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Till Eulenspiegel's funny Series

This wikibook deals with the striking similarity in the ratios of the vibrational frequencies of the Balmer Series with the ratios of the sound frequencies of the opening motif of the symphonic poem Till Eulenspiegels lustige Streiche by  (* 1864; † 1949).

There is no doubt that not only is superior musicality a prerequisite for such feats of compositional transfer, but that Richard Strauss also possessed the necessary genius and ingenuity to create such high-level works from basic physical material. It is based on a previous publication in the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Introduction

 is said to have been a roving cunning rogue in the 14th century, who played dumb and played many tricks on his fellow men.

The Balmer series, named after the Swiss mathematician and physicist  (* 1825; † 1898), describes a sequence of spectral lines in the spectrum of hydrogen atoms that can be described by specific frequencies or wavelengths of electromagnetic radiation and of which five lines are in the range of visible light. These were first detected in sunlight, since the chemical element hydrogen is the main component of stars and sunlight can be studied very well because of its great brightness. Later, the hydrogen lines were also detected in the light of bright stars, such as the brightest star in the night sky, namely Sirius (α Canis Majoris) in the constellation of the Great Dog.

At the time the composition was written, the composers Richard Strauss and the mathematician and composer  (* 1837; † 1922, actually Hans Zincken called Sommer), who was well versed in optics, became friends in Weimar. It could therefore well be that Richard Strauss learned about current physical discoveries at the time via his fatherly friend Hans Sommer, and thus also about the Balmer optical series. With its help, he could have used the frequencies of the five visible spectral lines transferred to sound frequencies - quasi mischievously - for the five notes c - f - g - gis - a of the opening motif of his symphonic poem Till Eulenspiegels lustige Streiche.

The Balmer series
The atoms or molecules of a gas can be excited by adding energy. In this process, the electrons in the atomic shell are brought to higher discrete energy levels. Through spontaneous emission, the electrons eventually fall back to a lower energy level by chance, whereby the energy released in the process is emitted as electromagnetic radiation in the form of a photon, whose oscillation frequency $$\nu$$ can be expressed via the speed of light $$c$$, which has a fixed magnitude of 299792458 metres per second:

 * $$\nu = \frac {c} {\lambda}$$

Here $$\lambda$$ is the wavelength of the light particle emitted by the atom, and each wavelength corresponds to a particular saturated colour. White light has a continuous spectrum with practically all visible wavelengths:

.

If atoms are excited by light, the photons with the wavelengths matching the energy levels are destroyed (absorbed) in the process, and a dark line can be seen in the corresponding absorption spectrum at the associated wavelength. If a gas is excited to form a plasma, photons are only produced (emitted) at the wavelengths that match the energy levels, and these can then be seen as bright lines in the emission spectrum.

The position of these lines on the screen of an optical spectrometer depends on the angles of refraction of the glass prism used or on the properties of the diffraction grating used, which depend on the wavelength of the light examined. The angles of refraction achieved are thus determined by the geometric arrangement in the optical devices used.

These wavelengths are also known from the absorption spectrum of the light of the stars, first and foremost our sun, as they consist to a very large extent of hydrogen and have a high energy turnover. The English physician, physicist and chemist  (* 1766; † 1828) was the first to describe such dark lines in the solar spectrum in 1802. They were also described independently of him in 1814 by the German optician and physicist  (* 1787; † 1826) and systematically investigated.

When the excited electron is in a hydrogen atom and passes from a higher energy level $$E_m$$ to the second lowest energy level $$E_2$$, the wavelengths $$\lambda_{m,2}$$ of the Balmer series result. The spectral lines of this series are named after the Swiss mathematician and physicist Johann Jakob Balmer (* 1825; † 1898), who was able to present the mathematical regularity 1884 by means of his empirically found, generalised Balmer formula for the wavelengths $$\lambda_{m,n}$$ in the Verhandlungen der Naturforschenden Gesellschaft in Basel:

 * $$\lambda_{m,n} = A \left( \frac {m^2} {m^2 - n^2} \right)$$ mit $$m > n$$ und $$n \ge 1$$

Here $$A = \lambda_{\infty,n} \approx 364.506820 \text{ nanometres}$$ is a constant wavelength in the ultraviolet. Balmer called it the fundamental number of hydrogen and gave its value as 3645.6 angstroms and 364.56 nanometres respectively, slightly higher due to the measurement accuracies of the time.

However, the existence of energy levels was not yet known at that time and the contemporary researchers were electrified by the lack of explanation. Johann Jakob Balmer wrote about this in his publication:

''Es sind besonders die numerischen Verhältnisse der Wellenlängen der ersten vier Wasserstofflinien, welche die Aufmerksamkeit reizen und fesseln. Die Verhältnisse dieser Wellenlängen lassen sich nämlich überraschend genau durch kleine Zahlen ausdrücken.''

Der Unterschied zwischen den berechneten und beobachteten Wellenlängen ist so klein, dass die Übereinstimmung im höchsten Grade überraschen muss.

Aus diesen Vergleichungen ergibt sich [...], dass die Formel auch für die fünfte [...] Wasserstofflinie zutrifft.

The coefficients $$k_{m,2}$$ of the Balmer series, which result from the bracket expression for $$n = 2$$, were also given by Johann Jakob Balmer:

 * $$k_{m,2} = \frac {m^2} {m^2 - 4}$$

These coefficients $$k_{m,2}$$ and the associated wavelengths $$\lambda_{m,2}$$ read with the additional definition $$r_{3} := 1$$ for the first five spectral lines of the Balmer series with $$3 \le m \le 7$$:

In his publication, Balmer gave, among other things, the wavelengths determined by  (* 1814; † 1874) in his work on the solar spectrum of 1862 experimentally determined wavelengths for the visible hydrogen lines:

Alternatively, these series can also be calculated with the Rydberg constant $$R_\infty = \frac {4} {A} \approx 1.09737316 \cdot 10^{7}\,\mathrm{m^{-1}}$$ for the frequencies $$\nu_{m,n}$$:

 * $$\nu_{m,n} = c \cdot R_\infty \left( \frac {1} {n^2} - \frac {1} {m^2} \right)$$ mit $$m > n$$ und $$n \ge 1$$

The frequency $$\nu_{A,2}$$, which results with the fundamental number of hydrogen for $$m \rightarrow \infty$$ in the Balmer series with $$n = 2$$, can therefore be calculated as follows:

 * $$\nu_{A,2} = \nu_{\infty,2} = c \cdot R_\infty \frac {1} {4} \approx 8.224605 \cdot 10^{14} \, \text{Hertz}$$

Till Eulenspiegel's Merry Pranks

Till Eulenspiegel's Merry Pranks was composed nine years after the discovery of the Balmer series by the German composer Richard Strauss (* 1864; † 1949). It is a tone poem for large orchestra with a performance duration of fifteen minutes, written between 1893 and 1894 after Richard Strauss had spent several months in Greece and Egypt recovering from the late effects of pneumonia and after his opera Guntram had been completed.

Originally, Richard Strauss had apparently planned to have the work begin immediately with the well-known and striking six-act Till Eulenspiegel motif. The five-bar prologue was only added by Richard Strauss as the beginning of the composition in the course of the revision. In the course of the composition, the Till-Eulenspiegel motif only reappears at the end.

The prologue was finally titled by the composer with the words "Es war einmal ein Schalknarr..." and "leisurely", the Till Eulenspiegel motif that immediately follows with the words "...Namens Till Eulenspiegel" and "allmählich lebhafter". .

The beginning of the composition is notated in F major, and the five notes of the Till Eulenspiegel motif are c - f - g - gis - a. They are repeated twice:

The symphonic poem Till Eulenspiegels lustige Streiche bears the opus number 28, and it was premiered under the direction of the German composer Franz Wüllner (* 1832; † 1902) on 5 November 1895 in Cologne.

Thus Spoke Zarathustra
In this context, it is noteworthy that the symphonic poem by Richard Strauss with the opus number 30, Also sprach Zarathustra, which was composed shortly afterwards, also has very clear references to physical number relationships and uses an opening motif with five tones.

The opening motif used in the introduction Sunrise is also derived from a physical number series, namely the natural tone series, whereby its first, second, third, fourth and fifth tones with the frequency ratios 1:1, 2:1, 3:2, 4:3 and 5:4 play a central role in relation to the fundamental.

Strauss again uses C as the basic note, which is held out as an octave on upper C and lower C, initially for four bars in the double basses, organ pedal and contrabassoon, before the four C trumpets join in with their distinctive ascending motif. For his trumpet motif, Richard Strauss uses the next four notes of the natural tone row c', g', c and e, so that finally a radiant C major chord is heard, played almost by the entire orchestra, which is immediately changed to C minor by the alteration of the third tone. After a brief triplet interlude by the timpani with the two fourths G and C, the trumpet motif is heard again with an inverted chord sequence of C minor and C major. As in the symphonic poem Till Eulenspielgel's Merry Pranks, the motif is played a third time and then developed further, whereby the built-up C major chord can be interpreted this time as the dominant to the F major that follows. After four bars, the introduction closes with a renewed and sustained C major in fortissimo, played by all instruments.

In the further course of the orchestral piece, Richard Strauss also plays with the number twelve, which is significant in astronomy, in the section The convalescent by introducing a twelve-tone row.

In the last section of the composition, entitled Nachtwandlerlied, he uses twelve midnight chimes. The twelve not only has close astronomical connections to the division of the night into the twelve night hours, but also to the twelve night signs of the ecliptic visible at night, which are traversed by the seven changing stars and in which the planet Jupiter resides for one year as seen from Earth.

This symphonic tone poem was premiered at the end of November 1896 by Richard Strauss in Frankfurt am Main.

There are only a few other physical number series that would be suitable for the pitches of musical motifs. Furthermore, it should be noted that Richard Strauss was neither a pioneer nor a supporter of twelve-tone music, so that the fact that he nevertheless works with it can be seen as symbolic and definitely also a little mischievous.

Three songs after poems by Otto Julius Bierbaum
Incidentally, the three poems Traum durch die Dämmerung, Schlagende Herzen and Nachtgang of the composition Drei Lieder nach Gedichten von Otto Julius Bierbaum from 1895 with the opus number 29, which was created between the two symphonic poems, also deal with the astronomical objects sun, moon and stars and their observation in nature.
 * Nummer 1: Traum durch die Dämmerung (Fis-Dur / B-Dur / Fis-Dur, sehr ruhig, 2/4-Takt)
 * ... im Dämmergrau, die Sonne verglomm, die Sterne ziehn,
 * ... in ein blaues, mildes Licht.
 * ... in ein mildes, blaues Licht.
 * Nummer 2: Schlagende Herzen (G-Dur, lebhaft und heiter, Allegro giocoso, 2/4-Takt)
 * ... du gold'ne Sonne in Himmelshöhn! (in strahlendem C-Dur)
 * Nummer 3: Nachtgesang (c-Moll, mäßig langsam, 3/4-Takt)
 * Der Mond goss silbernes Licht ...
 * ... rein wie die liebe Sonne.

Intervals
The light frequencies of the first five hydrogen lines ($$3 \le m \le 7$$) of the Balmer series visible to the human eye ($$n = 2$$) correspond quite exactly to the ratios of the tone frequencies of the five tones of the first motif of the symphonic poem Till Eulenspiegel's Merry Pranks repeated twice immediately afterwards:

The tone frequencies given in the second column thus refer to the concert pitch A with 440 hertz. The wavelengths given in the last column correspond almost exactly to the five wavelengths given in Balmer's 1884 publication for the hydrogen lines in the visible range (see above).

If the light frequencies $$\nu_{m,2}$$ are multiplied by the mean conversion factor $$u_{440} = 5.713 \cdot 10^{-13}$$, the corresponding sound frequencies $$f_m = u_{440} \cdot \nu_{m,2}$$ and the musical intervals of the ascending motif by Richard Strauss.

Tuning pitch
At this point it must still be stated that this conversion factor can be chosen arbitrarily, but must somehow be fixed. The tuning of musical instruments in ensemble music has always had to be carried out so that all instruments can produce harmonious sounds together. For this purpose, tuning tones or a concert pitch were also used, the pitch of which was fixed. The French scholar ' (* 1653; † 1716) and later the German physicist and astronomer ' (* 1756; † 1827) proposed to use the unit of one second as the fundamental time measure of a physical tuning for the determination of the keynote C. This means that the keynote C is the unit of one second. This means that the fundamental C0 has exactly the frequency 1 hertz and all octaves of this fundamental upwards always have exactly twice the frequency. This leads to the following series:

If a reference pitch is sought for the transmission of frequencies of electromagnetic waves to sound waves, it is obvious for physicists to orientate themselves on the keynote C of this physical tuning.

From a musical point of view, the first five tones of this series have only a theoretical meaning and are practically not perceptible to humans with a pitch. The tone A is a major sixth (respectively nine semitones) higher than the C directly below it or a minor third (respectively three semitones) lower than the C directly above it. If the A in the last octave between C8 and C9 in this row is set as a concert pitch, it has the same frequency ratio $${2}^{\frac {1} {12}}$$ in equal temperament (all twelve successive semitone intervals have the same frequency ratio $${2}^{\frac {1} {12}} \approx 1.059463$$) it has the following frequency:

 * $$f_A = f_{C8} \cdot {2}^{\frac {9} {12}} = f_{C9} \cdot {2}^{-\frac {3} {12}} \approx 430.539 \text { Hertz} \approx 431 \text { Hertz} $$

With a frequency of 430.539 hertz for the concert pitch A, the following pitches result for the five notes of the Till Eulenspiegel motif:

This results in a somewhat smaller mean factor $$u_{431} = 5.591 \cdot 10^{-13}$$ as the constant of proportionality between the light and sound frequencies $$f_m = u_{431} \cdot \nu_{m,2}$$ of the ascending motif by Richard Strauss.

These pitches were used, for example, in the early Paris tuning of 1829 in the 19th century. In the course of time, however, the reference pitch became higher and higher for practical reasons - stringed instruments sound fuller and louder when the strings are stretched a little more, although they acquire a higher natural frequency. Richard Strauss was aware of the problems involved and commented on the increased pitch of the concert pitch a few years before his death as follows:.

''Die hohe Stimmung unserer Orchester wird immer unerträglicher. Es ist doch unmöglich, dass eine arme Sängerin A-Dur-Koloraturen, die ich Esel schon an der äußersten Höhengrenze geschrieben habe, in H-Dur herausquetschen soll.''

Rückkehr zum Pariser A, bevor sich unsere armen Sänger die Paar letzten noch vorhanden Stimmen verschrien haben!

Coincidence or not coincidence
The question arises as to whether these physical and musical facts are coincidental, or whether it can be a matter of coincidence. Since there is apparently no reliable evidence for coincidence, the question cannot be answered.

At the very least, it must be stated that it would be an extremely remarkable coincidence. There are alone $${12}^{4} = 20736$$ possibilities of having exactly four tones from a stock of twelve tones follow any initial tone. The probability that exactly the five tones of the Till Eulenspiegel motif will be chosen at random from this supply of tones is therefore just under 0.00005. Other motifs have fewer or more than five tones, have a larger range of tones or have a different starting tone, which further and significantly increases the number of possibilities and reduces the probability of a coincidence accordingly.

The probability that the motif with the five notes c - f - g - gis - a, which unlike the many other motifs actually composed is not even diatonic but has chromatic components, is composed by chance and coincidence only a few years after the discovery of the Balmer series and its discussion in specialist circles, is even smaller.

Finally, it must be taken into account that the motif, notated in the key of F major, begins with the note C, whose pitch in the 18th and 19th  centuries was often determined purely physically with the help of the definition of the time unit of the second.

In the very extensive Répertoire International des Sources Musicales (RISM), there is no other example in more than one million music documents that begins with a motif consisting only of these five notes. Only two other examples in the key of C major with a date of origin before 1893 can be found:
 * Carl Czerny (* 1791; † 1857): Klavierübung in C-Dur (6/8-Takt), opus 599, Nummer 38
 * Anonymus: Ländler (3/4-Takt) C-Dur

According to these considerations, the probability that this motif was created at this point in time by pure chance is certainly not zero, but it is quite low.

Further examples

Let it be borne in mind that a rational sequence of five consecutive numbers, as in the Balmer series, need by no means result in aesthetically perceived pitch ratios. This is not the case with the higher orders in the Balmer series and, for example, with the Lyman series found in 1906 by the US physicist ' (* 1874; † 1954) or the Paschen series found in 1908 by the German physicist '. (* 1865; † 1947), for example, this is not the case.

Therefore, there are only very few known cases in which physical number sequences, especially if they can also still be directly perceived with the human senses, are reflected in musical tone sequences. A significant example is the natural overtone series, which results from whole-number ratios in vibrating strings or in air columns and was processed by Richard Strauss in the symphonic poem Also sprach Zarathustra (Engisch: "Thus Spoke Zarathustra", see above).

Another example with the four Pythagorean tones c' - f' - g' - c" is handed down in the legend of Pythagoras in the forge. The first three of these notes correspond to the first three notes of the Till Eulenspiegel motif. The integer relationship between the numbers four and three, which plays a role there among other things, describes the musical interval of the pure fourth, which appears both in the first interval of the Till Eulenspiegel motif and in the second interval of the Zarathustra motif.

The sounds caused by hammer blows were invented in 1690 by the French-German organist and composer  (* 1653; † 1704) with the organ composition Nova Cyclopeias Harmonica. This composition is framed by an aria, comprises eight variations on the theme Ad Malleorum Ictus Allusio (To allude to the blows of the hammers) and ends with the chant Summo Deo Gloria.

Two years later, the German violinist, composer and court kapellmeister ' (* 1646; † 1712), who like Georg Muffat was a pupil of the Italian composer ' (* 1632; † 1687), the seven orchestral suites dedicated to the Elector of Bavaria Maximilian II Emanuel:

 * Pythagorische Schmids=Fuencklein
 * Bestehend in unterschiedlichen Arien / Sonatinen / Ouverturen / Allemanden / Couranten / Gavotten / Sarabanden / Giquen / Menueten / &c.
 * Mit 4.Instrumenten und beygefügten General-Baß, Bey Tafel=Musicken / Comœdien / Serenaden / und zu anderen fröhlichen Zusammenkunfften zu gebrauchen.

The main keys of the seven suites for solo violin are F major, D major, G major, D minor, F major, D major and B flat major.

→ See also Wikibook Pythagoras in the Forge.

Even the Pythagoreans were of the opinion that the same numerical regularities were evident in astronomy as in music. Therefore, there were repeated attempts to connect the orbits of the planets with harmonic sounds, as for example by  (* 1571; † 1630) in his work Harmonices mundi libri V (five books on the harmonies of the world) of 1619, in which he transferred astronomical numerical ratios to musical intervals. The German composer  (* 1895; † 1963) took up this theme and created the symphony The Harmony of the World in 1951 with the three movements Musica instrumentalis, Musica humana and Musica mundana, as well as the libretto and music for the opera of the same name in five acts in 1957. In the introduction to his music-theoretical work Unterweisung im Tonsatz he writes:

''Ich weiß mich mit dieser Einstellung zum Handwerklichen des Tonsatzes einig mit den Anschauungen, die gültig waren lange vor der Zeit der großen klassischen Meister. Wir finden ihre Vertreter im frühen Altertum; weitblickende Künstler des Mittelalters und der Neuzeit bewahren die Lehre und geben sie weiter. Was war ihnen das Tonmaterial? Die Intervalle waren Zeugnisse aus den Urtagen der Weltschöpfung; geheimnisvoll wie die Zahl, gleichen Wesens mit den Grundbegriffen der Fläche und des Raumes, Richtmaß gleicherweise für die hörbare wie die sichtbare Welt; Teile des Universums, das in gleichen Verhältnissen sich ausbreitet wie die Abstände der Obertonreihe, so daß Maß, Musik und Weltall in eins verschmolzen.''

Quantum physical background
When the spectral lines had been discovered and even in 1884 when Johann Jakob Balmer had empirically found the mathematical regularities and described them with the Balmer formula, the physical and theoretical causes for the emergence of such discrete spectral lines were still completely unknown. It was not until the first half of the twentieth century that quantum mechanics made it possible to describe the structure of atoms and the interaction between charged matter particles (for example electrons) and light particles (photons) very precisely.

The electromagnetic light field reacts with other particles in such a way that a quantum with the energy $$E = h \cdot \nu$$ is either captured or emitted. The constant $$h$$ is the Planck's quantum of action and $$\nu$$ is the frequency of an electromagnetic wave.

The particle nature of light was concluded in the years from 1899 to 1905 by the German researcher ' (* 1858; † 1947) from the laws of thermal radiation and, in an intensified form, his younger colleague ' (* 1879; † 1955) in the publication Ueber einen heuristischen Gesichtspunkt betreffend die Erzeugung und Verwandlung des Lichtes from the photoelectric effect (or photoeffect). In 1918 Max Planck was awarded the Nobel Prize in Physics for the discovery of energy quanta and in 1921 Albert Einstein for the discovery of the law of the photoelectric effect. The buzzword of wave-corpuscle dualism has pervaded the difficult discussions about how to understand the physics of microscopic things ever since. Yes, even the sound waves of music have teeny tiny grains: the phonons with an energy proportional to the frequency according to the Planck-Einstein formula.

Even a particle with a rest mass, such as an electron, can be understood as a matter wave. The term "matter wave", named after the French physicist  (* 1892; † 1987) establish the relations between wavelength $$\lambda$$ and frequency $$\nu$$ for matter waves, which apply to particles with energy $$E$$ and momentum $$p$$ respectively:

 * $$\lambda = \frac {h} {p}$$

 * $$\nu = \frac {E} {h}$$

He was awarded the Nobel Prize in Physics in 1929 for his discovery of the wave nature of electrons.

When Max Planck introduced the quantum of action, he had no idea of the universal significance of the constant. It unambiguously and fundamentally links the scale of values of space and time with that of mass, energy and momentum. In the modern International System of Units, mechanics knows only one arbitrary quantity, the second. Metre and kilogram are derived from the second via the definitively fixed values of the universal speed of light $$c$$ (since 1983) and Planck's constant $$h$$ (since 2019). The latter has a 1967 decree nailed down to 9 192 631 770 periods of a quantum transition of cesium atoms.

Hydrogen atom
The formulae given concern the matter waves of free-flying particles. A little later, a brilliant theorist built the more general wave equations with forces that attract or repel particles. Make way for the prime example of this new wave mechanics!

The bond between the electron and the proton of the hydrogen atom takes a relatively time-stable, i.e. stationary, form precisely when there is a spatially concentrated standing wave that satisfies the famous Schrödinger equation, proposed in 1926 by the Austrian physicist  (* 1887; † 1961). As with large mechanical objects, such as strings and organ pipes, the possible frequencies of their various forms of vibration or standing waves are very well related to small integers. Since Pythagoras, we know that the harmonic intervals of tones are nothing other than such frequencies whose ratios are two integers. They readily appear as the natural frequencies of vibrating objects.

The natural frequencies of the hydrogen atom follow with $$n = 1, 2, 3, ...$$ the formula $$\nu_n = \frac {\nu_1} {n^2}$$, where $$\nu_1$$ is the highest occurring frequency. The binding energies of the atom are measured negatively with respect to an unbound pair of electron and proton. The result is $$E_n = -h \cdot \nu_n$$. Again, Planck's constant $$h$$ enters. The lowest energy for the principal quantum number 1 $$E_1 = -13.6 \text{ eV}$$ (eV stands for the energy unit electron volt) belongs to the spherical standing wave in which no nodes occur. The excited energies $$E_n$$ have a symmetrical spatial structure with nodal surfaces that divide the waveform into a number of 'lobes' or 'bellies'. Examples on the right of the picture for the main quantum numbers n from 1 to 4. You can see areas of constant amplitude and colours of constant phase.

Quantum mechanics explains how the hydrogen atom glows. It goes from the state $$E_m$$ to the lower state $$E_n$$ with $$m > n$$. The energy $$\Delta E = E_m - E_n$$ is thereby transferred to a photon whose frequency lies in the spectral series and can be determined with the following Rydberg formula:

 * $$\Delta E = h \cdot \nu_{m,n} = |E_1| \left( \frac 1{n^2}-\frac 1{m^2} \right)$$

One of the methods of physics is a perturbation calculation, according to which already a classically conceived electric oscillation causes the stimulated quantum transitions of hydrogen, exactly with the experimentally confirmed spectrum of frequencies. But to explain the spontaneous emission of light quanta, a thorough particle-like treatment of the light field was needed: the quantum field theory. With its calculation methods, it then correctly emerges that the excited states of the atom are not entirely stable. Photons can emerge from the vacuum and couple to the appropriate states of electrons.

We owe a profound interpretation of the hydrogen spectrum to the Austrian physicist ' (* 1900; † 1958), who worked with the physicists ' (* 1885; † 1962) from Denmark,  (* 1901; † 1976) from Germany, the above-mentioned Erwin Schrödinger and others contributed significantly to the scientific revolution of quantum mechanics. Niels Bohr was awarded the Nobel Prize in Physics in 1922 for his services to research into the structure of atoms and the radiation they emit. Werner Heisenberg was honoured with it in 1932 for founding quantum mechanics, the application of which led, among other things, to the discovery of the allotropic forms of hydrogen.

Wolfgang Pauli succeeded in 1926 in an algebraic formulation of the hydrogen model thanks to a self-contained group of symmetry operators. He was able to derive the spectrum entirely without the troublesome Schrödinger differential equations. The special form of Coulomb attraction between electron and proton allows a high degree of symmetry.

Since then, symmetry groups have played an increasing role in physics. Often such groups are represented in nature in connection with series of whole quantum numbers. Symmetry and harmony are obviously closely connected. The German theoretical physicist  (* 1868; † 1951) wrote in September 1919 in Munich in the preface of his book Atombau und Spektrallinien:

Seit der Entdeckung der Spektralanalyse konnte kein Kundiger zweifeln, daß das Problem des Atoms gelöst sein würde, wenn man gelernt hätte, die Sprache der Spektren zu verstehen. Das ungeheure Material, welches 60 Jahre spektroskopischer Praxis aufgehäuft haben, schien allerdings in seiner Mannigfaltigkeit zunächst unentwirrbar. Fast mehr haben die sieben Jahre Röntgenspektroskopie zur Klärung beigetragen, indem hier das Problem des Atoms an seiner Wurzel erfaßt und das Innere des Atoms beleuchtet wird. Was wir heutzutage aus der Sprache der Spektren heraus hören, ist eine wirkliche Sphärenmusik des Atoms, ein Zusammenklingen ganzzahliger Verhältnisse, eine bei aller Mannigfaltigkeit zunehmende Ordnung und Harmonie. Für alle Zeiten wird die Theorie der Spektrallinien den Namen Bohrs tragen. Aber noch ein anderer Name wird dauernd mit ihr verknüpft sein, der Name Plancks. Alle ganzzahligen Gesetze der Spektrallinien und der Atomistik fließen letzten Endes aus der Quantentheorie. Sie ist das geheimnisvolle Organon, auf dem die Natur die Spektralmusik spielt und nach dessen Rhythmus sie den Bau der Atome und Kerne regelt.

Auditory perception
The human ear is a remarkable measuring instrument that has a logarithmic sensitivity to both the amplitudes and frequencies of sound waves. This means that for us, loudness increases by the same increment when amplitude is multiplied by the same factor - not when the same amount is added. In the same way, the pitch increases for us by the same step when the frequency is increased by a factor. The factor of two is perceived particularly clearly, namely as an octave. At intervals of octaves, the melodies sound so similar that the notes are given the same name, accompanied by dashes or numbers, if you want to indicate the octave position.

Many people are familiar with the logarithmic measure with which we describe the ratios of amplitudes. A factor of ten was defined as 20 decibels. defined. An amplitude factor $$q$$ can be converted into a level $$Q$$ in the unit of measurement decibel (dB) with the decadic logarithm $$\operatorname{lg}$$ or $$\log_{10}$$ or with a logarithm to any base $$\log$$:

 * $$Q\,\text{in dB} = 20\,\text{dB} \cdot \log_{10}\;q = 20\,\text{dB} \cdot \frac {\log\;q} {\log\;10}$$

The same principle works for the frequencies or the perceived pitches. A factor of two was defined as 1200 cents. Every twelve semitones - or every small second - have a size of 100 cents by definition in music theory for equal temperament. Twelve semitones of the same size directly above each other make an octave, which consequently has a size of 1200 cents. The pitch difference $$c$$ in the unit of measurement cent (C) at a given frequency ratio $$q$$ can also be calculated with the following formula with a logarithm to any base $$\log$$ or with the logarithm dualis $$\operatorname{ld}$$ or $$\log_{2}$$ respectively:

 * $$c\,\text{in Cent} = 1200\,\text{C} \cdot \log_{2}\;{q} = 1200\,\text{C} \cdot \frac {\log\;{q}} {\log\;{2}}$$

Intervals
Two tones that are in a whole-number vibration ratio to each other as a musical interval are perceived as sounding consonant. This is particularly clear in the case of the prime and the octave, where the frequency ratio is 1:1 and 1:2 respectively. The larger the two integers become, the smaller the effect of perceived consonance and thus of harmonic perception. For trained ears, this effect can still be heard well with the pure fifth (frequency ratio 2:3) and with the pure fourth (frequency ratio 3:4). Instruments without fixed pitches, such as strings or trombones, as well as vocal ensembles, can also intonate major and minor thirds (frequency ratio 4:5 and 5:6) or minor and major sixths (frequency ratio 5:8 and 3:5) purely. Particularly in the case of major and minor triads, which are composed of three notes spaced a major third, minor third or fifth apart, the use of pure intervals produces a particularly harmonious sound.

The following table shows the frequency ratios for musical intervals from the prime to the octave in pure tuning and in relation to the fourth, fifth and octave in Pythagorean tuning:

The tritone is therefore the interval between the fourth and the fifth. You can read off from the large numerical ratio 25:36 without having heard the diphthong: a dissonance comes out. In equal temperament, the two tones have the following irrational frequency ratio (also not a gentle sound):

 * $$2^{- \frac {6} {12}} = 2^{- \frac {1} {2}} = \sqrt {\frac {1} {2}}$$.

Positive deviations in the last column with the unit of measurement cent mean that the tone of the pure tuning is higher than the tone of the equal temperament, and negative deviations mean that the tone of the pure tuning is lower than the tone of the equal temperament.

It is noteworthy that in the whole-number ratios given in the table above, among the numbers with only one digit, seven alone does not appear, which underlines that it is a special number from the point of view of some medieval authors.

The next table compares the corresponding musical pure intervals with the frequency ratios in the Balmer series:

Finally, a table comparing the corresponding musical intervals from the Balmer series with the frequency ratios at equal temperament:

.

Hans Sommer

The composer and natural scientist Hans Sommer may have conveyed the facts to Richard Strauss via the acoustic-musical variant of the optical-physical Balmer series. It is easy to understand that he was well informed about discoveries in optics and physics.

Hans Sommer was named as the son of Otto Gustav Zincken (* 1809; † 1940) was actually Hans Friedrich August Zincken called Sommer. The father of Otto Gustav Zincken was the Duchy of Brunswick Court Medical Officer Julius Leopold Theodor Friedrich Zincken (* 1770; † 1856), who in turn was the son of the judicial officer Carl Friedrich Wilhelm Zincken (* 1729; † 1806) and his wife Sophie Schläger. This marriage was divorced and both spouses remarried. Sophie Schläger married in 1782 Johann Christoph Sommer (* 1741; † 1802), who was a court councillor and professor of anatomy at the Anatomical-Surgical Institute in Brunswick and whose surname was henceforth used by her Zincken family descendants.

Hans Sommer's father died when he was two and a half years old. His widowed mother, Nanny Langenheim (* 1813; † 1902), married the entrepreneur, optician and pioneer of photography Peter Wilhelm Friedrich von Voigtländer in 1845. (* 1812; † 1878). His father was the optician Johann Friedrich Voigtländer (* 1779; † 1859), who had run the company J. F. Voigtländer, Werkstätte für optische und feinmechanische Instrumente since 1808 and was the descendant of the optician and inventor Johann Christoph Voigtländer (* 1732; † 1797).

Hans and Antonie Sommer had two sons Otto and Richard.

Hans Sommer was trained as a mathematician, and he was also taught in Göttingen by the physicist  (* 1804; † 1891). Already in 1866 he became professor of mathematics at the polytechnic Collegium Carolinum in Braunschweig. Twelve years later he was appointed rector, and he was involved in the transfer of the Collegium into the Herzogliche Technische Hochschule Carolo-Wilhelmina, which is now the Technische Universität Braunschweig. He conducted research in Braunschweig until 1884, particularly in the field of applied optics.

As one of the pioneers in applied optics, he also helped his stepfather Peter Wilhelm Friedrich Ritter von Voigtländer, who, together with his father Johann Friedrich Voigtländer, ran the optical works Voigtländer & Sohn in Braunschweig as an entrepreneur from 1849. .

In 1858, Hans Sommer published his inaugural dissertation in Göttingen to obtain a doctorate in philosophy with the title Zur Bestimmung der Brechungsverhältnisse. In it, he also devotes himself in detail to refraction at the prism, with the help of which white light can be split spectrally in order to be able to recognise and measure spectral lines, for example. .

In 1870, Hans Sommer published a book in Braunschweig entitled On the Dioptics of Lens Systems, in which he also discusses the refractive effect.

Hans Sommer and Richard Strauss jointly founded the Anstalt für musikalische Aufführungsrechte (AFMA) in 1903, which is considered the first predecessor organisation of the later Gesellschaft für musikalische Aufführungs- und mechanische Vervielfältigungsrechte (GEMA).

In 2019, the correspondence between Richard Strauss and Hans Sommer was published in book form.

Epilogue
Without the manifold research results and remarkable discoveries from the 19th century, it would have been very difficult to develop quantum theory. And without the deeper understanding of quantum mechanics, there would probably be no semiconductors, which are present in almost every household today. In explicit reference to light emission, it should be noted that through knowledge of the many different discrete energy levels of semiconductors, light-emitting diodes can be produced today that emit monochromatic light at practically any wavelength in a wide spectral range from the infrared to the ultraviolet. Today, light-emitting diodes are used as energy-efficient light sources in lighting technology and in almost all screens.

So far, no written sources are known for the interdisciplinary hypothesis put forward here that the discovery of the integer ratios in the optical-physical Balmer series might have inspired the transformation into an acoustic-musical motif. However, this story was passed on orally by generations of physicists. The main author had heard it in the mid-1980s in a course at the Technical University of Berlin taught by Professor Gerd Koppelmann (* September 5, 1929; † September 21, 1992). The main author thanks his doctoral supervisor Professor Heinz Niedrig for the nice and informative obituary of his colleague Gerd Koppelmann.

Summary of the project

 * Target audience: Musicians and natural scientists
 * Learning objectives: Number ratios at light and sound frequencies.
 * Book sponsorship/contact person: User:Bautsch
 * Are co-authors currently desired? Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
 * Guidelines for co-authors: Wikimedia-like.

title: Till Eulenspiegel's funny Series


text:
User:ReadyKW

I'm here for the Electric Vehicle conversion Wikibook. Hope I can help. We'll see.......

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

title: User:ReadyKW


text:
Rust for the Novice Programmer

Rust for the Novice Programmer
This is a guide to the Rust programming language for people who have little to no experience with programming in any language.

If you do have experience programming in other languages, other resources such as the official Rust Book may be more useful for you, as this book will try to explain a lot of basic programming concepts and ideas that may already be familiar to you.

With that, here's a basic introduction to programming and Rust!

Contents

 * The basics
 * Introduction
 * Getting set up
 * Hello World!
 * Numbers
 * Variables
 * Comments
 * Numbers and data types
 * Functions
 * If statements and booleans
 * Loops and Iteration
 * Arrays and Vectors
 * Number Conversions
 * Tuples and Structs
 * Enums
 * Pattern Matching
 * Basics of References and borrowing
 * Strings
 * First program – Basic Maths Program
 * Getting Started
 * Traits and Displaying
 * User Input
 * Converting a String to a Number
 * Checking the Numbers
 * Randomness and External Crates
 * Full Source Code and Potential Extensions
 * Introduction - Simple CSV Analysis Program
 * Misc things

title: Rust for the Novice Programmer


text:
Rust for the Novice Programmer/Setup

Setup
We will need to do some setup when just starting out. Before you can run Rust code, you will need to setup a workspace. This workspace is often called a "Development Environment". The easiest way to get started with Rust is to follow the instructions at the official Rust language website..
 * You need to understand how to open a terminal and the basics of using one.
 * You will also need some text editor for editing code files.

Next, we can move on to actually doing stuff.

title: Rust for the Novice Programmer/Setup


text:
Module:Wikilisp/doc/use-count/doc

99,000+

title: Module:Wikilisp/doc/use-count/doc


text:
Template:Evalx/doc/use-count/doc

99,000+

title: Template:Evalx/doc/use-count/doc


text:
Salu, Jon!

Salu, Jon!

This is the start of the Mini translation of "Salute, Jonathan!", a book by Mithridates which teaches a language in a the language, using easy vocab and building up in a fun way. This is work a in progress, so please be patient.

1

Man e en siti. Man i note a dia-buku. Man i mira a siti.

Man e en siti? Ja, si e en siti.

Man e en... man? No, si e no en man. Si e en siti.

Man e en... rela? No, si e no en rela. Si e en siti. Man e en siti.

Man i note a dia-buku? Ja, si i note a dia-buku?

Dia-buku i note a man? No, dia-buku i no note a man. Dia-buku i no note. Man i note. Man i note a dia-buku.

Siti i mira a man? No, siti i no mira a man. Siti i no mira. Man i mira. Man i mira a siti? Ja, man i mira a siti.

Man e ve? Si e en siti. Man i note a ke? Si i note a dia-buku. Man i mira a ke? Si i mira a siti. Si e en siti, an si i note a dia-buku, an si i mira a siti.

Man e en dia-buku? No, si i note en dia-buku. Man i mira a man? No, si i no mira a man; si i mira a siti.

Man e mega. Man e bon, an man e inteli. Si i pensa. Si i pensa de siti. Si i pensa: "Siti e ke? Siti e bon? Siti e mega?"

Man i note en dia-buku? No, si i no note a siti; siti e mega. Si i note a dia-buku; dia-buku e no mega.

Dia-buku i pensa de man? No, dia-buku i no pensa. Man i pensa. Si i pensa de dia-buku, an si i pensa de siti.

An si i note en dia-buku. Si i note de siti. Si i note: "Siti e bon, an siti e mega".

Si i pensa: "Siti e bon"; si i pensa ke siti e bon. Si i pensa: "Siti e mega"; si i pensa ke siti e mega. Si i pensa de siti, an si i pensa de dia-buku.

Dia-buku a ke? Dia-buku e ve man i note; si i note en dia-buku. En dia-buku, man i note de siti. En dia-buku, man i no note de rela; man i ereki en siti, no en rela. Si i no pensa de rela; si i pensa de siti, siti Munich (Munik).

Siti a ke? Si a Munich. Munich e ve? Si e ve man i ereki. Man e ve? Si e en Munich.

Ja, Munich a mega siti, an a bon siti. Man i pensa ke Munich a bon siti, an si i pensa ke si a mega siti. Man e inteli. Si i dire: "Salu, Munich!". Si a bon man!

Man i ereki an pensa: "Rela e ve?" Si i mira... si i mira a rela! Si i pensa: "Rela!" Nun si i no pensa de dia-buku an i no pensa de siti, si i pensa de rela!

title: Salu, Jon!


text: Category:Book:Salu, Jon!

title: Category:Book:Salu, Jon!


text:
Rust for the Novice Programmer/Hello World

Hello World!
It is a tradition in programming that the first program people see is one that prints 'Hello World'.

Make a folder to store your learning exercises in and inside make a folder. You can call it anything you want but an example name might be 'hello_world'.

 * 1) Open the terminal inside this folder.
 * 2) Type 'cargo init'
 * 3) You should notice the following things inside the folder:
 * .git (you might not see this if you have hidden items turned off)
 * .gitignore
 * Cargo.toml
 * src folder, inside which is main.rs
 * 1) For now we will only concern ourselves with main.rs which is inside src so open this in your text editor.

Inside you will see:

What does this mean?
 * 'fn' is short for 'function'. We will revisit this later, for now just know it is a collection of code.
 * 'fn main' means it is a function called 'main'.
 * The main function is special as it is the start point of your program.
 * The braces { and } indicate the beginning and end of the function.
 * Then the line 'println!("Hello, world!");' is made up of different parts:
 * 'println!' is the name of the function being called. We don't need to know that much except the stuff between ( and ) will be printed to the terminal.
 * Then the " " quotes around "Hello, world!" indicates the stuff inside is some text.

To run this, type 'cargo run' in the terminal. You should see the lines showing the code being compiled, then the text Hello, world! being printed to the terminal!

You can try changing the text within the "" quotes to see different messages being printed!

Next - numbers in Rust


text:
User talk:98.163.58.53

Hi 98.163.58.53 ! You added a line to the paragraph at line 17 ofClock and Data Recovery/Introduction/Acquisition, tracking and jitter performances, Requirements. I created that page, and now I do not understand what it is and what it means. Can you please send me some explanations?

In the mean time the addition will be suspended, and the page reverted to its former state.

Waiting for your comment, as you never end learning..

By for now BORGATO Pierandrea (discuss • contribs) 11:22, 24 January 2022 (UTC)

title: User talk:98.163.58.53


text:
Free Knowledge Culture Calendar/November 17

Today in 1999 SourceForge.net was launched; the first significant product of its kind: an Internet platform that provides infrastructure for Free Software development, complete with hosting as well as development and collaboration tools.

Under frequently changing ownership and management, SourceForge was working on its own demise with questionable monetization attempts that cost trust and got the site onto on advertising and malware blocklists. For example, it was known for misleading advertisements with fake “download” buttons that were bigger and in more prominent locations than the real one, and incentivized developers to bundle their software with installers for pseudo-free spyware. When one day SourceForge seized several popular accounts and offered the software in question bundled with spyware, masses of users migrated to the now-dominant competitor Github.


text: Flora of New York/Crassiclitellata

title: Flora of New York/Crassiclitellata


text: Training the Search and Rescue Dog/Printable version


text: Goats/Printable version

title: Goats/Printable version


text: Wine/Printable version

title: Wine/Printable version


text:
Wine/Wine Regions

Terroir
A number of regions worldwide are noted for their wine production. The unique environmental and social factors involved in a region's wine production are called Terroir by wine enthusiasts.

Because of this concept many wines associated with a region, especially wines that share a name with a region, are considered to only be true examples of that sort of wine if they originate there. For example, if one made "Champagne" on the moon, it would not be considered Champagne by many, and in many countries Champagne can only legally be called "Champagne" if it originated in the Champagne region of France.

France

 * Champagne
 * Burgandy
 * Bordeaux
 * Rhône

Italy

 * Tuscany
 * Veneto

United States

 * Napa Valley
 * Finger Lakes
 * Lake Erie

title: Wine/Wine Regions


text: User talk:205.189.94.9

title: User talk:205.189.94.9


text: Removal of Skunk Odor/Printable version

title: Removal of Skunk Odor/Printable version


text:
I Ching/Octal Bagua

The eight trigrams of ancient China provide a portal to elemental thinking. A trigram is symmetric when the top and bottom lines agree, regardless of the middle line. The symmetric trigrams with yang at core are Air ☰ and Water ☵. The symmetric trigrams with yin at core Earth ☷ and Fire ☲. These are the elements of Empedocles, adopted by Aristotle and Plato. The asymmetric trigrams with yang at core are Lake ☱ and Wood/Wind ☴. The asymmetric trigrams with yin at core are Thunder ☳ and Mountain ☶.

As the trigrams or bagua are derived from the yin and yang duality, there is always the correspondence of a trigram with its complement. Fire and Water are complements, as are Mountain and Lake. It is absorbtion of yin and yang to the sixth degree that is manifested in the I Ching, and the entrance to structure comes by way of the bagua. The sixty-four hexagrams correspond to a complete, directed graph on the eight bagua. The graph includes eight loops, one at each of the bagua, which correspond to doubled trigrams, so eight articles of the I Ching illuminate intensity of a doubled bagua, and the names of these articles can be used  to designate them.

G.W. Leibniz showed in 1703 that a yin line can be taken as zero (0) and a yang line as one (1) so that an arbitrary line is a binary digit (bit). Three lines are then three bits or an octal number. The octal digits 0 to 7 provide a brief designation of a bagua. The complementation is n → 7−n. For example:

Fire = ☲ = 101 = 5 → 2 = 010 = ☵ = Water.

Hexagram sequence
The Book of Changes is a tool of divination and is consulted on a chance-selection basis, traditionally with yarrow stalks or coins. Nevertheless, the sixty-four hexagrams have an order, starting from absolute separation of yin and yang and ending with complete intermingling of yin and yang lines. The rationale for the order remains a mystery. The hexagrams of doubled trigrams have the following order:

 * ☰ air (1), ☷ earth (2), ☲ fire (29), ☵ water (30).
 * ☳ thunder (51), ☶ mountain (52), ☴ wood (57), ☱ lake (58).

Consecutive hexagrams can be compared by counting their differences. This count is called the Hamming distance in coding theory of bit strings. For consecutive hexagrams the maximum distance is 6, corresponding to the case where each trigram is replaced by its complement. Of the sixty-three consecutive pairs, 23 are at distance 2, 17 are at distance 4, and 12 at distance 3. The nine cases of distance 6 (complementary hexagrams) are these: 1 & 2, 11 & 12, 17 & 18, 27 & 28, 29 & 30, 38 & 39, 53 & 54, 61 & 62, 63 & 64. There are only two pairs at a distance of 1, and none at distance 5.

For the twelve months of the year expressed with hexagrams there is a definite algorithm: The darkest month corresponds to the all yin hexagram (#2). Then light begins to appear one month later with a yang line at the base (#24). The next month two yang lines (#19) invade the hexagram from below. Each month a new yang line appears until all lines are yang, so after six months the brightest days appear (summer solstice, #1). The algorithm continues by introducing yin lines: #44 has one yin line at the base represents one month into summer. With advancing yin lines from below the darkest month #2 returns.

Can an algorithm for the sequence from 1 to 64 be found? It represents a depolarization from the first pair to the last pair. The derived sequence of Hamming distances has some features: absence of 5 and few distances 1 and 3, so the distance is mostly even in the steps through the I Ching. In terms of the bagua, six instances of transposition occur: 5 to 6, 7 to 8, 11 to 12, 13 to 14, 35 to 36, and 63 to 64. In 19 cases just one trigram changes place. Only six steps have a constant trigram in place. As the commentaries lean heavily on trigram images, these characteristics are noted in the text.

The transpositions
Only the Greek elements partake in transposition in the sequence.
 * Water ☵ above and below Air ☰ are #5 and #6.
 * Earth ☷ above and below Water ☵ are #7 and #8.
 * Earth ☷ above and below Air ☰ are #11 and #12.
 * Fire ☲ above and below Air ☰ are #13 and #14.
 * Fire ☲ above and below Earth ☷ are #35 and #36.
 * Fire ☲ above and below Water ☵ are #63 and #64.

Group theory
The meaning of this section may be enhanced by reference to the wikibook Abstract Algebra, where the basic concepts of binary operation, abstract group and cyclic group are presented.

The interpretation of bagua as binary numbers according to Leibnitz associates them with the cyclic group of order eight $$C_8 .$$ In fact, there are five different groups of order eight that might have their elements represented by bagua.

For instance, the three lines of a trigram may be considered a binary register with three parallel operations, performed with binary arithmetic of the integers modulo 2 where 1 + 1 = 0. As a group with this operation, the bagua represent
 * $$C_2 \times C_2 \times C_2 ,$$ the cube of the cyclic group of order 2.

Another group structure on the set of 8 bagua may use the subset {☷, ☳, ☵, ☶} or {earth, thunder, water, mountain} to represent the cyclic group of order 4: $$C_4 .$$ The other four bagua arise as complements of these four, with complementation generating the cyclic group of order 2. With these operations the bagua represent the product group $$C_4 \times C_2 .$$

The groups $$C_8, \ C_4 \times C_2, \ (C_2)^3$$ represent the commutative groups of order 8 (see link to Groupprops below). But in group theory the binary operation may not be symmetric: order matters: some groups have elements p and q where pq ≠ qp. There are two of these of order 8 called the quaternion group and the dihedral group.

The complement of a bagua will represent the negative element. Take ☰ to represent 1, then its complement ☷ represents −1. Since bagua represent three bits of information, they can be combined with logical AND and OR operations. Multiplying by ☰ uses ^ (AND), and multiplying by ☷ means invoking the complementary bagua, written T.

It is natural to associate ☳ = i, ☵ = j, and ☶ = k by reason of order, thus connecting to traditional representation of quaternion and dihedral groups. The v (OR) operation is used to obtain the product of two distinct bagua, thus obtaining a product with two yang lines. The quaternion group takes the complement of this product in order to obtain one of i, j, or k. The dihedral group does not use the complement in the case jk = ☵ v ☶ = ☴ = T☳ = − i. The order of the two distinct factors determines whether the product is simple or complemented, so kj = i in the dihedral group.

Since the negative of an element is its complement, the OR of an element with its negative gives ☰ = 1. For the quaternion group the minus can be moved to the other side of the equation giving −1 for the square of i, j, or k. However, in the dihedral group j2 = +1 = k2. The associative property of group operations can be confirmed with either notation for elements. These groups are known for the four-dimensional algebras they underpin, quaternions and split-quaternions.

In summary, the eight bagua, with their structures of order and complementation, provide the means to represent any of the groups of order eight. The cyclic group has no particular claim on bagua for representation. In fact, these bagua structures can be taken as precursors to abstract algebra, as well as to binary numeration.

title: I Ching/Octal Bagua


text:
Cookbook talk:Menu Ideas

New category?
Should this page be classified as a recipe? It shows up in Category:Recipes, which doesn't seem quite right. -- Nostriker (discuss • contribs) 01:10, 25 January 2022 (UTC)

title: Cookbook talk:Menu Ideas


text: PermaBlitz/Printable version

title: PermaBlitz/Printable version


text: Growing Edible Sprouts/Printable version


text: Raising Chickens/Printable version

title: Raising Chickens/Printable version


text: Marine Aquaria/Printable version

title: Marine Aquaria/Printable version


text: Dogs/Printable version

title: Dogs/Printable version


text: Dog Care/Printable version

title: Dog Care/Printable version


text: Field Repairs of Tractors and Trailers for the Driver/Printable version


text: Organic Horticulture in the Mid-Atlantic/Printable version


text: Permaculture Design/Printable version

title: Permaculture Design/Printable version


text:
User:Nrco0e/sandbox

The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory.

This guide is solely about how to use the MPC.

Essentials

 * Objectives
 * Rules & Ethics
 * Formatting Astrometry
 * Notes
 * Observatory Codes
 * Astronomical images: FITS files
 * Submission

History

 * Online services such as the Minor Planet Circular (MPC) and Extended Computer Service (ECS) were made freely available after 1 October 2009 thanks to funding by NASA. The ECS was formerly a paid service that provided subscribers astrometric data on specific comets and minor planets. (MPC 66141)

MPECs

 * A. U. Tomatic = automatically published MPECs
 * M. P. C. Staff = manually published MPECs

NEOCP notes
A tracklet is a single-night set of astrometry submitted by a single observer. Individual tracklets can contain 1 or more lines of astrometry, all from the same submitter, obs code, and date.

Preliminary designations for tracklets (trksubs) are temporary names given by the observer who originally submitted them. trksub = "Observer-assigned tracklet identifier, unique within submission batch. Observers are encouraged to use values with seven or fewer alphanumeric characters. (https://minorplanetcenter.net//iau/info/IAU2015_ADES.pdf)

In the NEOCP, you may come across a variety of preliminary designations for NEO candidates. Generally major surveys such as Mt Lemmon, ATLAS, and Pan-STARRS use their own scheme of preliminary designations for their discoveries.

 * Common preliminary designations (in alphanumerical order):
 * # = numbers only, x = alphanumerical including upper/lower-cases, X = alphanumerical, capitalized only, Y = last two digits of the year 20YY
 * MAP, San Pedro de Atacama (W94) = XXxxxxx
 * ATLAS-HKO (T05) = A##xxxx
 * ATLAS-MLO (T08) = A##xxxx
 * ATLAS-South Africa (M22) = A##xxxx
 * Kitt Peak-Bok (V00) = Cxxxxxx (first two x digits are lesser than G96) (changes by year)
 * Mt Lemmon Survey (G96) = Cxxxxxx (first two x digits are lesser than 703) (changes by year)
 * Catalina Sky Survey (703) = Cxxxxxx (changes by year)
 * Multa Observatory (N82) = ELE####
 * MARGO (L51) = gb#####
 * Siding Spring-Janess-G, JAXA (Q66) = JNS###
 * Pleasant Groves Observatory (W16) = MHX####
 * WISE (C51) = N00xxxx (W00xxxx before NEOWISE)
 * Klet Observatory (246) = NK####
 * Observatorio Campo dos Amarais (X74) = O######
 * Pan-STARRS 1 (F51) = P1#xxxx
 * Pan-STARRS 2 (F52) = P2#xxxx
 * SONEAR Observatory (Y00) = S######
 * GINOP-KHK (K88) = Sar####
 * Santa Maria de Montmagastrell (B74) = SMYYxxx
 * JPL SynTrack (U68) = STYYxxx
 * Steward Observatory (691) = SWxxxx
 * Tokyo-Kiso (381) = TMG####
 * Purple Mountain Observatory (D29) = X#####
 * Zwicky Transient Facility (I41) = ZTFxxxx (PTFxxxx during Palomar Transient Factory)

Publication of new asteroids
For a newly discovered object to be officially published by the Minor Planet Center and receive a provisional designation (such as 2022 AB123) there are various conditional requirements, which mostly vary depending on the asteroid's orbital type.

Some small, near-Earth asteroids (especially those with an absolute magnitude less than 30 or those that impacted Earth or made a very close approach) may be confirmed with as few as two tracklets over an arc of only an hour.

More distant asteroids, such as Jupiter Trojans or Centaurs, may require more than three tracklets over three days before being confirmed.

Typically, if the observations of an asteroid were only made by one observatory, it is less likely to be confirmed than if many separate observatories made the observations.

For asteroids classified by the MPC as Atira, Aten, Apollo, or Amor, a Minor Planet Electronic Circular (MPEC) will be published. This can also happen for Centaurs and Trans-Neptunian Objects if they have been observed for more than one opposition. Either way, the resulting orbit will be published in the next day's Daily Orbit Update (DOU) MPEC, and observations will be published in the following Minor Planet Supplement (MPS), which is published semi-weekly.

Astrometry

 * By default, observers cannot submit a single position (single line of astrometry). Doing so would result in a rejected batch (given by the bounceback email)
 * MPC maintains a list of measurers from whom single position measurements are allowed.
 * Published astrometry can only be replaced and remeasured upon request by the original submitter.

Miscellaneous
Critical-list objects are numbered minor planets still in need of additional observations due to increasing uncertainty in the near future. The critical list is now deprecated. (https://www.minorplanetcenter.net/iau/Ephemerides/CritList/index.html)
 * https://www.minorplanetcenter.net/iau/TheIndex.html
 * A "dark run" is a period of observations taking place during a new moon. (https://www.planetary.org/articles/3418)

title: User:Nrco0e/sandbox


text:
User talk:85.119.196.254

YOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO I AM ME BAAAAABEH

title: User talk:85.119.196.254


text: Wikijunior:Countries A-Z/Togo

title: Wikijunior:Countries A-Z/Togo


text: Wikijunior:Countries A-Z/Togo/Image

title: Wikijunior:Countries A-Z/Togo/Image


text: Wikijunior:Countries A-Z/Tonga

title: Wikijunior:Countries A-Z/Tonga


text: Wikijunior:Countries A-Z/Tonga/Image

title: Wikijunior:Countries A-Z/Tonga/Image


text: Wikijunior:Countries A-Z/Trinidad and Tobago


text: Wikijunior:Countries A-Z/Trinidad and Tobago/Image


text:
A-level Computing/WJEC (Eduqas)/Component 1/Program construction

1. Pre-processor

 * The pre-processor takes the source code and performs tasks such as merging the contents of one or more files and replacing string patterns with another
 * The pre-processor scans the code for pre-processor directives

2. Lexical Analysis

 * Comments and unneeded spaces are removed
 * Converts the source code into a token stream - Numeric representation of the code and stores in a table
 * Keywords, constants and identifiers are replaced by 'tokens' (generally hex numbers)
 * 2 tables are created

 * 1) Reserved word table
 * 2) Identifier table

Examples of lexical analysis tables:

 * Reserved Word
 * User Identifier

3. Syntax Analysis

 * Tokens are checked to see if they match the spelling and grammar expected. This is done by parsing each token to determine if it uses the correct syntax for the programming language
 * If syntax errors are found, error messages are produced

4. Semantic Analysis

 * Variables are checked to ensure that they have been properly declared and used
 * Variables are checked to ensure they are of the correct data type
 * Operations are checked to ensure that they are legal for the type of variable being used

5. Code generation

 * Where code specific to the target machine is generated. Several lines of machine code tend to be generated for each high level instruction

6. Code optimisation
"➡ EXAM QUESTION - what are the objectives of code optimisation?"
 * High-level general programming constructs are replaced by codes
 * The replacement codes are very efficient low-level programming codes
 * 1) Achieve the required output of the program
 * 2) Increase the speed of the program
 * 3) Decrease demand on resources
 * 4) Not delay the overall compilation process


text: Wikijunior:Countries A-Z/Tunisia

title: Wikijunior:Countries A-Z/Tunisia


text: Wikijunior:Countries A-Z/Tunisia/Image

title: Wikijunior:Countries A-Z/Tunisia/Image


text:
Cookbook:Digestive Biscuit

The digestive biscuit, also called wholemeal biscuit, is a type of biscuit (UK) or cookie (US) originating in the United Kingdom.

Characteristics
Digestives, at their most basic, are light brown in color, with a mild nutty flavor and crisp, crumbly texture. They typically use whole wheat flour, which contributes to their heartiness and flavor. Some brands may sell digestives with additional components, like chocolate or caramel coatings.

Use
Digestives are very popular in the United Kingdom, where they may be eaten on their own or used as an ingredient in other food preparations. They are commonly eaten with tea, ground to crumbs for use in desserts, or made into tiffin.

Substitution
If you can't get your hands on digestives, graham crackers make a reasonable substitute, especially if using them only for crumbs.

title: Cookbook:Digestive Biscuit


text: Wikijunior:Countries A-Z/Somalia

title: Wikijunior:Countries A-Z/Somalia


text: Wikijunior:Countries A-Z/Somalia/Image

title: Wikijunior:Countries A-Z/Somalia/Image


text: Wikijunior:Countries A-Z/Spain

title: Wikijunior:Countries A-Z/Spain


text: Wikijunior:Countries A-Z/Spain/Image

title: Wikijunior:Countries A-Z/Spain/Image


text: Wikijunior:Countries A-Z/Switzerland

title: Wikijunior:Countries A-Z/Switzerland


text: Wikijunior:Countries A-Z/Switzerland/Image


text: Wikijunior:Countries A-Z/Sweden

title: Wikijunior:Countries A-Z/Sweden


text: Wikijunior:Countries A-Z/Sweden/Image

title: Wikijunior:Countries A-Z/Sweden/Image


text: User talk:Muratbasargnc

title: User talk:Muratbasargnc


text: User talk:WinnieBaehr43

title: User talk:WinnieBaehr43


text:
User talk:Henrylucas07

Blocked
You cannot edit for indefinite due to disruptive editing. You are welcome to contribute constructively when editing privileges are restored. You may appeal this action with a reasonable argument by adding the text below. —Atcovi (Talk - Contribs) 03:29, 26 January 2022 (UTC)

title: User talk:Henrylucas07


text: Horticultural Gardening Tips/Printable version


text: Horticulture/Printable version

title: Horticulture/Printable version


text: Hydroculture/Printable version

title: Hydroculture/Printable version


text: Wikijunior:Countries A-Z/Sudan

title: Wikijunior:Countries A-Z/Sudan


text: Wikijunior:Countries A-Z/Sudan/Image

title: Wikijunior:Countries A-Z/Sudan/Image


text:
Micronations/Future

There are a number of future possibilities for micronations, both conceptual and for existing microstates.

Short term future
It is likely that micronations which do not attempt to establish themselves on the same level as existing states will continue to operate as normal. As the idea of micronations continues to proliferate, it's likely that recognized nation states will begin to crack down more on microstates which conduct certain activities, such as establishing borders.

title: Micronations/Future


text:
Micronations/Rationale

Micronations are often founded as a fun side project.

Some micro nations are founded for more serious reasons. For example some micronations are founded to attempt to claim terra nullis, or to make a political statement.

title: Micronations/Rationale


text: Windows Add-ons/Printable version

title: Windows Add-ons/Printable version


text: Windows Movie Maker/Printable version

title: Windows Movie Maker/Printable version


text:
Rust for the Novice Programmer/Numbers

Numbers
Computers are great at handling numbers. Here, we will see how to do basic mathematical operations in Rust.

First, let's copy the steps from the previous page to ensure we have a clean slate to begin with:
 * 1) Make a new folder in your projects folder, call it something like "numbers"
 * 2) Open a terminal inside that new folder and type 'cargo init'
 * 3) Open src/main.rs in your text editor

Now to print out a number, we have to change the println!("Hello, world!"); line. To print out a number we will change it to If we type 'cargo run' in the terminal now, we will see 5 in the terminal.

What is this? Note there aren't "" quotes around 5, that means it is a number. However, println! expects some text(normally called a 'string') inside it, so instead we put "{}" where the {} braces are a placeholder for the next input. That next input being the 5 we put after the comma. Thus, the 5 is printed to the terminal. This might seem odd but it is helpful for doing more complex things.

Now, what maths things can we do? The most obvious things to do are the basic operations: addition, subtraction, multiplication and division.

Addition is simple: put a + between two numbers, so will print out 9 to the terminal. Subtraction is equally simple with a - between the numbers, will print out 4 to the terminal.

Multiplication is the same idea, but uses the * symbol: will print out 15 to the terminal.

Division uses the / symbol but functions a little strangely: will print out 5 to the terminal.

However, doesn't print out 2.5 but instead prints out 2. The reason for this will be explained later, but the mechanic is that the number is rounded down to the nearest whole number.

But this is quite inconvenient to include these calculations in the println! statements, what's a better way to organise our code?

Next: Variables

title: Rust for the Novice Programmer/Numbers


text:
Wikimedia/Wikidata

Wikidata is a wikimedia project intended to manage structured, computer readable data, about items.

One notable use of Wikidata items is the linking of articles across languages in a centralized way.

title: Wikimedia/Wikidata


text:
Wikimedia/The History of Wikivoyage

The early history of Wikivoyage is quite complex. Wikivoyage joined Wikimedia in early 2013.

title: Wikimedia/The History of Wikivoyage


text:
Wikimedia/The Culture of Wikivoyage

In part because of it's long operational stint prior to becoming a Wikimedia project, Wikivoyage culture differs significantly from other Wikimedia projects. The style of writing on Wikivoyage tends to be more relaxed and direct then on other Wikimedia projects.

title: Wikimedia/The Culture of Wikivoyage


text:
Wikimedia/Wikivoyage

Wikivoyage is a wiki dedicated to developing free travel guides.

title: Wikimedia/Wikivoyage


text:
Wikimedia/Wikispecies

Wikispecies is a wiki that intents to be a directory of all species.

title: Wikimedia/Wikispecies


text:
Wikimedia/Wikiversity

Wikiversity hosts educational materials.

Wikiversity also includes several peer reviewed journals.

title: Wikimedia/Wikiversity


text:
Wikimedia/Meta

Meta-Wiki serves to coordinate and plan efforts across wikimedia.

title: Wikimedia/Meta


text:
Wikimedia/Wikimedia Incubator

Wikimedia Incubator is where new languages of Wikimedia projects are developed before being deployed for general use.

Wikimedia Incubator, inaugurated on June 2, 2006, stands as a remarkable digital crucible within the expansive landscape of the Wikimedia Foundation's endeavors. This wiki-based platform, orchestrated by the non-profit Wikimedia Foundation, shares its virtual home with celebrated projects like Wikipedia, and yet, it carves out a distinctive niche of its own. The Incubator, as its name suggests, serves as the nurturing ground for aspiring language editions of Wikimedia projects, a hallowed space where linguistic diversity finds expression, and cultures take root in the global digital encyclopedia. It is a testament to the Wikimedia Foundation's commitment to inclusivity and the preservation of linguistic heritage that the Incubator was conceived and continues to thrive.

title: Wikimedia/Wikimedia Incubator


text:
UNIX Basics/Commands

Here are some typical commands in UNIX. (Using a BASH Shell or similar)
 * pwd - "Print Working Directory" shows you your current location.
 * ls - List contents of current directory.
 * cd - Changes directory
 * man - Brings up a manual page for assistance in using a command.

title: UNIX Basics/Commands


text:
UNIX Basics/Permissions

UNIX operating systems manage file access and execution with permissions.

Permissions include:
 * Read - Lets user read a file.
 * Write - Lets user edit a file.
 * Execute - Lets user execute a file as software.

title: UNIX Basics/Permissions


text: UNIX Basics/Printable version

title: UNIX Basics/Printable version


text:
Rust for the Novice Programmer/Variables

Variables
What if we wanted to not have the calculation inside the println! line.

To solve this, we can use variables. Variables allow you to store values for later use.

Here's an example: First, 'let' is a special keyword that means the thing is a variable. 'number' here is the name of the variable so we can refer to it later. '=' shows we are putting in the variable. the 5 is the value we are putting in. This means that later when we put in 'number', it will refer to the number 5. The semicolon ; is as before, indicating the end of the line.

For example will print out the 5 we put into 'number'.

Why is this good?

The advantage of this is now we have the value separated out it is easier to change. So we could change the line to and the println line would print out the new value of number, which is 9.

Changing Variables
One interesting thing about Rust is that variables are immutable by default. This means they cannot be changed after being created. However, you can set a variable as mutable(able to be changed) by using the 'mut' keyword. This will print out the value 8 since it overrides the value 5 and in fact the value 5 does absolutely nothing here. If you run this, the compiler will tell you 'warning: value assigned to `number` is never read' since it is quite smart but also quite pedantic in checking for strange things in your code.

If you did want to put the variable earlier but put the value later you could do this: Note that number no longer needs to have 'mut' since it is only being set to a value once so it is not being changed. However, you cannot use it before you set a value to it. For example, This will give you an error when you run 'cargo run' on it since the compiler works out number doesn't have a value in it when you try to print it.

Here are some things you can do with changing variables:

You can do mathematical equations in order: Here we have a variable with the name 'x'. This is shorter and easier to type, but is less descriptive so might not tell you or other people what this number is for later. Also note we are using the last value of x each time we set the next value of x. This computes x2 + 5 with x = 10, which gives 105. You can change the x value and it will compute the same equation for the different values of x.

One last thing for now: that x = x * number or x = x + number is quite a common thing to do, so there is a way to shorten this: This program does the same thing. For the basic operations, you can use them with '=' to shorten it.

Next up: Comments


text:
Skaneateles Conservation Area/Invasive species/Adelgidae

The hemlock woolly adelgid (HWA, Adelges tsugae) is a native of Asia that infests and kills hemlock trees, including the eastern hemlock, which is a keystone species at the Skaneateles Conservation Area (SCA). The HWA is present in the vicinity of the SCA. It was documented at Baltimore Woods in Marcellus, within 2 miles to the east on January 19, 2022. HWA has not been documented at the SCA as of January 27, 2022, but there is a good chance that it is present. Monitoring and early treatment is important.

The balsam woolly adelgid (Adelges piceae) is a native of Europe that infests and kills fir (Abies) trees. The North American balsam fir (Abies balsamea) and Fraser fir (Abies fraseri) are thought to be most sensitive to this this insect. Of these two firs, only balsam fir is native to the northeast

 * }

Adelges tsugae
Although the HWA (Adelges tsugae) is considered to be native to eastern Asia, it may have been present in western North America for thousands of years, feeding on western hemlock (Tsuga heterophylla), which is apparently more resistant to the HWA than eastern hemlocks. Predatory insects such as Laricobius nigrinus also help to keep western HWA under control and are being used experimentally on eastern HWA infestations.

The HWA is present in the vicinity of the SCA. It was documented at Baltimore Woods in Marcellus, within 2 miles to the east on January 19, 2022. HWA has not been documented at the SCA as of January 27, 2022, but there is a good chance that it is present. Monitoring and early treatment is important.


text: Flora of New York/Hemiptera

title: Flora of New York/Hemiptera


text: Umbraco/Printable version

title: Umbraco/Printable version


text:
The Stack Exchange User Guide/Asking a Question

Before asking a question, it is good etiquette to check to see if your question has been asked before.

Include as much relevant information about what your exact question is. This not only helps people better answer your question, it helps those that follow know if the answer to your question is also the answer to their question.


text:
The Stack Exchange User Guide/The Stack Exchange Sites

Stack Exchange is organized into a number of smaller subsites, each focusing on a specific topic or field. The programming focused Stack Overflow is perhaps the best known of these sites, however there are a number of other Q&A sites on the platform including topics from sciences and the humanities.


text:
The Stack Exchange User Guide/How Reputation Is Earned

Reputation on Stack Exchange sites is a quantified measure of a user's place in the community. In some instances reputation can be spent as a currency to enable certain actions.

Reputation is generally earned by asking useful questions and giving useful answers. Reputation is typically lost by doing the inverse.


text:
Scribus/Installation

Scribus can be installed in a number of ways.

On GNU/Linux
Scribus can typically be installed via a package manager.

title: Scribus/Installation


text:
Rust for the Novice Programmer/Comments

Comments
We're beginning to have a few lines in our code, and sometimes it can awkward to organise properly. One useful tip is that you can put notes or annotations to your code that explains what it does. This is done with comments, with which there are two ways in Rust:
 * 1) using '//' before a line means that the rest of the line is ignored by the compiler
 * 2) using '/*' means that all the rest of the code will be ignored by the compiler until it finds '*/'

For example: Here the comment is used to annotate what the next section does. We can also use these comments to stop code from being run temporarily without removing it entirely. For example: Here the value 10 will be printed since the two lines have the /* */ comments around them so they are not run.

Comments are just generally useful and will be used in later sections to annotate code.

Next: More numbers and data types

title: Rust for the Novice Programmer/Comments


text: Physical Chemistry/Printable version

title: Physical Chemistry/Printable version


text: Solutions to General Chemistry (Linus Pauling)/Printable version


text: VCE Chemistry/Printable version

title: VCE Chemistry/Printable version


text: AP Chemistry/Printable version

title: AP Chemistry/Printable version


text: OCR Advanced GCE in Chemistry/Printable version


text: General Chemistry Laboratory/Printable version


text:
Card Games/Bridge

Contract Bridge is a popular game in the United Kingdom.

For more information on Bridge, see the Wikibook on Bridge.

title: Card Games/Bridge


text: Bridge/Printable version

title: Bridge/Printable version


text:
Card Games/Poker

Poker is a family of card games. Poker is popular in the United States of America.

For more information on Poker, see the Poker Wikibook.

title: Card Games/Poker


text: Chemistry for Idiots, Humans and Rebels/Periodic Table of Elements


text: Chemistry for Idiots, Humans and Rebels/Printable version


text: Chemistry 101/Printable version

title: Chemistry 101/Printable version


text: Analytical Chemiluminescence/Printable version


text: Advanced Inorganic Chemistry/Printable version


text: Statistical Thermodynamics and Rate Theories/Printable version


text: Molecular Simulation/Printable version

title: Molecular Simulation/Printable version


text: Forensic sciences: Effects of stress and perturbations on soil communities/Printable version


text: Chemistry Friends/Printable version

title: Chemistry Friends/Printable version


text: Computational Chemistry/Printable version


text: Introductory Chemistry Online/Printable version


text:
User talk:Bokica 395

Welcome
Welcome, Bokica 395!

If you have any questions, you can ask in the assistance reading room or possibly contact me personally. -Gifnk dlm 2020 From Middle English Wikipedia 📜📖💻 (talk) 10:37, 28 January 2022 (UTC)

title: User talk:Bokica 395


text:
User talk:Beritat343

Welcome
Welcome, Beritat343!

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title: User talk:Beritat343


text:
User talk:Merajul Islam Book

Welcome
Welcome, Merajul Islam Book!

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title: User talk:Merajul Islam Book


text:
User talk:Blahsville1985

Welcome
Welcome, Blahsville1985!

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title: User talk:Blahsville1985


text:
User talk:RUSSEL JOHN BOADO

Welcome
Welcome, RUSSEL JOHN BOADO!

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title: User talk:RUSSEL JOHN BOADO


text:
User talk:Alej27

Welcome
Welcome, Alej27!

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title: User talk:Alej27


text:
User talk:Hmlfm

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Welcome, Hmlfm!

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title: User talk:Hmlfm


text:
User talk:Itti

Welcome
Welcome, Itti!

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title: User talk:Itti


text:
User talk:Henry.Lucas.TG

Welcome
Welcome, Henry.Lucas.TG!

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title: User talk:Henry.Lucas.TG


text:
User talk:Dbrnjhbysx

Welcome
Welcome, Dbrnjhbysx!

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title: User talk:Dbrnjhbysx


text:
User talk:Wayne49188890582

Welcome
Welcome, Wayne49188890582!

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title: User talk:Wayne49188890582


text:
User talk:Igicko

Welcome
Welcome, Igicko!

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title: User talk:Igicko


text:
User talk:Sagar Kumar agrahri

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Welcome, Sagar Kumar agrahri!

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title: User talk:Sagar Kumar agrahri


text:
User talk:IamprickedPhentee

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Welcome, IamprickedPhentee!

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title: User talk:IamprickedPhentee


text:
User talk:Raffat5

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Welcome, Raffat5!

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title: User talk:Raffat5


text:
User talk:64Miki2009

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Welcome, 64Miki2009!

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title: User talk:64Miki2009


text:
User talk:Leo3g7

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Welcome, Leo3g7!

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title: User talk:Leo3g7


text:
User talk:Eltayeb TJ

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Welcome, Eltayeb TJ!

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title: User talk:Eltayeb TJ


text:
User talk:Nawaaltjie

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Welcome, Nawaaltjie!

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title: User talk:Nawaaltjie


text:
User talk:Yasminswz

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title: User talk:Yasminswz


text:
User talk:Ninacvj

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title: User talk:Ninacvj


text:
User talk:Scott111tut

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title: User talk:Scott111tut


text:
User talk:Vivekpantdelhi

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title: User talk:Vivekpantdelhi


text:
User talk:JJLiu112

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title: User talk:JJLiu112


text:
User talk:Tar loyd

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Welcome, Tar loyd!

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title: User talk:Tar loyd


text:
User talk:FauktilarPhentee

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title: User talk:FauktilarPhentee


text:
User talk:Kismat allex

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Welcome, Kismat allex!

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title: User talk:Kismat allex


text:
User talk:I am a Leaf

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Welcome, I am a Leaf!

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title: User talk:I am a Leaf


text:
User talk:Mcmmssi

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title: User talk:Mcmmssi


text:
User talk:Coreytrity

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title: User talk:Coreytrity


text:
User talk:Daraklitsas

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title: User talk:Daraklitsas


text:
User talk:Proud Serb

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title: User talk:Proud Serb


text:
User talk:Jamesviady

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title: User talk:Jamesviady


text:
User talk:AkiAkisaS

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title: User talk:AkiAkisaS


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User talk:23Axel23

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title: User talk:23Axel23


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User talk:Chaoborus

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title: User talk:Chaoborus


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User talk:Bencemac

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title: User talk:Bencemac


text:
User talk:Nba cofi

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Rust for the Novice Programmer/Functions

Functions
Let's say we wanted to do the x2 + 5 again but to y with a starting value of 7. What would that look like? This will work, but say we wanted to change it to x2 + 8 for both of them. This means we would have to change it in both locations. This doesn't seem like too much effort here, but these patterns of wanting to do the same things, just with slightly different values is very common. Therefore we can separate this out into a function. Let's take one of them first: This will work, except the point of a function is to take in input(s) that can have different values. The way to put an input into a function is to put it inside the brackets. You put the name of the input, similar to the name of a variable, then you have to explicitly write out the type of the input.

Note that we set value to the value of input. This means since we use the 'mut' keyword, we can modify the value inside the function. Alternatively we could put mut before 'input' and then input would be modifiable. Next we can change main to This means the code is a lot easier to modify and separate out, making it easier to read and edit later.

An alternative way might be if we wanted to not print out inside the function and instead output the different value, so that the main function can print it out or do anything it wants with it. This is called returning and you specify the return value with '-> type'. For example, the function will return an integer, so we return i32 by adding '-> i32' after the but before the {}. Then we print the values in the main function as so: This lets us do more interesting things, like chaining function calls together. For example we can change the main to: This does the x2 + 5 on 10 twice, so we get 11030. This has just been a rough overview of functions and their usefulness.

Next: if statements and booleans