the starship must use other means of propulsion to get to and from planets. Other writers have limited access to hyperspace by requiring a very large expenditure of energy in order to open a link (sometimes called a jump point) between hyperspace and regular space; this effectively limits access to hyperspace to very large starships, or to large stationary jump gates that can open jump points for smaller vessels.An example of this is the "jump" technology as seen in Babylon 5. Another would be the star gate seen in Arthur C. Clarke's 2001: A Space Odyssey (1968). The reasons given for such restrictions are usually technobabble, but their existence is just a plot device allowing for interstellar policies to actually form and exist. Science fiction author Larry Niven published his opinions to that effect in N-Space. According to him, such an unrestricted technology would give no limits to what heroes and villains could do. Limiting the places a ship can appear in means that they will meet each other most often around contested planets or space stations, allowing for narratively satisfying battles or other encounters. On the other hand, less restricted hyperdrive may also allow for dramatic escapes as the pilot "jumps" to hyperspace in the midst of battle to avoid destruction.

James P. Hogan observed that (as of 1999) hyperspace still remains underutilized in science-fiction writing, treated too often as a plot-enabling gadget rather than as a fascinating, world-changing item, noting that for example there are next to no works that discuss the topic of how hyperspace has been discovered and how such discovery subsequently changed the world.

While generally associated with science fiction, hyperspace-like concepts exist in some works of fantasy, particularly ones which involve travel between different worlds or dimensions. In such works, such travel, usually done through portals rather than vehicles, is usually explained through the existence of magic.

Hermann Minkowski

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hermann_Minkowski 

 

Hermann Minkowski
Born	22 June 1864 Aleksotas, Suwałki Governorate, Kingdom of Poland (now in Kaunas, Lithuania)
Died	12 January 1909 (aged 44) Göttingen, German Empire
Citizenship	Russian Empire or Germany
Alma mater	Albertina University of Königsberg
Known for	Geometry of numbers Minkowski content Minkowski diagram Minkowski's question-mark function Minkowski space Work on the Diophantine approximations
Spouse(s)	Auguste Adler
Children	Lily (1898–1983), Ruth (1902–2000)
Scientific career
Fields	Mathematics, physics, philosophy
Institutions	University of Göttingen and ETH Zurich
Doctoral advisor	Ferdinand von Lindemann
Doctoral students	Constantin Carathéodory Louis Kollros Dénes Kőnig
Signature

Hermann Minkowski (/mɪŋˈkɔːfski, -ˈkɒf-/; German: [mɪŋˈkɔfski]; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. In different sources Minkowski's nationality is variously given as German, Polish, or Lithuanian-German, or Russian. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.

Minkowski is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein's special theory of relativity (1905) could be understood geometrically as a theory of four-dimensional space–time, since known as the "Minkowski spacetime".

Personal life and family

Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher Oskar (born 1858).

To escape persecution in the Russian Empire the family moved to Königsberg in 1872, where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys (he operated his firm Lewin Minkowski & Son with his eldest son Max).

Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zurich (1896–1902), and finally in Göttingen from 1902 until his death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; the electrical engineer and inventor Reinhold Rudenberg was his son-in-law.

Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909. David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians (translated):

Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.

Max Born delivered the obituary on behalf of the mathematics students at Göttingen.

The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor.

Education and career

Minkowski in 1883, at the time of being awarded the Mathematics Prize of the French Academy of Sciences

Minkowski was educated in East Prussia at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms. He also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski (1858–1931), was a well-known physician and researcher.

Minkowski taught at the universities of Bonn, Königsberg, Zürich, and Göttingen. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers.

Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve.

In 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there.

Work on relativity

Further information: Minkowski space and Minkowski diagram

By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$ (see History of special relativity).

The mathematical basis of Minkowski space can also be found in the hyperboloid model of hyperbolic space already known in the 19th century, because isometries (or motions) in hyperbolic space can be related to Lorentz transformations, which included contributions of Wilhelm Killing (1880, 1885), Henri Poincaré (1881), Homersham Cox (1881), Alexander Macfarlane (1894) and others (see History of Lorentz transformations).

The beginning part of his address called "Space and Time" delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908) is now famous:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Publications

Relativity

• Minkowski, Hermann (1915) [1907]. "Das Relativitätsprinzip" . Annalen der Physik. 352 (15): 927–938. Bibcode:1915AnP...352..927M. doi:10.1002/andp.19153521505.
• Minkowski, Hermann (1908). "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" . Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111.
  • English translation: "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". In: The Principle of Relativity (1920), Calcutta: University Press, 1–69.
• Minkowski, Hermann (1909). "Raum und Zeit" . Jahresbericht der Deutschen Mathematiker-Vereinigung. 18: 75–88. Bibcode:1909JDMaV..18...75M.
  • Various English translations on Wikisource: "Space and Time".
• Blumenthal O. (ed): Das Relativitätsprinzip, Leipzig 1913, 1923 (Teubner), Engl tr (W. Perrett & G. B. Jeffrey) The Principle of Relativity London 1923 (Methuen); reprinted New York 1952 (Dover) entitled H. A. Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, The Principle of Relativity: A Collection of Original Memoirs.
• Space and Time – Minkowski's Papers on Relativity, Minkowski Institute Press, 2012 ISBN 978-0-9879871-3-6 (free ebook).

Diophantine approximations

• Minkowski, Hermann (1907). Diophantische Approximationen: Eine Einführung in die Zahlentheorie. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016.

Mathematical (posthumous)

• Minkowski, Hermann (1910). "Geometrie der Zahlen". Leipzig-Berlin: B. G. Teubner Verlag. MR 0249269. Retrieved 28 February 2016. 
• Minkowski, Hermann (1911). Gesammelte Abhandlungen 2 vols. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016. Reprinted in one volume New York, Chelsea 1967.