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Nanowire

A nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 metres). More generally, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. At these scales, quantum mechanical effects are important—which coined the term "quantum wires".

Many different types of nanowires exist, including superconducting (e.g. YBCO ), metallic (e.g. Ni, Pt, Au, Ag), semiconducting (e.g. silicon nanowires (SiNWs), InP, GaN) and insulating (e.g. SiO2, TiO2).

Molecular nanowires are composed of repeating molecular units either organic (e.g. DNA) or inorganic (e.g. Mo6S9−xIx).

Characteristics

Typical nanowires exhibit aspect ratios (length-to-width ratio) of 1000 or more. As such they are often referred to as one-dimensional (1-D) materials. Nanowires have many interesting properties that are not seen in bulk or 3-D (three-dimensional) materials. This is because electrons in nanowires are quantum confined laterally and thus occupy energy levels that are different from the traditional continuum of energy levels or bands found in bulk materials.

A consequence of this quantum confinement in nanowires is that they exhibit discrete values of the electrical conductance. Such discrete values arise from a quantum mechanical constraint on the number electronic transport channels at the nanometer scale, and they are often approximately equal to integer multiples of the quantum of conductance:

 * $$\frac{2e^2}{h} \simeq 77.41\; \mu S$$

This conductance is twice the reciprocal of the resistance unit called the von Klitzing constant,, defined as RK = h/e2 and named for Klaus von Klitzing, the discoverer of the integer quantum Hall effect.

Examples of nanowires include inorganic molecular nanowires (Mo6S9−xIx, Li2Mo6Se6), which can have a diameter of 0.9 nm and be hundreds of micrometers long. Other important examples are based on semiconductors such as InP, Si, GaN, etc., dielectrics (e.g. SiO2,TiO2), or metals (e.g. Ni, Pt).

There are many applications where nanowires may become important in electronic, opto-electronic and nanoelectromechanical devices, as additives in advanced composites, for metallic interconnects in nanoscale quantum devices, as field-emitters and as leads for biomolecular nanosensors.

Synthesis

There are two basic approaches to synthesizing nanowires: top-down and bottom-up. A top-down approach reduces a large piece of material to small pieces, by various means such as lithography, milling or thermal oxidation. A bottom-up approach synthesizes the nanowire by combining constituent adatoms. Most synthesis techniques use a bottom-up approach. Initial synthesis via either method may often be followed by a nanowire thermal treatment step, often involving a form of self-limiting oxidation, to fine tune the size and aspect ratio of the structures. After the bottom-up synthesis, nanowires can be integrated using pick-and-place techniques.

Nanowire production uses several common laboratory techniques, including suspension, electrochemical deposition, vapor deposition, and VLS growth. Ion track technology enables growing homogeneous and segmented nanowires down to 8 nm diameter. As nanowire oxidation rate is controlled by diameter, thermal oxidation steps are often applied to tune their morphology.

Suspension
A suspended nanowire is a wire produced in a high-vacuum chamber held at the longitudinal extremities. Suspended nanowires can be produced by:

 * The chemical etching of a larger wire
 * The bombardment of a larger wire, typically with highly energetic ions
 * Indenting the tip of a STM in the surface of a metal near its melting point, and then retracting it

VLS growth
A common technique for creating a nanowire is vapor–liquid–solid method (VLS), which was first reported by Wagner and Ellis in 1964 for silicon whiskers with diameters ranging from hundreds of nm to hundreds of μm. This process can produce high-quality crystalline nanowires of many semiconductor materials, for example, VLS–grown single crystalline silicon nanowires (SiNWs) with smooth surfaces could have excellent properties, such as ultra-large elasticity. This method uses a source material from either laser ablated particles or a feed gas such as silane.

VLS synthesis requires a catalyst. For nanowires, the best catalysts are liquid metal (such as gold) nanoclusters, which can either be self-assembled from a thin film by dewetting, or purchased in colloidal form and deposited on a substrate.

The source enters these nanoclusters and begins to saturate them. On reaching supersaturation, the source solidifies and grows outward from the nanocluster. Simply turning off the source can adjust the final length of the nanowire. Switching sources while still in the growth phase can create compound nanowires with super-lattices of alternating materials. For example, a method termed ENGRAVE (Encoded Nanowire GRowth and Appearance through VLS and Etching) developed by the Cahoon Lab at UNC-Chapel Hill allows for nanometer-scale morphological control via rapid in situ dopant modulation.

A single-step vapour phase reaction at elevated temperature synthesises inorganic nanowires such as Mo6S9−xIx. From another point of view, such nanowires are cluster polymers.

Similar to VLS synthesis, VSS (vapor-solid-solid) synthesis of nanowires (NWs) proceeds through thermolytic decomposition of a silicon precursor (typically phenylsilane). Unlike VLS, the catalytic seed remains in solid state when subjected to high temperature annealing of the substrate. This such type of synthesis is widely used to synthesise metal silicide/germanide nanowires through VSS alloying between a copper substrate and a silicon/germanium precursor.

Solution-phase synthesis
Solution-phase synthesis refers to techniques that grow nanowires in solution. They can produce nanowires of many types of materials. Solution-phase synthesis has the advantage that it can produce very large quantities, compared to other methods. In one technique, the polyol synthesis, ethylene glycol is both solvent and reducing agent. This technique is particularly versatile at producing nanowires of gold, lead, platinum, and silver.

The supercritical fluid-liquid-solid growth method can be used to synthesize semiconductor nanowires, e.g., Si and Ge. By using metal nanocrystals as seeds, Si and Ge organometallic precursors are fed into a reactor filled with a supercritical organic solvent, such as toluene. Thermolysis results in degradation of the precursor, allowing release of Si or Ge, and dissolution into the metal nanocrystals. As more of the semiconductor solute is added from the supercritical phase (due to a concentration gradient), a solid crystallite precipitates, and a nanowire grows uniaxially from the nanocrystal seed.

Liquid Bridge Induced Self-assembly
Protein nanowires in spider silk have been formed by rolling a droplet of spider silk solution over a superhydrophobic pillar structure.

Non-catalytic growth

The vast majority of nanowire-formation mechanisms are explained through the use of catalytic nanoparticles, which drive the nanowire growth and are either added intentionally or generated during the growth. However, nanowires can be also grown without the help of catalysts, which gives an advantage of pure nanowires and minimizes the number of technological steps. The mechanisms for catalyst-free growth of nanowires (or whiskers) were known from 1950s.

The simplest methods to obtain metal oxide nanowires use ordinary heating of the metals, e.g. metal wire heated with battery, by Joule heating in air can be easily done at home. Spontaneous nanowire formation by non-catalytic methods were explained by the dislocation present in specific directions or the growth anisotropy of various crystal faces. More recently, after microscopy advancement, the nanowire growth driven by screw dislocations or twin boundaries were demonstrated. The picture on the right shows a single atomic layer growth on the tip of CuO nanowire, observed by in situ TEM microscopy during the non-catalytic synthesis of nanowire.

Atomic-scale nanowires can also form completely self-organised without need for defects. For example, rare-earth silicide (RESi2) nanowires of few nm width and height and several 100 nm length form on silicon(001) substrates which are covered with a sub-monolayer of a rare earth metal and subsequently annealed. The lateral dimensions of the nanowires confine the electrons in such a way that the system resembles a (quasi-)one-dimensional metal. Metallic RESi2 nanowires form on silicon(hhk) as well. This system permits tuning the dimensionality between two-dimensional and one-dimensional by the coverage and the tilt angle of the substrate.

DNA-templated metallic nanowire synthesis
An emerging field is to use DNA strands as scaffolds for metallic nanowire synthesis. This method is investigated both for the synthesis of metallic nanowires in electronic components and for biosensing applications, in which they allow the transduction of a DNA strand into a metallic nanowire that can be electrically detected. Typically, ssDNA strands are stretched, whereafter they are decorated with metallic nanoparticles that have been functionalised with short complementary ssDNA strands.

Crack-Defined Shadow Mask Lithography
A simple method to produce nanowires with defined geometries has been recently reported using conventional optical lithography. In this approach, optical lithography is used to generate nanogaps using controlled crack formation. These nanogaps are then used as shadow mask for generating individual nanowires with precise lengths and widths. This technique allows to produce individual nanowires below 20 nm in width in a scalable way out of several metallic and metal oxide materials.

Conductivity

Several physical reasons predict that the conductivity of a nanowire will be much less than that of the corresponding bulk material. First, there is scattering from the wire boundaries, whose effect will be very significant whenever the wire width is below the free electron mean free path of the bulk material. In copper, for example, the mean free path is 40 nm. Copper nanowires less than 40 nm wide will shorten the mean free path to the wire width. Silver nanowires have very different electrical and thermal conductivity from bulk silver.

Nanowires also show other peculiar electrical properties due to their size. Unlike single wall carbon nanotubes, whose motion of electrons can fall under the regime of ballistic transport (meaning the electrons can travel freely from one electrode to the other), nanowire conductivity is strongly influenced by edge effects. The edge effects come from atoms that lay at the nanowire surface and are not fully bonded to neighboring atoms like the atoms within the bulk of the nanowire. The unbonded atoms are often a source of defects within the nanowire, and may cause the nanowire to conduct electricity more poorly than the bulk material. As a nanowire shrinks in size, the surface atoms become more numerous compared to the atoms within the nanowire, and edge effects become more important.

The conductance in a nanowire is described as the sum of the transport by separate channels, each having a different electronic wavefunction normal to the wire. The thinner the wire is, the smaller the number of channels available to the transport of electrons. As a result, wires that are only one or a few atoms wide exhibit quantization of the conductance: i.e. the conductance can assume only discrete values that are multiples of the conductance quantum G0 = 2e2/h (where e is the charge of the electron and h is the Planck constant). (See also the Quantum Hall effect). This quantization has been observed by measuring the conductance of a nanowire suspended between two electrodes while pulling it progressively longer: as its diameter reduces, its conductivity decreases in a stepwise fashion and the plateaus correspond approximately to multiples of G0.

The quantization of conductivity is more pronounced in semiconductors like Si or GaAs than in metals, because of their lower electron density and lower effective mass. It can be observed in 25 nm wide silicon fins, and results in increased threshold voltage. In practical terms, this means that a MOSFET with such nanoscale silicon fins, when used in digital applications, will need a higher gate (control) voltage to switch the transistor on.

Welding
To incorporate nanowire technology into industrial applications, researchers in 2008 developed a method of welding nanowires together: a sacrificial metal nanowire is placed adjacent to the ends of the pieces to be joined (using the manipulators of a scanning electron microscope); then an electric current is applied, which fuses the wire ends. The technique fuses wires as small as 10 nm.

For nanowires with diameters less than 10 nm, existing welding techniques, which require precise control of the heating mechanism and which may introduce the possibility of damage, will not be practical. Recently scientists discovered that single-crystalline ultrathin gold nanowires with diameters ≈3–10 nm can be "cold-welded" together within seconds by mechanical contact alone, and under remarkably low applied pressures (unlike macro- and micro-scale cold welding process). High-resolution transmission electron microscopy and in situ measurements reveal that the welds are nearly perfect, with the same crystal orientation, strength and electrical conductivity as the rest of the nanowire. The high quality of the welds is attributed to the nanoscale sample dimensions, oriented-attachment mechanisms and mechanically assisted fast surface diffusion. Nanowire welds were also demonstrated between gold and silver, and silver nanowires (with diameters ≈5–15 nm) at near room temperature, indicating that this technique may be generally applicable for ultrathin metallic nanowires. Combined with other nano- and microfabrication technologies, cold welding is anticipated to have potential applications in the future bottom-up assembly of metallic one-dimensional nanostructures.

Mechanical properties
The study of nanowire mechanics has boomed since the advent of the atomic force microscope (AFM), and associated technologies which have enabled direct study of the response of the nanowire to an applied load. Specifically, a nanowire can be clamped from one end, and the free end displaced by an AFM tip. In this cantilever geometry, the height of the AFM is precisely known, and the force applied is precisely known. This allows for construction of a force vs. displacement curve, which can be converted to a stress vs. strain curve if the nanowire dimensions are known. From the stress-strain curve, the elastic constant known as the Young's Modulus can be derived, as well as the toughness, and degree of strain-hardening.

Young's modulus

The elastic component of the stress-strain curve described by the Young's Modulus, has been reported for nanowires, however the modulus depends very strongly on the microstructure. Thus a complete description of the modulus dependence on diameter is lacking. Analytically, continuum mechanics has been applied to estimate the dependence of modulus on diameter: $$E=E_{0}[1+4(E_{0}/E_{s}-1)(r_{s}/D-r_{s}^{2}/D^{2})]$$ in tension, where $$E_{0} $$ is the bulk modulus, $$r_{s}$$ is the thickness of a shell layer in which the modulus is surface dependent and varies from the bulk, $$E{s}$$ is the surface modulus, and $$D$$ is the diameter. This equation implies that the modulus increases as the diameter decreases. However, various computational methods such as molecular dynamics have predicted that modulus should decrease as diameter decreases.

Experimentally, gold nanowires have been shown to have a Young's modulus which is effectively diameter independent. Similarly, nano-indentation was applied to study the modulus of silver nanowires, and again the modulus was found to be 88 GPa, very similar to the modulus of bulk Silver (85 GPa) These works demonstrated that the analytically determined modulus dependence seems to be suppressed in nanowire samples where the crystalline structure highly resembles that of the bulk system.

In contrast, Si solid nanowires have been studied, and shown to have a decreasing modulus with diameter The authors of that work report a Si modulus which is half that of the bulk value, and they suggest that the density of point defects, and or loss of chemical stoichiometry may account for this difference.

Yield strength
The plastic component of the stress strain curve (or more accurately the onset of plasticity) is described by the yield strength. The strength of a material is increased by decreasing the number of defects in the solid, which occurs naturally in nanomaterials where the volume of the solid is reduced. As a nanowire is shrunk to a single line of atoms, the strength should theoretically increase all the way to the molecular tensile strength. Gold nanowires have been described as ‘ultrahigh strength’ due to the extreme increase in yield strength, approaching the theoretical value of E/10. This huge increase in yield is determined to be due to the lack of dislocations in the solid. Without dislocation motion, a ‘dislocation-starvation’ mechanism is in operation. The material can accordingly experience huge stresses before dislocation motion is possible, and then begins to strain-harden. For these reasons, nanowires (historically described as 'whiskers') have been used extensively in composites for increasing the overall strength of a material. Moreover, nanowires continue to be actively studied, with research aiming to translate enhanced mechanical properties to novel devices in the fields of MEMS or NEMS.

Electronic devices

Nanowires have been proposed for use as MOSFETs (MOS field-effect transistors). MOS transistors are used widely as fundamental building elements in today's electronic circuits. As predicted by Moore's law, the dimension of MOS transistors is shrinking smaller and smaller into nanoscale. One of the key challenges of building future nanoscale MOS transistors is ensuring good gate control over the channel. Due to the high aspect ratio, if the gate dielectric is wrapped around the nanowire channel, we can get good control of channel electrostatic potential, thereby turning the transistor on and off efficiently.

Due to their one-dimensional structure with unusual optical properties, the nanowire are of interest for photovoltaic devices. Compared with its bulk counterparts, the nanowire solar cells are less sensitive to impurities due to bulk recombination, and thus silicon wafers with lower purity can be used to achieve acceptable efficiency, leading to the reduction on material consumption.

After p-n junctions were built with nanowires, the next logical step was to build logic gates. By connecting several p-n junctions together, researchers have been able to create the basis of all logic circuits: the AND, OR, and NOT gates have all been built from semiconductor nanowire crossings.

In August 2012, researchers reported constructing the first NAND gate from undoped silicon nanowires. This avoids the problem of how to achieve precision doping of complementary nanocircuits, which is unsolved. They were able to control the Schottky barrier to achieve low-resistance contacts by placing a silicide layer in the metal-silicon interface.

It is possible that semiconductor nanowire crossings will be important to the future of digital computing. Though there are other uses for nanowires beyond these, the only ones that actually take advantage of physics in the nanometer regime are electronic.

In addition, nanowires are also being studied for use as photon ballistic waveguides as interconnects in quantum dot/quantum effect well photon logic arrays. Photons travel inside the tube, electrons travel on the outside shell.

When two nanowires acting as photon waveguides cross each other the juncture acts as a quantum dot.

Conducting nanowires offer the possibility of connecting molecular-scale entities in a molecular computer. Dispersions of conducting nanowires in different polymers are being investigated for use as transparent electrodes for flexible flat-screen displays.

Because of their high Young's moduli, their use in mechanically enhancing composites is being investigated. Because nanowires appear in bundles, they may be used as tribological additives to improve friction characteristics and reliability of electronic transducers and actuators.

Because of their high aspect ratio, nanowires are also suited to dielectrophoretic manipulation,  which offers a low-cost, bottom-up approach to integrating suspended dielectric metal oxide nanowires in electronic devices such as UV, water vapor, and ethanol sensors.

Due to their large surface-to-volume ratio, physico-chemical reactions are facilitated on the surface of nanowires.

Single nanowire devices for gas and chemical sensing
The high aspect ratio of nanowires makes this nanostructures suitable for electrochemical sensing with the potential for ultimate sensitivity. One of the challenge for the use of nanowires in commercial products is related to the isolation, handling, and integration of nanowires in an electrical circuit when using the conventional and manual pick-and-place approach, leading to a very limited throughput. Recent developments in the nanowire synthesis methods now allow for parallel production of single nanowire devices with useful applications in electrochemistry, photonics, and gas- and biosensing.

Nanowire lasers

Nanowire lasers are nano-scaled lasers with potential as optical interconnects and optical data communication on chip. Nanowire lasers are built from III–V semiconductor heterostructures, the high refractive index allows for low optical loss in the nanowire core. Nanowire lasers are subwavelength lasers of only a few hundred nanometers. Nanowire lasers are Fabry–Perot resonator cavities defined by the end facets of the wire with high-reflectivity, recent developments have demonstrated repetition rates greater than 200 GHz offering possibilities for optical chip level communications.

Sensing of proteins and chemicals using semiconductor nanowires
In an analogous way to FET devices in which the modulation of conductance (flow of electrons/holes) in the semiconductor, between the input (source) and the output (drain) terminals, is controlled by electrostatic potential variation (gate-electrode) of the charge carriers in the device conduction channel, the methodology of a Bio/Chem-FET is based on the detection of the local change in charge density, or so-called “field effect”, that characterizes the recognition event between a target molecule and the surface receptor.

This change in the surface potential influences the Chem-FET device exactly as a ‘gate’ voltage does, leading to a detectable and measurable change in the device conduction. When these devices are fabricated using semiconductor nanowires as the transistor element the binding of a chemical or biological species to the surface of the sensor can lead to the depletion or accumulation of charge carriers in the "bulk" of the nanometer diameter nanowire i.e. (small cross section available for conduction channels). Moreover, the wire, which serves as a tunable conducting channel, is in close contact with the sensing environment of the target, leading to a short response time, along with orders of magnitude increase in the sensitivity of the device as a result of the huge S/V ratio of the nanowires.

While several inorganic semiconducting materials such as Si, Ge, and metal oxides (e.g. In2O3, SnO2, ZnO, etc.) have been used for the preparation of nanowires, Si is usually the material of choice when fabricating nanowire FET-based chemo/biosensors.

Several examples of the use of silicon nanowire(SiNW) sensing devices include the ultra sensitive, real-time sensing of biomarker proteins for cancer, detection of single virus particles, and the detection of nitro-aromatic explosive materials such as 2,4,6 Tri-nitrotoluene (TNT) in sensitives superior to these of canines. Silicon nanowires could also be used in their twisted form, as electromechanical devices, to measure intermolecular forces with great precision.

Limitations of sensing with silicon nanowire FET devices
Generally, the charges on dissolved molecules and macromolecules are screened by dissolved counterions, since in most cases molecules bound to the devices are separated from the sensor surface by approximately 2–12 nm (the size of the receptor proteins or DNA linkers bound to the sensor surface). As a result of the screening, the electrostatic potential that arises from charges on the analyte molecule decays exponentially toward zero with distance. Thus, for optimal sensing, the Debye length must be carefully selected for nanowire FET measurements. One approach of overcoming this limitation employs fragmentation of the antibody-capturing units and control over surface receptor density, allowing more intimate binding to the nanowire of the target protein. This approach proved useful for dramatically enhancing the sensitivity of cardiac biomarkers (e.g. Troponin) detection directly from serum for the diagnosis of acute myocardial infarction.

Nanowire assisted transfer of sensitive TEM samples
For a minimal introduction of stress and bending to transmission electron microscopy (TEM) samples (lamellae, thin films, and other mechanically and beam sensitive samples), when transferring inside a focused ion beam (FIB), flexible metallic nanowires can be attached to a typically rigid micromanipulator.

The main advantages of this method include a significant reduction of sample preparation time (quick welding and cutting of nanowire at low beam current), and minimization of stress-induced bending, Pt contamination, and ion beam damage. This technique is particularly suitable for in situ electron microscopy sample preparation.

Corn-like nanowires
Corn-like nanowire is a one-dimensional nanowire with interconnected nanoparticles on the surface, providing a large percentage of reactive facets. TiO2 corn-like nanowires were first prepared by a surface modification concept using surface tension stress mechanism through a two consecutive hydrothermal operation, and showed an increase of 12% in dye-sensitized solar cell efficiency the light scattering layer. CdSe corn-like nanowires grown by chemical bath deposition and corn-like γ-Fe2O3@SiO2@TiO2 photocatalysts induced by magnetic dipole interactions have been also reported previously.

title: Nanowire


text:
Talk:Magic square

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Initial text
The tag for superscripts is <sup> which is closed with </sup>

The tag for subscripts is (similarly) - sub — Preceding unsigned comment added by Marj Tiefert (talk • contribs) 16:19, 16 May 2002 (UTC)

M+N, M*N, kM
I believe that if two matrices M, N are magic squares, then M+N, M*N, kM are also magic squares. If I'm right, could someone add this?

Also, out of interest -- let M be the set of all n x n magic squares, for some n. Let V be the real vector space (M,+), where + is matrix addition. Does V exist? How many dimensions has V? Can you give a basis for V? (I am asking this question for all n, but would appreciate answers even for a particular n > 1). -- SJK

Addition of magic squares produces magic squares (ignoring that the coefficients are not consecutive integers). Multiplication of magic squares, however, does not: 8163574922 is 91 67 67 67 91 67 67 67 91. -phma

To SJK:
Provided your definition of a magic square is just that the columns, rows and long diagonals have the same sum, then the magic squares with real entries do form a vector space over R.

For n = 2, the dimension is 1, because all the entries must be equal.

For n = 3, the dimension is 3, because you can choose the first row freely, but this then determines all the remaining entries. The following three matrices form a basis:

/ 3 0  0 \      / 0  3  0 \      / 0  0  3 \ |-2  1  4 |      | 1  1  1 |      | 4  1 -2 | \ 2  2 -1 /      \ 2 -1  2 /      \-1  2  2 /

I don't know how to determine the dimension in general, although it must be at least n2-2n-1 and no more than n2. --Zundark 14:51 Sep 13, 2002 (UTC)

The dimension is well known to be n2-2n-1. Zaslav 03:17, 23 January 2006 (UTC)
 * Actually, it's n2-2n The math works out this way, and I can give bases for the 3x3 and 4x4 vector space (of size 3 and 8 resp.) --128.61.115.86 (talk) 05:11, 24 April 2009 (UTC)
 * Figured out the confusion: the well-known result (the extra minus one) is WITH the restriction to the first consecutive n2 integers. This restriction fixes the magic number, reducing the number of unknowns by one, but it also destroys the vector spaceness of the set.  —Preceding unsigned comment added by Quintopia (talk • contribs) 18:04, 24 April 2009 (UTC)

Loh-Shu magic square
Isn't there a legend that the Loh-Shu magic square was first seen written on the shell of a turtle? -- Tarquin 21:34 Feb 1, 2003 (UTC)
 * See Cheers Chas zzz brown 07:59 Feb 2, 2003 (UTC)

 * See Lo Shu Square. —Herbee 17:53, 20 Mar 2005 (UTC)

The following section is not very enlightening, so I'm taking it out until someone can elaborate.
 * Euler showed how to derive magic squares from Latin squares.

—Herbee 18:01, 20 Mar 2005 (UTC)

Yes, It is a legend. So that I personally believe that it is not a truth.--Meavel 06:05, 8 July 2007 (UTC)

Definition

 * Halló! I have some problems with the definition: Both according to Mathworld and to Harvey Heinz (this link should be included at #External links) the numbers start with 1 and end with n2.
 * I would suggest that Wikipedia should use the same definition. To my understanding the "other" squares are somehow related or just simple "magic patterns".
 * For analysis purposes more equivalent representations can be given: numbers from 0 to n2-1; odd numbers from -n2+1 to n2-1 and probably others.
 * I developed a set of templates to be used to ilustrate many properties of 4x4 type magic squares at meta:Category:4x4 type square. Please let me know if you have some time to work on this subject and to help to make a wikibook at en: in other languages. We should make some documenation first and a list of items to work on. Thanks in advance! Best regards Gangleri | [ Th] | T 22:40, 28 July 2005 (UTC)

The term "magic square" covers many different though related ideas. They all have in common that the rows and columns have the same "magic sum". The diagonals should be included, but research mathematicians have often been very sloppy about that and other requirements; properly, a square that is magic for rows and columns, ignoring the diagonals, is "semimagic", not "magic". Recreational mathematicians (a different group from research mathematicians) have always included the diagonal sums in the requirements. I also have never heard of any magic squares whose entries are not integers, except in the vector-space generalization alluded to in another comment. Historically, squares with negative entries seem not to have appeared, but squares with 0 have appeared (centuries ago) and squares with entries that are consecutive integers but not starting with 1 have also appeared for centuries (usually, I think, as tools to make larger squares out of smaller ones). Squares with entries that are nonconsecutive integers have been studied in the twentieth century and possibly the nineteenth. Thus, your two references are excessively narrow and not historically justified.

Subtracting or adding any constant to the entries of a magic square, e.g., 1, will always give a magic square. This is so obvious that it seems silly to discuss it at length. (It might be worth a mention in the appropriate subsection, if there is one.)

Zaslav 03:28, 23 January 2006 (UTC)

Yang Hui's square and "table of Jupiter"

 * See: meta:User:Gangleri/tests/4x4 type square/generating magic squares with T000. Regards Gangleri | [ Th] | T 23:23, 28 July 2005 (UTC)

Constructing a magic square of doubly even order
It is necessary write the numbers from right to left? (Keyword: symmetry)

Western occult

 * Durer's Melancholia square, noted in the article, is reportedly based on the magic squares of his contemporary, occultist Heinrich Cornelius Agrippa. Agrippa apparently assigned magic squares of the orders of 3 through 9 to the planets (and astrological/alchemical figures) Saturn, Jupiter, Mars, Sol (the sun), Venus, Mercury, and Luna (the moon) respectively, hence the "table of Jupiter" noted above. (See Agrippa's text for more.) I believe his "tables" would be inscribed on magic talismans, in hopes of harnessing the various virtues and powers of each particular planet.

Historic image available
If anyone's interested, I just uploaded Image:16th century arabic magic square.jpg. Couldn't see an obvious place to add it, so I thought I'd just let you know. — Laura Scudder ☎ 01:40, 10 February 2006 (UTC)

Ben Franklin and others
Mr. Franklin was a great fan of magic squares, and created quite a few HUGE (16x16, 24x24) magic squares with a number of interesting attributes.

I also recall a magic square that had to do with the 365 days in our year. After a brief web search, I found on this site: http://www.jainmathemagics.com/page/1/default.asp that it says "This 27 x 27 Magic Square Calendar has, as its central cell, the number 365 which is the number of days in a solar year. It has 364 dark cells which represent the number of nights, and 365 white cells which represent the number of days. The Magic Sum of the inner and central 3x3 square is 1,095 being the number of days in a 3 year period. The Magic Sum of the 9x9 square is 3,285 being the number of days in a 9 year period. The Magic Sum of the whole 27x27 square is 9,855 being the number of days in a 27 year period."

I also recall a few other crazy interesting magic squares (http://mathworld.wolfram.com/PrimeMagicSquare.html is neat), and know that there have been some famous people (besides Franklin and Durer) that dabbled with/used magic squares.

If anyone wants to run with this idea to add a section on a list of famous people who played with magic squares or who wants to add a section talking about the many connections with various occultist/whatever things ... go ahead. Email me at cht13er a t gmaildotcom if you like :-) cheater 14:03, 15 March 2006 (UTC)

Magic Squares of odd order
Perhaps I have missed something, but aren't the first and second paragraphs of this section saying the same thing? Bio 18:08, 22 November 2006 (UTC)

Source of info
I saw this site on smagic quares and I think the construction of magic squares there is easier. Can someone investigate this? --Blonkm 02:38, 17 December 2006 (UTC)

Islam
Islam is a religion, I think the part titled "Islam" should be renamed. It is about arabic uses of magic squares. It should be called "Arabs".

What about mid-eastern? Yongke 17:09, 11 March 2007 (UTC)

More "See also"
I moved 2 cube links to end, then added 3 hypercube links.Harvey Heinz 17:11, 12 September 2007 (UTC)

"Middle Number" and "Last Number"
In the section describing how to construct a magic square of odd order, the article posted two assertions about the properties of "Middle Number" and "Last Number", namely:

The "Middle Number" is always in the diagonal bottom left to top right.

The "Last Number" is always opposite the number 1 in an outside column or row. Queries. Counterexample for a magic square of order 5:

The "Middle Number" is 13, the "Last Number" is 25, but 13 (Middle Number) is not always in the diagonal bottom left to top right and 25 (Last Number) is not always opposite 1 in an outside column or row for an order 5 (odd order) magic square.

A very good site on magic square, where I found the counterexample from, is http://www.magic-squares.de/general/general.html.

In my opinion, an order n magic square must be a square array of natural numbers from 1 to n2 arranged in a way such that every row, every column, and the two diagonals all sum to a constant. Other so-called magic squares that do not fit to this definition are not considered "magical" and beautiful for me. But again, that's just my opinion.

By the way, this article is hard to follow and seriously needs a lot of decoration, the magic squares look horrible.

Wei Cheng (talk) 10:30, 14 June 2008 (UTC)

 * The section is called "A method for constructing a magic square of odd order". The statements about the middle and last number only apply to magic squares constructed with that method. You are right that there exist other magic squares not having the same property. PrimeHunter (talk) 11:14, 14 June 2008 (UTC)

 * Thank you, PrimeHunter, for pointing out my mistakes. I skimmed the article, so I didn't pay close attention to the wording. I would delete the formulae under that section since it is kind of trivial, insignificant and useless. We are already told an easy method of constructing odd order magic squares, so why bother with the more complicated formulae, which only help locate two numbers? Also, for the history of magic squares, why not just directly link from the Lo Shu Square article, with the pictorial graph of order 3 magic square rather than the modern representation? Wei Cheng (talk) 10:08, 15 June 2008 (UTC)

Siamese method
Here a nice animation of the Siamese method, which I guess could be used also in the main Magic square article. Cheers PHG (talk) 04:05, 19 August 2008 (UTC)

Special Case for 3x3
I was combing the internet trying to find a method to make an arbitrary 3x3 magic square -- not a normalized one. Meaning the sum doesn't have to be 15. I remember in grade school there was a simple method they taught to do it, but I couldn't remember it.

I worked out the trick, or a method. In a 3x3 magic square the center is always 1/3 of the sum. Knowing that it's easy to make an arbitrary magic square. Just choose the top 3 numbers of the square and make sure the sum is a multiple of 3 greater than 15. Say

7 9 2

We know the center square is 6 since the sum is 18. From that you can work out the whole square easily.

Anyway I thought it would be a nice addition to the methods of construction, maybe someone wants to add it in.

David Ashley —Preceding undated comment was added at 03:04, 8 November 2008 (UTC).

Read about this what properties does it have?!?
I saw this "magic square" somewhere, what properties does it have?!? It's not magic! 89 3 34 8 21 55 13 144 5 Professor M. Fiendish, Esq. 03:23, 13 September 2009 (UTC)

A New Interpretation of Odd Magic Squares in the Lo Shu format
FYI A New Interpretation of Odd Magic Squares in the Lo Shu format has been prodded now - see Wikipedia talk:WikiProject Mathematics. Rd232 talk 11:30, 19 September 2009 (UTC)

file:2009.10.23 quaternion square 01.png

·לערי ריינהארט·T·m:Th·T·email me· 23:09, 22 October 2009 (UTC)

Is This a Spam Link?
Should http://Magic-squares.net be linked to this page? Whie posted on Wikipedia's Magic Squares page, this discussion applies to all magic square catagory articles.

The powers-that-be think that this is a SPAM LINK. but
 * It is one of the oldest sites on magic squares on the Internet.(http://www.geocities.com/~harveyh before being forced to move in late October 2009). The new site URL is http://www.magic-squares.net/
 * It is one of the largest, most comprhensive, and well researched sites on magic hypercubes on the Internet (over 70 MB in 4 interconnected sites).
 * It contains extensive bibliographies, historical quotations and examples, statistics, a timeline, etc.
 * It can supply much additional material and many visual examples in support of this article and the other articles in the Magic Squares catagory).

Links to the above site have now all been removed from all magic square catagory articles. Because of how quickly this happened after my posting, I must assume it was done without proper review. Is that fair? I do not wish to get into a discussion of which web sites are better. But I do see existing links on these pages which clearly do not add to the material in the article at hand.

What do you think? --Harvey Heinz (talk) 23:56, 5 November 2009 (UTC)
 * Good site. Definitely not a spam link. Machine Elf 1735 (talk) 00:01, 1 January 2010 (UTC)

incorrect information
I think that some of the information in this article is incorrect. (1) There is no reference to magic squares of planets that I could find in either Agrippa's Three Books of Occult Philosophy or Magic or in Agrippa's Fourth Book of Occult Philosophy. It appears that the information in the wiki article comes from a blog of Mark Swaney. That is certainly not a primary source. If it is indeed from one of the Agrippa books, then that book and the page numbers in the book should be referenced.

(2) The magic square given for the Sun is not the same as that given on page 395 in Budge's book Amulets and Superstitions, Oxford University Press, 1930. —Preceding unsigned comment added by Haralick (talk • contribs) 05:37, 30 May 2010 (UTC)

More "references"
Added the reference Demirörs, Rafraf, and Tanik (1992). Scribbleink (talk) 22:40, 3 October 2010 (UTC)

Simplified explanation of Melancholia I magic square
For this magic square it could be noted that any combination of four numbers that exhibit a radial symmetry about the center add up to 34. In other words any four box locations that are in the same locations when the whole square is rotated 180 degrees or sometimes just 90 degrees will add up to 34. This is of course in addition to the other asymmetrical arrangements such as in columns, rows, and the three-square groupings' corners. — Preceding unsigned comment added by Carlsez (talk • contribs) 17:31, 20 January 2011 (UTC)

Cheating dice
There is a connection between magic squares and cheating dice. For example, use the rows of a 3x3 square as sides of triangular dice A, B and C:

 * A: 8, 1, 6
 * B: 3, 5, 7
 * C: 4, 9, 2

Each dice has the same total number of points. But if you throw any pair of dice:

 * A beats B 5 times out of 9.
 * B beats C 5 times out of 9.
 * C beats A 5 times out of 9.

I have not investigated whether this is true of all additive squares.

nrp@dog-days.co.uk —Preceding unsigned comment added by 87.112.8.225 (talk) 11:30, 16 March 2011 (UTC)

Unsolved problems in Mathematics?
I am not sure if Magic Squares should be in the "Unsolved problems in Mathematics" category.

Bande-Ali (talk) 06:08, 27 June 2011 (UTC)

Geomagic squares
To editor Arthur Rubin: I notice that you reverted my reference to the two new kinds of squares introduced by Lee Sallows, saying, "new generalization is not notable". I want to make a case for geomagic squares, at least, being notable. In Magic squares are given a whole new dimension, The Observer, April 3, 2011 we find
 * Peter Cameron, professor of mathematics at Queen Mary, University of London, calls geomagic squares "a wonderful new piece of recreational maths, which will delight non-mathematicians and give mathematicians food for thought". Despite the inherent whimsy of the field, he said: "I have no doubt that there is serious maths to be done here, too."

and in Ancient puzzle gets new lease of 'geomagical' life, New Scientist, January 24, 2011 we find
 * So could geomagic squares have applications outside the study of puzzles? Cameron certainly thinks so. "You can ask these questions in much more general terms," he says.

See at Peter Cameron's blog for more remarks along these lines.

If you truly feel that geomagic squares are not notable, I will not contest it.

Thank you for your time —  Foobarnix (talk) 03:32, 6 July 2011 (UTC)

 * Seems to be a neologism, which has not yet met the test of time. However, it appears to have appeared in the popular press (although New Scientist has been known to publish press releases as news articles.)  Cameron's speculation on uses or applications are not much more notable than my speculations would be.  I think, perhaps, a sentence in "generalizations" would not be undue weight.
 * Which is about what you added. My only suggestion is to put the New Scientist article as a reference, rather than as an external link.  — Arthur Rubin  (talk) 16:35, 8 July 2011 (UTC)

 * Thank you Arthur Rubin for your thoughtful response. I think you said it is all right to put the reference back in and I have done so.  However, I did remove the mention of Sallows's alphamagic squares which I would agree with you are not mathematically significant.  I cited the Guardian/Observer article rather than New Scientist because I consider it more reliable.  In my opinion, the idea of geomagic squares is more than a neologism—it is a true generalization.  The best case for the significance of this generalization was made by Sallows himself who has said, "Traditional magic squares featuring numbers are then revealed as that particular case of 'geomagic' squares in which the elements are all one-dimensional."  See Lee Sallows website for a more complete explanation of this point. — Foobarnix (talk) 05:09, 9 July 2011 (UTC)

Repetition
The section on the first 5 and 6 order squares is repeated literally. make choice on where to delete 145.100.125.133 (talk) 07:00, 6 December 2011 (UTC)

The section "Cultural Significance" is redundant and unreferenced: it should be removed.91.92.179.172 (talk) 10:05, 19 August 2012 (UTC) It is not certain that magic square have any significance that is not cultural, so this brief section 'cultural significance' was removed:

"Magic squares have fascinated humanity throughout the ages, and have been around for over 4,120 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases."91.92.179.172 (talk) 13:59, 2 October 2012 (UTC)

Related problems
Besides magic squares of primes, there is a longstanding unsolved problem relating a 3x3 magic square of perfect squares. The case 4x4 was solved by Euler in 1770. But please refer to http://www.multimagie.com/English/SquaresOfSquares.htm for more magic square of squares examples.

Also note that http://www.multimagie.com/English/Records.htm shows a table of known multimagic squares, where a square with entries {s(a,b) with 1 <= a,b <= k} is n-multimagic if each square formed by {s(a,b)^i with 1 <= a,b <= k} is magic, for i = 1, ..., n. Bcurfs (talk) 00:55, 4 June 2012 (UTC)

Simple magic square
The page Simple magic square appears in WP:DUSTY and hasn't been edited since 2007 and is unreferenced. What do people think should happen to the article?--Salix (talk): 15:36, 30 September 2012 (UTC)
 * I would say that a merge is in order, with perhaps a slightly higher (earlier in the article) introduction of pandiagonal magic squares. The "simple" adjective doesn't add anything and I've never seen it used (but I am no authority on the subject). Bill Cherowitzo (talk) 17:30, 30 September 2012 (UTC)

Number of magic squares
I managed to calculate the number of 1x1, 2x2, ..., 5x5, etc. size magic squares. I believe that the number of 5x5 is incorrectly posted as 275,305,224. It should be 275,305,810 according to my formulas. If you'd like see the solutions, then please visit my website... www.oddperfectnumbers.com. I'm not trying to break any guidelines, just provide you with the correct answer.99.142.32.118 (talk) 18:07, 16 November 2012 (UTC)

 * According to OEIS, the number of 5×5 magic squares was calculated by Rich Schroeppel, but they don't give a reference. This suggests that the result is unpublished, and it's possible that Schroeppel made a mistake. Perhaps you could contact him and discuss it? —Mark Dominus (talk) 06:48, 17 November 2012 (UTC)

 * You might also want to check a more recent (Schroeppel's work dates to 1973) article which provides a generating function for the number of magic squares. Beck, Cohen, Cuomo and Gribelyuk, The number of "magic" squares, cubes and hypercubes, Amer. Math Monthly 110(2003), 707-717. Bill Cherowitzo (talk) 18:05, 17 November 2012 (UTC)

 * Other computations agree with Rich Schroeppel. A Google search on quickly found, , . I haven't found other reports of your number so I suspect you have an error. Thousands of sources mention the accepted count. Your alternative count should not be mentioned in the article without being accepted by a serious peer-reviewed math journal. PrimeHunter (talk) 01:38, 18 November 2012 (UTC)

 * oh, ok... if all the (programs) agree, then I should doubt my formulas. NO! They all share the same algorithm; I have sent an e-mail to Rich. They reason that I contest his (count) is because, for n=4, 880 has the factor of 11 as does my algebraic formula, and that 11 comes from my generator function "portion". Richard's (computation) does NOT contain the factor 149 which clearly has to be present, since it's in my generator function "portion" for n=5. Our answers are too close not to suppose that Richard's isn't correct and mine could be... I'm 4 for 4... N= 1, 0, 1, 880, and I hope that I'm 5 for 5... N= 275,305,810.99.142.44.250 (talk) 02:53, 18 November 2012 (UTC)

 * I finally did confirm that Richard's computation was correct. It turned out that when 'n' is odd, I have to subract ((n-d)^2)*2^n from my calculation where n= (order) and d= (dimension). we agree at 275, 305, 224. 99.142.24.207 (talk) 19:05, 3 March 2013 (UTC)

Genetic Algorithms
The idea of using genetic algorithms to construct magic squares seems dubious, given that different solutions have unrelated arrangements of numbers. The page used as a reference certainly does not support the idea: I have not seen the java program it provides successfully generate a single 4x4 magic square! — Preceding unsigned comment added by Tyson Jacobs (talk • contribs) 07:40, 15 January 2013 (UTC)

Modulo vs. times
I stand by my good faith changes to the section here on medjig method, and would like to undo the revert by Arthur Rubin (see below). My students here at Canisius College found the current discussion confusing, and helped me word my editing. Here is a summary: The desired square is produced with entries x+9y, where x comes from an N x N square, and y comes from the solution to a medjig puzzle. Referring to "modulo" might be saying that once the desired square is produced, one can recover the x entries by working modulo 9. But it is confusing not to point out that x and y entries must first be found, with the y then multiplied by 9.

18:43, 17 April 2013 Arthur Rubin (talk | contribs). . (56,941 bytes) (-13). . (Reverted good faith edits by 138.92.105.245 (talk): I'm not sure which explanation is more accurate (and/or sourced), but it is really is "modulo". (TW)) (undo) — Preceding unsigned comment added by 138.92.105.245 (talk)

 * I'm still not sure which construction is more clear for the 6×6 square; I'm not qualified to do that, being a long-time mathematician. However, it really is modulo N^2, rather than times N^2.  In fact, the latter is clearly wrong.  — Arthur Rubin  (talk) 18:58, 18 April 2013 (UTC)

Franklin's chess-knight 64-square
Anyone have any idea how Ben Franklin set up his 8x8 magic square in which a chess knight can jump from square 1 to square 64 in sequence? I've been wondering about this ever since I first saw the square--which doesn't seem to be in this article--at the age of 9. Thanks in advance! [signed] FLORIDA BRYAN — Preceding unsigned comment added by 2601:3:1000:5B1:9227:E4FF:FEF0:BBDE (talk) 20:42, 24 April 2014 (UTC)
 * Maybe it is covered at Knight's tour. All the best: Rich Farmbrough, 06:49, 29 April 2014 (UTC).

Quadrasquares
When any kind of prime numbers are plotted (in this example) to 5x5 "quadrasquare", they seem to always exhibit even numbers as opposed to other numbers or integer series which give both even and odd numbers.

The example can be downloaded at http://www.2shared.com/document/NIVoia0b/Quadrasquares_with_primes_and_.html as PDF-file. It demonstrates how a "quadrasquare" is assembled. -VET — Preceding unsigned comment added by 85.156.10.88 (talk) 13:24, 28 October 2014 (UTC)

India
I changed the list of sub-squares to include the corners of 4x2 and 2x4 rectangles, and removed the two special cases that are actually included in the set of these rectangles. I'd appreciate it if someone could review my change to make sure it's appropriate. 50.240.201.161 (talk) 17:40, 18 April 2015 (UTC)Bob S.

Missing "normal"
In the Types and Construction section, it claims that it is impossible to construct a magic square of order 2. I think it should say "normal magic square", since it's trivial to construct an order 2 magic square if duplicate numbers are allowed. Based on this and other statements, such as those referring to the difficulty of construction, I suspect most or all of this section really means "normal magic square" whenever it says "magic square" but I don't have the expertise to fully correct this. Mnudelman (talk) 19:06, 25 April 2015 (UTC)

 * Normal magic squares are those using the integers 1 to n^2. In any magic square, normal or not, the numbers have to be all different, so the statement that squares of order 2 are impossible is correct. 2.25.149.104 (talk) 14:59, 20 April 2016 (UTC)

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Date of publication of Benjamin Franklin's magic square
Does anyone know when Franklin's magic square was published? I know that he constructed them around 1750 and privately communicated them in 1770s. But when did the world know about it? Manoguru (talk) 19:39, 15 December 2018 (UTC)

Governor
I will Do 8 if you Create 2023 i will Make you 2 2409:4056:282:67B6:2C27:4F65:A7DB:9DB6 (talk) 18:30, 17 July 2022 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 20:07, 1 August 2022 (UTC)
 * Euler knight tour semimagic square.svg

So what is the "magic" behind magic squares?
What is the "magic" behind magic squares? Why does it work? Is there a mathematical formula explaining why it works? Ramanujan did one for his birthday (December 22, 1887), and it worked; I did one for my birthday, and it worked. So why? StevenTorrey (talk) 20:35, 8 February 2023 (UTC)
 * Referring to the diagram attached, Latin squares are matrices in which each row and column (but not necessarily diagonal) has exactly one of each item in a set (the background colours in the example). The diagonals of the one shown also have this property.
 * The property is retained when the square is transposed, and also when such squares are added together.
 * One can thus replace each colour in each square with a mathematical expression or constant: D−1, M−4, C−3, Y−2, 1, 4, 3 and 2 to make the top row have the values wanted: D, M, C and Y.
 * Does this explain why it works? cmɢʟee⎆τaʟκ 11:53, 27 July 2023 (UTC)

Help-Editing question
Hello. I don't intend to modify the tag, but I can't find the "This article may be too long to read and navigate comfortably" tag (at the head of the article) in the editor. How can I get at it? MultiScrivner (talk) 02:57, 9 October 2023 (UTC)

The Goat
Hello I am Cristiano Ronaldo

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Magic square


text:
Superellipse

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.

In two dimentional Cartesian coordinate system, a superellipse is defined as the set of all points $$(x,y)$$ on the curve that satisfy the equation$$\left|\frac{x}{a}\right|^n\!\! + \left|\frac{y}{b}\right|^n\! = 1,$$where $$a$$ and $$b$$ are positive numbers referred to as semi-diameters or semi-axes of the superellipse, and $$n$$ is a positive parameter that defines the shape. When $$n=2$$, the superellipse is an ordinary ellipse. For $$n>2$$, the shape is more rectangular with rounded corners, and for $$0<n<2$$, it is more pointed.

In the polar coordinate system, the superellipse equation is (the set of all points $$(r,\theta)$$ on the curve satisfy the equation):$$r = \left(\left|\frac{\cos(\theta)}{a}\right|^n\!\! + \left|\frac{\sin(\theta)}{b}\right|^n\!\right)^{-1/n}\!.$$

Specific cases
This formula defines a closed curve contained in the rectangle $−a ≤ x ≤ +a$ and $−b ≤ y ≤ +b$. The parameters $$a$$ and $$b$$ are the semi-diameters or semi-axes of the curve. The overall shape of the curve is determined by the value of the exponent $$n$$, as shown in the following table:

If $$n<2$$, the figure is also called a hypoellipse; if $$n>2$$, a hyperellipse. When $$n\geq1$$ and $$a=b$$, the superellipse is the boundary of a ball of $$\R^2$$ in the $$n$$-norm. The extreme points of the superellipse are ($$\pm a,0 $$) and ($$0,\pm b$$), and its four "corners" are ($$\pm s_{a}$$,$$\pm s_{b}$$), where $$s = 2^{-1/n}$$ (sometimes called the "superness" ).

Mathematical properties
When n is a positive rational number $$p/q$$ (in lowest terms), then each quadrant of the superellipse is a plane algebraic curve of order $$p/q$$. In particular, when $$a=b=1$$ and n is an even integer, then it is a Fermat curve of degree n. In that case it is non-singular, but in general it will be singular. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.

The curve is given by the parametric equations (with parameter $$t$$ having no elementary geometric interpretation)$$\left. \begin{align} x\left(t\right) &= \plusmn a\cos^{\frac{2}{n}} t \\ y\left(t\right) &= \plusmn b\sin^{\frac{2}{n}} t \end{align} \right\} \qquad 0 \le t \le \frac{\pi}{2} $$where each $$\pm$$ can be chosen separately so that each value of $$t$$ gives four points on the curve. Equivalently, letting $$t$$ range over $$0\le t < 2\pi,$$$$ \begin{align} x\left(t\right) &= {\left|\cos t\right|}^{\frac{2}{n}} \cdot a \sgn(\cos t) \\ y\left(t\right) &= {\left|\sin t\right|}^{\frac{2}{n}} \cdot b \sgn(\sin t) \end{align} $$where the sign function is$$ \sgn(w) = \begin{cases} -1, & w < 0 \\ 0, & w = 0 \\ +1, & w > 0. \end{cases}$$Here $$t$$ is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals $$y/x$$ while in the parametric expressions $\frac{y}{x} = \frac{b}{a} (\tan t)^{2/n} \neq \tan t.$

Area
The area inside the superellipse can be expressed in terms of the gamma function as$$ \mathrm{Area} = 4 a b \frac{\left(\Gamma \left(1+\tfrac{1}{n}\right)\right)^2}{\Gamma \left(1+\tfrac{2}{n}\right)}, $$or in terms of the beta function as
 * $$ \mathrm{Area} = \frac{4 a b}{n} \Beta\!\left(\frac{1}{n},\frac{1}{n}+1\right) . $$

Perimeter
The perimeter of a superellipse, like that of an ellipse, does not admit closed-form solution purely using elementary functions. Exact solutions for the perimeter of a superellipse exist using infinite summations; these could be truncated to obtain approximate solutions. Numerical integration is another option to obtain perimeter estimates at arbitrary precision.

A closed-form approximation obtained via symbolic regression is also an option that balances parsimony and accuracy. Consider a superellipse centered on the origin of a 2D plane. Now, imagine that the superellipse (with shape parameter $$n$$) is stretched such that the first quadrant (e.g., $$x>0$$, $$y>0$$) is an arc from $$(1, 0)$$ to $$(0, h)$$, with $$h \geq 1$$. Then, the arc length of the superellipse within that single quadrant is approximated as the following function of $$h$$ and $$n$$:

This single-quadrant arc length approximation is accurate to within ±0.2% for across all values of $$n$$, and can be used to efficiently estimate the total perimeter of a superellipse.

Pedal curve
The pedal curve is relatively straightforward to compute. Specifically, the pedal of$$\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1,$$is given in polar coordinates by $$(a \cos \theta)^{\tfrac{n}{n-1}}+(b \sin \theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1}}.$$

Generalizations
The generalization of these shapes can involve several approaches.The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.

Higher Dimensions
The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.

 * A superellipsoid extends the superellipse into three dimensions, creating shapes that vary between ellipsoids and rectangular solids with rounded edges. The superellipsoid is defined as the set of all points $$(x,y,z)$$ that satisfy the equation:$$\left|\frac{x}{a}\right|^n\!\! + \left|\frac{y}{b}\right|^n\! + \left|\frac{z}{c}\right|^n\! = 1,$$where $$a,b$$ and $$c$$ are positive numbers referred to as the semi-axes of the superellipsoid, and $$n$$ is a positive parameter that defines the shape.
 * A hyperellipsoid is the $$d$$-dimensional analogue of an ellipsoid (and by extension, a superellipsoid). It is defined as the set of all points $$(x_1,x_2,\ldots, x_d)$$ that satisfy the equation:$$\left|\frac{x_1}{a_1}\right|^n\!\! + \left|\frac{x_2}{a_2}\right|^n\! +\ldots+ \left|\frac{x_d}{a_d}\right|^n\! = 1,$$where $$a_1,a_2,\ldots, a_d$$ are positive numbers referred to as the semi-axes of the hyperellipsoid, and $$n$$ is a positive parameter that defines the shape.

Different Exponents
Using different exponents for each term in the equation, allowing more flexibility in shape formation.

For two-dimentional case the equation is $$\left|\frac{x}{a}\right|^m\!\! + \left|\frac{y}{b}\right|^n\! = 1;m,n>0,$$ where $$m$$ either equals to or differs from $$n$$. If $$m=n$$, it is the Lamé's superellipses. If $$m\neq n$$, the curve possesses more flexibility of behavior, and is better possible fit to describe some experimental information. For the three-dimensional case, three different positive powers $$m$$, $$n$$ and $$p$$ can be used in the equation $$\left|\frac{x}{a}\right|^m\!\! + \left|\frac{y}{b}\right|^n\! + \left|\frac{z}{c}\right|^p\! = 1$$. If $$m=n=p$$, a super-ellipsoid is obtained. If any two or all three powers differ from each other, a solid is obtained that may possess more flexibility in representing real structural data than the super ellipsoid. A three-dimensional super-ellipsoid with $$m=n=2.2$$, $$p=2.4$$ and the semi-diameters $$a=b=1$$,  $$c=0.5$$ represents the structure of the National Centre for the Performing Arts in China.

In the general $$N$$–dimensional case, the equation is $$\left|\frac{x_1}{a_1}\right|^{N_1}\!\! + \left|\frac{x_2}{a_2}\right|^{N_2}\! +\ldots+ \left|\frac{x_N}{a_N}\right|^{N_N}\! = 1$$, where In general, $$n_1,n_2,\ldots, n_N$$ may differ from each other. It is the superellipsoid only if $$n_1=n_2=\ldots= n_N=n$$.

Related shapes
Superquadrics are a family of shapes that include superellipsoids as a special case. They are used in computer graphics and geometric modeling to create complex, smooth shapes with easily adjustable parameters. While not a direct generalization of superellipses, hyperspheres also share the concept of extending geometric shapes into higher dimensions. These related shapes demonstrate the versatility and broad applicability of the fundamental principles underlying superellipses.

Anisotropic Scaling
Anisotropic scaling involves scaling the shape differently along different axes, providing additional control over the geometry. This approach can be applied to superellipses, superellipsoids, and their higher-dimensional analogues to produce a wider variety of forms and better fit specific requirements in applications such as computer graphics, structural design, and data visualization. For instance, anisotropic scaling allows the creation of shapes that can model real-world objects more accurately by adjusting the proportions along each axis independently.

History
The general Cartesian notation of the form comes from the French mathematician Gabriel Lamé (1795–1870), who generalized the equation for the ellipse.

Hermann Zapf's typeface Melior, published in 1952, uses superellipses for letters such as o. Thirty years later Donald Knuth would build the ability to choose between true ellipses and superellipses (both approximated by cubic splines) into his Computer Modern type family.

The superellipse was named by the Danish poet and scientist Piet Hein (1905–1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in Stockholm, Sweden announced a design challenge for a roundabout in their city square Sergels Torg. Piet Hein's winning proposal was based on a superellipse with n = 2.5 and a/b = 6/5. As he explained it: "Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity."

Sergels Torg was completed in 1967. Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc. By rotating a superellipse around the longest axis, he created the superegg, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a novelty toy.

In 1968, when negotiators in Paris for the Vietnam War could not agree on the shape of the negotiating table, Balinski, Kieron Underwood and Holt suggested a superelliptical table in a letter to the New York Times. The superellipse was used for the shape of the 1968 Azteca Olympic Stadium, in Mexico City.

The second floor of the original World Trade Center in New York City consisted of a large, superellipse-shaped overhanging balcony.

Waldo R. Tobler developed a map projection, the Tobler hyperelliptical projection, published in 1973, in which the meridians are arcs of superellipses.

The logo for news company The Local consists of a tilted superellipse matching the proportions of Sergels Torg. Three connected superellipses are used in the logo of the Pittsburgh Steelers.

In computing, mobile operating system iOS uses a superellipse curve for app icons, replacing the rounded corners style used up to version 6.

title: Superellipse


text:
Original Dixieland Jass Band

The Original Dixieland Jass Band (ODJB) was a Dixieland jazz band that made the first jazz recordings in early 1917. Their "Livery Stable Blues" became the first jazz record ever issued. The group composed and recorded many jazz standards, the most famous being "Tiger Rag". In late 1917, the spelling of the band's name was changed to Original Dixieland Jazz Band.

The band consisted of five musicians who had played in the Papa Jack Laine bands.

ODJB billed itself as the Creators of Jazz. It was the first band to record jazz commercially and to have hit recordings in the genre. Band leader and cornetist Nick LaRocca argued that ODJB deserved recognition as the first band to record jazz commercially and the first band to establish jazz as a musical idiom or genre.

Origins
In early 1916, a promoter from Chicago approached clarinetist Alcide Nunez and drummer Johnny Stein about bringing a New Orleans-style band to Chicago, where the similar Brown's Band From Dixieland, led by trombonist Tom Brown, was enjoying success. They then assembled trombonist Eddie Edwards, pianist Henry Ragas, and cornetist Frank Christian. Shortly before they were to leave, Christian backed out, and Nick LaRocca was hired as a last-minute replacement.

On March 3, 1916 the musicians began their job at Schiller's Cafe in Chicago under the name Stein's Dixie Jass Band. The band was a hit and received offers of higher pay elsewhere. Since Stein as leader was the only musician under contract by name, the rest of the band broke off, sent to New Orleans for drummer Tony Sbarbaro, and on June 5, started playing under the name, The Dixie Jass Band. LaRocca and Nunez had personality conflicts, and on October 30 Tom Brown's Band and ODJB agreed to swap clarinetists, bringing Larry Shields into the Original Dixieland Jass Band. The band attracted the attention of theatrical agent Max Hart, who booked the band in New York City. At the start of 1917 the band began an engagement playing for dancing at Reisenweber's Cafe, on Columbus Circle, in Manhattan.

First recordings

While a couple of other New Orleans bands had passed through New York City slightly earlier, they were part of vaudeville acts. ODJB, on the other hand, played for dancing and hence, were the first "jass" band to get a following of fans in New York and then record at a time when the American recording industry was essentially centered in the northeastern United States, primarily in New York City and Camden, New Jersey.

Shortly after arriving in New York, a letter dated January 29, 1917, offered the band an audition for the Columbia Graphophone Company. The session took place on Wednesday, January 31, 1917. Nothing from this test session was issued.

The band then recorded two sides for the Victor Talking Machine Company, "Livery Stable Blues" and "Dixieland Jass Band One-Step", on February 26, 1917 at Victor's New York studios. These titles were released as Victor 18255 in May 1917, the first issued jazz record. The band's recordings, first marketed as a novelty, were a surprise hit, and gave many Americans their first taste of jazz. Musician Joe Jordan sued, since the "One Step" incorporated portions of his 1909 ragtime composition "That Teasin' Rag". The record labels subsequently were changed to "Introducing 'That Teasin' Rag' by Joe Jordan". A court case dispute over the authorship of "Livery Stable Blues" resulted in the judge declaring the tune in the "public domain".

In the wake of the group's success of the Victor record, the ODJB returned to Columbia in May, recording two selections of popular tunes of the day chosen for them by the label (possibly hoping to avoid the copyright problems which arose after Victor recorded two of the band's supposedly original compositions) "Darktown Strutters' Ball" and "(Back Home Again in) Indiana" as catalogue #A-2297.

Numerous jazz bands were formed in the wake of the success of ODJB that copied and replicated its style and sound. Also bands were brought from Chicago and California (such as the Frisco Jass Band) in an attempts to join the jazz craze. Established bands of different types and bandleaders such as Wilbur Sweatman began billing their groups as "jass" or "jazz" bands. Earl Fuller, bandleader at a competing New York venue, was ordered by management to form a "jass" band.

W. C. Handy recorded one of the earliest cover versions of an ODJB tune when he released a recording of "Livery Stable Blues" by Handy's Orchestra of Memphis for Columbia in 1917.

In 1918, the song "When You Hear That Dixieland Jazz Band Play" by Shelton Brooks, "the King of Ragtime Writers", was published by Will Rossiter in Chicago. It was a tribute to the Original Dixieland Jazz Band, who were featured on the cover.

Original New Orleans Jazz Band
When the New Orleans Jazz style swept New York by storm in 1917 with the arrival of the Original Dixieland Jazz Band, Jimmy Durante was part of the audience at Reisenweber's Cafe on Columbus Circle when ODJB played that venue. Durante was very impressed with the band and invited them to play at a club called the Alamo in Harlem where Jimmy played piano.

Durante had his friend, Johnny Stein (the previous drummer and leader of the group), assemble a group of like-minded New Orleans musicians to accompany his act at the Alamo. Stein did so, with a band consisting of fellow veterans of the Laine bands in New Orleans, other than pianist Durante. In late 1918 they recorded two sides for Okeh under the name of the New Orleans Jazz Band. They recorded the same two numbers a couple of months later for Gennett under the name of Original New Orleans Jazz Band, and in 1920 the same group recorded again for Gennett as Jimmy Durante's Jazz Band. They later billed themselves as "Durante's Jazz and Novelty Band".

Sicilian influence
Both LaRocca and Sbarbaro were children of immigrants from the Italian region of Sicily. The Sicilian capital of Palermo had long held cotton and citrus fruit trade with New Orleans. This resulted in the establishment of a direct shipping line between the two port cities which enabled a vast number of Sicilians to migrate to New Orleans, and other American cities, between the late 1800s and early 1900s. With this migration, Sicilian sound was brought to New Orleans and integrated with regional African-American music. The band would capitalize on this growing integration.

Later history of the band
After their initial recording for the Victor Company, the ODJB recorded for Columbia Records (after the first Victor session, not before as has sometimes been reported) and Aeolian-Vocalion in 1917, then returned to Victor the following year, while enjoying continued popularity in New York. Trombonist Edwards was drafted for World War I in 1918 and replaced by Emile Christian, and pianist Henry Ragas died of influenza in the 1918 flu pandemic the following year and he was replaced by pianist and composer J. Russel Robinson.

Robinson's compositions for the band recorded and released in 1920, include the classic "Margie" and "Palesteena (Lena from Palesteena)", were among the most popular and best-selling hits of 1920. "Aggravatin' Papa" was composed with lyricist Roy Turk and Addie Britt and was recorded by Alberta Hunter in 1923 with Fletcher Henderson's Dance Orchestra and also by Bessie Smith, Sophie Tucker, Florence Mills, Lucille Hegamin, and Pearl Bailey. Robinson also collaborated with Roy Turk on the compositions "Sweet Man O' Mine" and "A-Wearin' Away the Blues", and he wrote "Mama Whip! Mama Spank! (If Her Daddy Don't Come Home)" for blues and jazz singer Mamie Smith and her Jazz Band in 1921, which were released on the Okeh label. Robinson was a member of the band until it broke up in 1923. He rejoined the band when it reformed in 1936.

"Margie", composed by J. Russel Robinson with Con Conrad, with lyrics added by Benny Davis, has been covered over a hundred times. "Margie" has been recorded by Louis Armstrong, who also covered the band's "Tiger Rag", Ray Charles, Al Jolson, Duke Ellington and His Orchestra in 1935, the Billy Kyle Swing Club Band, Claude Hopkins, Red Nichols, Django Reinhardt, George Paxton, the Dutch Swing College Band, Fats Domino, Sidney Bechet, Don Redman, Cab Calloway, Jim Reeves, Gene Krupa, and Benny Goodman.

"Margie" was a no. 9 hit for ODJB in 1921 with J. Russel Robinson on piano. Eddie Cantor had the biggest hit version of the ODJB classic, spending five weeks at no. 1 in 1921. The song also was featured in the movie The Eddie Cantor Story and was the theme of the television series of the same name in 1961–1962. Cantor also recorded ODJB's "Palesteena (Lena from Palesteena)". Gene Rodemich and His Orchestra reached no. 7 with their version in 1920. Ted Lewis and His Band reached no. 4 in 1921. Frank Crumit had a no. 7 hit in 1921. Claude Hopkins and His Orchestra reached no. 5 in 1934 with Orlando Peterson on vocals. Don Redman and His Orchestra got to no. 15 in 1939 with a cover of the ODJB song. Dave Brubeck, Bix Beiderbecke, Bing Crosby, Jo Stafford, Erroll Garner, Oscar Peterson, Charlie Shavers, Jimmy Smith, Joe Venuti, Ray Barretto, and Shelly Manne also have recorded the song. Jimmie Lunceford recorded the song in 1938 with a Sy Oliver arrangement that featured Trummy Young.

London tour
Other New Orleans musicians, including Nunez, Tom Brown, and Frank Christian, followed ODJB's example and went to New York to play jazz as well, giving the band competition. LaRocca decided to take the band to London, where they would once again enjoy being the only authentic New Orleans jazz band in the metropolis, and again present themselves as the Originators of Jazz because they were the first band to record the new genre of music dubbed jass or jazz. The band's April 7, 1919 appearance in the revue Joy Bells at the London Hippodrome was the first official live jazz performance by any band in the United Kingdom and was followed by a command performance for King George V at Buckingham Palace. The concert did not start auspiciously, with the assembled aristocracy, which included French Marshal Philippe Pétain, peering through opera glasses at the band "as though there were bugs on us", according to LaRocca. The audience loosened up, however, after the king laughed and loudly applauded their rendition of "The Tiger Rag". The British tour ended with the band being chased to the Southampton docks by Lord Harrington, who was infuriated that his daughter was being romanced by the lead singer of the band. In London, they made twenty more recordings for the British branch of Columbia. While in London, they recorded the second, more commercially successful, version of their hit song "Soudan" (also known as "Oriental Jass").

The band returned to the United States in July 1920 and toured for four years. This version of the band played in a more commercial style, adding a saxophone to the arrangements in the manner of other popular orchestras. In 1927 LaRocca was replaced by 19-year-old trumpeter Henry Levine, who later brought this kind of repertoire to the NBC radio show The Chamber Music Society of Lower Basin Street. Jazz pianist and composer Frank Signorelli, who collaborated on the jazz standards "A Blues Serenade", recorded by Glenn Miller and Duke Ellington, "Gypsy", and "Stairway to the Stars", joined ODJB for a brief time in 1921.

Break-up
The band broke up in the late 1920s and its originators scattered. During the Depression, trombonist Eddie Edwards was discovered operating a newsstand in New York City. Newspaper publicity resulted in Edwards fronting a local nightclub band.

In 1936, the musicians played a reunion performance on network radio. Victor invited them back into the recording studio, and over the next two years the band recorded 25 sides for Victor  as "The Original Dixieland Five." The group toured briefly before disbanding again. Clarinetist Larry Shields received particularly positive attention on this tour, and Benny Goodman has commented that Shields was an important early influence.

In the 1940s and 1950s, Edwards and Sbarbaro both formed bands without other original members under the ODJB name; Teddy Roy was one of the players in Edwards's version of the band. In 1944, a new version of "Tiger Rag" was released as V-Disc 214 by the reformed band. "Sensation Rag" also was released as V-Disc 214B2. V-Discs were non-commercial recordings issued only to the U.S. armed forces.

Back in New Orleans, LaRocca licensed bandleader Phil Zito to use the ODJB name for many years. Nick LaRocca's son, Jimmy LaRocca, continues to lead bands under the name The Original Dixieland Jazz Band.

In 1960 the book, The Story of the Original Dixieland Jazz Band, was published. Writer H. O. Brunn based it on Nick LaRocca's recollections, which sometimes differ from that of other sources.

Influence
ODJB was the first band to record jazz successfully, establishing and creating jazz as a new musical idiom and genre of music. Bix Beiderbecke was influenced by the ODJB to become a jazz musician and was heavily influenced by Nick LaRocca's cornet and trumpet style. Louis Armstrong acknowledged the importance of ODJB:

"Only four years before I learned to play the trumpet in the Waif's Home, or in 1909, the first great jazz orchestra was formed in New Orleans by a cornet player named Dominick James LaRocca. They called him 'Nick' LaRocca. His orchestra had only five pieces but they were the hottest five pieces that had ever been known before. LaRocca named this band 'The Old Dixieland Jass Band'. He had an instrumentation different from anything before, an instrumentation that made the old songs sound new. Besides himself at the cornet, LaRocca had Larry Shields, clarinet, Eddie Edwards, trombone, Ragas, piano, and Sbarbaro, drums. They all came to be famous players and the Dixieland Band has gone down now in musical history."

Film appearances
In 1917, the band made the first appearance of a jazz band in a motion picture, a silent movie entitled, The Good for Nothing (1917), directed by Carlyle Blackwell, who also played the lead role as Jack Burkshaw. Written by Alexander Thomas, it also featured Evelyn Greeley and Kate Lester and was produced by William Brady. Nick LaRocca, Larry Shields, Tony Sbarbaro, and Henry Ragas appeared in the film as a band, with LaRocca on cornet, Shields on clarinet, Ragas on piano, and Sbarbaro on drums. The film was released on December 10, 1917, produced by Peerless Productions, and distributed by World Pictures.

Nick LaRocca and the reunited Original Dixieland Jass Band performed "Tiger Rag" in The March of Time newsreel segment titled "Birth of Swing," released to U.S. theaters February 19, 1937.

"Tiger Rag"
The band's 1917 composition "Tiger Rag" became one of the most popular and ubiquitous of jazz standards. There were 136 cover versions of ODJB's copyright jazz standard and classic "Tiger Rag" by 1942. It has been standard ever since.

Their first release, "Livery Stable Blues", featured instruments doing barnyard imitations and the fully loaded trap set, wood blocks, cowbells, gongs, and Chinese gourds. This musical innovation represented one of the first experimental exercises in jazz. At the time, their music was liberating; the barnyard sounds were experiments in altering the tonal qualities of the instruments, and clattering wood blocks broke up the rhythm. The music was very lively when compared to the pop music of the time.

Many of the tunes first composed and recorded by the Original Dixieland Jass Band, such as "Tiger Rag" and "Margie", were recorded by many of the major jazz bands and orchestras of the twentieth century, black and white. "Tiger Rag" was recorded by many artists, from Louis Armstrong to Duke Ellington to Glenn Miller to Benny Goodman. "Tiger Rag", in particular, became popular with many colleges and universities having a tiger as a mascot. In the biography John Coltrane: His Life and Music, published in 1999, Lewis Porter noted that ODJB's classic, "Margie", was a "specialty" of John Coltrane, a song he performed regularly in his early career. "Tiger Rag", "Margie", "Clarinet Marmalade", "At The Jazz Band Ball", "Sensation Rag", and "Fidgety Feet" remain much played classics in the repertory of contemporary Dixieland and traditional jazz bands. Their tunes were published as collaborations by some or all of the entire ensemble, including band leader Nick La Rocca.

The Original Dixieland Jazz Band recording of "Tiger Rag" was no. 1 for two weeks on the U.S. Hit Parade charts beginning on December 11, 1918. The Mills Brothers recorded "Tiger Rag" in 1931 with lyrics and spent four weeks at no. 1 on the charts in 1931–1932 with their version of the ODJB song.

The Eddie Edwards composition "Sensation Rag" (aka "Sensation") was performed at the 1938 landmark Benny Goodman jazz concert at Carnegie Hall released on the album The Famous 1938 Carnegie Hall Jazz Concert.

Compared to later jazz, the ODJB recordings have only modest improvisation in mostly ensemble tunes. Clarinetist Larry Shields is perhaps the most interesting player, showing a good fluid tone, and if his melodic variations and breaks now seem overly familiar, this is because they were imitated widely by musicians who followed in the band's footsteps.

Their concept of arrangement was somewhat limited, and their recordings can seem rather repetitive. The lack of a bass player is scarcely compensated for by the piano on their earlier, acoustically recorded sessions. Nonetheless, ODJB arrangements were wild, impolite, and definitely had a jazz feel, and that style still is referred to as the style of music known as Dixieland.

Covers
ODJB's songs were recorded by other musicians, such as Fletcher Henderson and His Orchestra, one of the most popular and influential jazz bands of the 1920s.
 * "Beale Street Mama" – by J. Russel Robinson, recorded by Henderson in 1923 as an instrumental on Paramount.
 * "Clarinet Marmalade" – recorded in 1926, released on Vocalion and Brunswick. In 1931, Henderson recorded a new version for Columbia.
 * "Livery Stable Blues" – recorded in 1927, released on Columbia
 * "Fidgety Feet" – by Nick LaRocca, recorded in 1927, released on Vocalion
 * "Sensation" – recorded in 1927, released on Vocalion
 * "Tiger Rag" – recorded in 1931, released on Crown
 * "Aggravatin' Papa" – collaboration with Robinson, recorded in 1923 with Alberta Hunter on vocals
 * "Singin' the Blues (Till My Daddy Comes Home)" – recorded in 1931 with Rex Stewart on cornet

Bix Beiderbecke recorded nine compositions associated with the ODJB from 1924 to 1930: "Fidgety Feet", his first recording in 1924, "Tiger Rag", "Sensation", "Lazy Daddy", "Ostrich Walk", "Clarinet Marmalade", "Singin' the Blues" with Frankie Trumbauer and Eddie Lang, "Margie", and "At the Jazz Band Ball".

Recordings
The band's seminal 78-rpm recordings include the following (on Victor, Columbia, and Aeolian Vocalion):

1917–1920

 * 1) "Dixie Jass Band One-Step"/"Introducing That Teasin' Rag"/"Livery Stable Blues", 1917, Victor 18255. This was the second pressing. The original title of the A side was "Dixieland Jass Band One-Step".
 * 2) "At the Jazz Band Ball"/"Barnyard Blues", 1917, Aeolian Vocalion A1205
 * 3) "Ostrich Walk"/"Tiger Rag", 1917, Aeolian Vocalion A1206
 * 4) "Reisenweber Rag/Look at 'Em Doing It Now", 1917, Aeolian Vocalion 1242
 * 5) "Darktown Strutters' Ball"/"(Back Home Again in) Indiana", 1917, Columbia A2297; the ODJB recording of "Darktown Strutters' Ball" was inducted into the Grammy Hall of Fame on February 8, 2006
 * 6) "At the Jazz Band Ball" (1918 version)/"Ostrich Walk" (1918 version), 1918, Victor 18457
 * 7) "Skeleton Jangle"/"Tiger Rag" (1918 version), 1918, Victor 18472
 * 8) "Bluin' the Blues"/"Sensation Rag", 1918, Victor 18483
 * 9) "Mournin' Blues"/"Clarinet Marmalade", 1918, Victor 18513, "Mournin' Blues" also appeared as "Mornin' Blues" on some releases. The full B side title was "Clarinet Marmalade Blues".
 * 10) "Fidgety Feet (War Cloud)"/"Lazy Daddy", 1918, Victor 18564
 * 11) "Lasses Candy"/"Satanic Blues", 1919, Columbia 759
 * 12) "Oriental Jazz" (or "Jass"), 1919, recorded November 24, 1917 and issued as Aeolian Vocalion 12097 in April 1919 with "Indigo Blues" by Ford Dabney's Band
 * 13) "At the Jazz Band Ball" (1919 version)/"Barnyard Blues" (1919 version), 1919, recorded in London, England, April 16, 1919, English Columbia 735
 * 14) "Soudan" (also known as "Oriental Jass" and "Oriental Jazz"), 1920, recorded in London, England, in May 1920 and released as English Columbia 829; was composed by Czech composer Gabriel Sebek in 1906 as "In the Soudan: A Dervish Chorus" or "Oriental Scene for Piano, Op. 45". The B side was "Me-Ow" by the London Dance Orchestra
 * 15) "Margie"/"Singin' the Blues"/"Palesteena", 1920, Victor 18717
 * 16) "Broadway Rose"/"Sweet Mama (Papa's Getting Mad)"/"Strut, Miss Lizzie", 1920, Victor 18722

1921–1946

 * 1) "Home Again Blues"/"Crazy Blues"/"It's Right Here For You (If You Don't Get It, Tain't No Fault O' Mine)", 1921, Victor 18729
 * 2) "Tell Me/Mammy o' Mine", 1921, recorded in the UK and released as Columbia 804
 * 3) "I'm Forever Blowing Bubbles"/"My Baby's Arms", 1921, Columbia 805
 * 4) "I've Lost My Heart in Dixieland"/"I've Got My Captain Working for Me Now", 1921, Columbia 815
 * 5) "Sphinx/Alice Blue Gown", 1921, Columbia 824
 * 6) "Jazz Me Blues/St. Louis Blues", 1921, Victor 18772
 * 7) "Royal Garden Blues"/"Dangerous Blues", 1921, Victor 18798
 * 8) "Bow Wow Blues (My Mama Treats Me Like a Dog)", 1922, Victor 18850. The B side was "Railroad Blues" by the Benson Orchestra of Chicago under pianist and composer Roy Bargy
 * 9) "Toddlin' Blues"/"Some of These Days", 1923, Okeh 4738
 * 10) "You Stayed Away Too Long/Slipping Through My Fingers", 1935, Vocalion 3099
 * 11) "Original Dixieland One-Step/Barnyard Blues" (new version of "Livery Stable Blues"), 1936, Victor 25502
 * 12) "Who Loves You?"/"Did You Mean It?", 1936, Victor 25420, featuring vocals by Chris Fletcher and trumpet by Nick LaRocca
 * 13) "Old Joe Blade", 1936, Victor 26039-B, Nick LaRocca and the Original Dixieland Band featuring J. Russel Robinson on vocals
 * 14) "Good-Night, Sweet Dreams, Good-Night"/"In My Little Red Book", 1938, Bluebird B-7444, which featured vocals by Lola Bard
 * 15) "Tiger Rag" (1943 version), 1944, V-Disc 214B1, issued June, 1944, with Eddie Edwards and Tony Sbarbaro
 * 16) "Sensation" (1943 version), 1944, V-Disc 214B2, with Eddie Edwards and Tony Sbarbaro
 * 17) "Shake It and Break It"/"When You and I Were Young, Maggie", 1946, Commodore C-613

Soundtracks
The soundtrack album to the 2011 Boardwalk Empire series on HBO includes performances of three songs recorded by the Original Dixieland Jazz Band: "Livery Stable Blues", "Mournin' Blues", and "Margie", performed by Vince Giordano and the Nighthawks Orchestra. The soundtrack won the Grammy Award for Best Compilation Soundtrack for Visual Media at the 54th Grammy Awards.

Honors
In 1977, the ODJB classic "Singin' the Blues", co-written by ODJB pianist J. Russel Robinson, was inducted into the Grammy Hall of Fame in a landmark 1927 recording by Frankie Trumbauer and His Orchestra featuring Bix Beiderbecke on cornet and Eddie Lang on guitar, as Okeh 40772-B, recorded on February 4, 1927.

On April 3, 1992, the City Council of New Orleans issued a proclamation honoring the members of the band. In 2003, the 1918 ODJB recording of "Tiger Rag" was placed on the U.S. Library of Congress National Recording Registry. In 2006, the Original Dixieland Jazz Band's recording of "Darktown Strutters' Ball", released in 1917 as Columbia single A2297, was inducted into the Grammy Hall of Fame.

The Original Dixieland Jazz Band was posthumously inducted into the Grammy Hall of Fame, which is a special Grammy award established in 1973 to honor recordings that are at least 25 years old and that have "qualitative or historical significance."

title: Original Dixieland Jass Band


text:
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I'm kragen@canonical.org; my home page is at http://canonical.org/~kragen/

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I tend to edit computer science articles a lot; I've been working on algorithm lately.

I really like Wikipedia. I've written a bunch of computer-science articles for it.

I'm worried about the stuff The Cunctator brings up about the GFDL, although I wish he would be more polite about it.

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Thanks. I've been reading the Wikipedia-L archives and considering joining. -- Kragen

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User:Kvikeg

Hi, my name is Philip Derbeko and i'm from Israel. Here is my homepage (if you want to spend some spare time) : I think this is a really cool project.

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title: User:Kvikeg