## Roundel

A roundel in heraldry is any circular shape; the term is also commonly used to refer to a type of national insignia used on military aircraft, generally circular in shape and usually comprising concentric rings of different colours. In heraldry, a roundel is a circular charge. Roundels are among the oldest charges used in coats of arms, dating from at least the twelfth century. Roundels in British heraldry have different names depending on their tincture. Thus, while a roundel may beblazoned by its tincture, e.g., a roundel vert (literally “a roundel green”), it is more often described by a single word, in this case pomme (literally “apple”, from the French).

The first use of a roundel on military aircraft was during the First World War by the French Air Service.[citation needed] The chosen design was the French national cockade, which consisted of a blue-white-red emblem mirroring the colours of the Flag of France. Similar national cockades, with different ordering of colours, were designed and adopted as aircraft roundels by their allies, including the British Royal Flying Corps and the US Army Air Service. After the First World War, many other air forces adopted roundel insignia, using different colours or numbers of concentric rings to distinguish them.

Some corporations and other organizations also make use of roundels in their branding; employing them as a trademark, or logo.

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## Round Table

The Round Table is King Arthur’s famed table in the Arthurian legend, around which he and his Knights congregate. As its name suggests, it has no head, implying that everyone who sits there has equal status. The table was first described in 1155 by Wace, who relied on previous depictions of Arthur’s fabulous retinue. The symbolism of the Round Table developed over time; by the close of the 12th century it had come to represent the chivalric order associated with Arthur’s court.

During the Middle Ages festivals called Round Tables were celebrated throughout Europe in imitation of Arthur’s court. These events featured jousting, dancing, and feasting, and in some cases attending knights assumed the identities of Arthur’s knights. The earliest of these was held in Cyprus in 1223 to celebrate a knighting. Round Tables were popular in various European countries through the rest of the Middle Ages and were at times very elaborate; René of Anjou even erected an Arthurian castle for his 1446 Round Table.

The artifact known as the “Winchester Round Table,” a large tabletop hanging in Winchester Castle bearing the names of various knights of Arthur’s court, was probably created for a Round Table tournament.[10] The current paintwork is late; it was done by order of Henry VIII of England for Holy Roman Emperor Charles V’s 1522 state visit, and depicts Henry himself sitting in Arthur’s seat above a Tudor rose. The table itself is considerably older, dating perhaps to the reign of Edward I. Edward was an Arthurian enthusiast who attended at least five Round Tables and hosted one himself in 1299, which may have been the occasion for the creation of the Winchester Round Table

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## The letter O

O is the fifteenth letter of the modern Latin alphabet. Its name in English (pronounced /oʊ/) is spelled o; the plural is oes, though this is rare.

The letter was derived from the Semitic Ayin (eye), which represented a consonant, probably the voiced pharyngeal fricative (IPA: [ʕ]), the sound represented by the Arabic letter ع called Ayn. This Semitic letter in its original form seems to have been inspired by a similar Egyptian hieroglyph for “eye”.

The Greeks are thought to have come up with the innovation of vowel characters, and lacking a pharyngeal consonant, employed this letter as the Greek O to represent the vowel /o/, a sound it maintained in Etruscan and Latin. In Greek, a variation of the form later came to distinguish this long sound (Omega, meaning “large O”) from the short o (Omicron, meaning “small o”).

Its graphic form has also remained fairly constant from Phoenician times until today. Indeed, even alphabets constructed “from scratch”, i.e. not derived from Semitic, usually have similar forms to represent this sound — for example the creators of theAfaka and Ol Chiki scripts, each invented in different parts of the world in the last century, both attributed their vowels for ‘O’ to the shape of the mouth when making this sound.

O is most commonly associated with the close-mid back rounded vowel [o] in many languages. This form is colloquially termed the “long o” in English, but it is actually a most often adiphthong /oʊ/ (realized dialectically anywhere from [o] to [əʊ]).

In English there is also a “short O”, which also has several pronunciations. In most dialects of English English, it is an open back rounded vowel [ɒ]; in North America, it is most commonly an unrounded back to central vowel [ɑː] to [a].

Common digraphs include OO, which represents either /ʊ/ or /uː/; OI which typically represents the diphthong /ɔɪ/; and OA, OE, and OU represent a variety of pronunciations depending on context and etymology.

Other languages use O for various values, usually back vowels which are at least partly open. Derived letters such as Ö and Ø have been created for the alphabets of some languages to distinguish values that were not present in Latin and Greek, particularly rounded front vowels.

In the International Phonetic Alphabet, [o] represents the close-mid back rounded vowel.

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## Notitia Dignitatum

The Notitia Dignitatum is a unique document of the Roman imperial chanceries. One of the very few surviving documents of Roman government, it details the administrative organisation of the eastern and westernempires, listing several thousand offices from the imperial court down to the provincial level. It is usually considered to be up to date for the Western empire in the 420s, and for the Eastern empire in 400s. However, no absolute date can be given, and there are omissions and problems.

There are several extant fifteenth and sixteenth-century copies (plus a colour-illuminated 1542 version). All the known and extant copies of this late Roman document are derived, either directly or indirectly, from a codexthat is known to have existed in the library of the cathedral chapter at Speyer in 1542 but which was lost before 1672 and cannot now be located. That book contained a collection of documents, of which the ‘Notitia’ was the last and largest document, occupying 164 pages. that brought together several previous documents of which one was of the 9th century. The heraldry in illuminated manuscripts of Notitiae is thought to copy or imitate no other examples than those from the lost Codex Spirensis. The most important copy of the Codex is that made for Pietro Donato (1436), illuminated by Peronet Lamy.

The Notitia presents four main problems, as regards the study of the Empire’s military establishment:

1. The Notitia depicts the Roman army at the end of the 4th century. Therefore its development over the 4th century from the Principate structure is largely conjectural, due to the lack of other evidence.
2. It was compiled at two different times. The Eastern section apparently dates from c395 AD; the Western from considerably later, c420. Furthermore, each section is probably not a contemporaneous “snapshot”, but relies on data stretching back as far as twenty years. The Eastern section may contain data from as early as 379, the start of the rule of Theodosius I. The Western section contains data from as early as c400: for example, it shows units deployed in Britain, which must date from before 410, when Roman troops were evacuated from the island. In consequence, there is substantial duplication, with the same unit often listed under different commands. It is impossible to ascertain whether these were detachments of the same unit in different places at the same time, or the same whole unit at different times. Also, it is likely that some units only existed on paper or contained just a skeleton personnel.
3. The Notitia has many sections missing and lacunae (gaps) within sections. This is doubtless due to accumulated text losses and copying errors as it was repeatedly copied over the centuries: the earliest manuscript we possess today dates from the 15th century. The Notitia cannot therefore provide a comprehensive listing of all units in existence.
4. The Notitia does not contain any personnel figures. Therefore, the size of individual units, and of the various commands, cannot be ascertained, as we have little other evidence of unit sizes at this time. In turn, this makes it impossible to assess accurately the overall size of the army. Depending on the strength of units, the late 4th century army may, at one extreme, have equalled the size of the 2nd century force (i.e. over 400,000 men); at the other extreme, it may have been far smaller. For example, the forces deployed in Britain c400 may have been just 18,000 against c55,000 in the 2nd century.

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## Taijitu

Taijitu (a Chinese word that translates roughly as ‘diagram of ultimate power’) is a term which refers to any of the Chinese symbols for the concept of yin yang, and is sometimes extended to similar geometric patterns used historically by various cultures. The most recognized form is composed of two semi-circular teardrop-shaped curves of different colors, or a circle separated by an S-shaped line, where each half is marked with a dot in the opposite (or different) color. Symbols of this type are found as Celtic art forms and coat of arms for several Western Roman army units in Late Antiquity.[1][2][3] Taoist philosophy adopted equivalent symbols several hundred years later, as representations of yin and yang, from which the most common modern usage of the symbol and the name ‘taijitu’ arise. There is no academically established relationship between the Taoist and the earlier ancient Roman symbols.

Symbols with a partial resemblance to the later Taoist diagram appeared in Celtic art from the 3rd century BC onwards, showing groups of leaves separated by an S-shaped line.[1] The pattern lacked the element of mutual penetration, though, and the two halfs were not always portraited in different colours.[1] A mosaic in a Roman villa in Sousse, Tunisia, features different colors for the two halves of the circle, but here, too, the little circles of opposite color are absent.[1]

The earliest depiction of the diagram which today is known as Taijitu or Taiji  appears in the Roman Notitia Dignitatum, an ancient collection of shield patterns of the Roman army dated to ca. AD 430.[1][2][3] The emblem of an infantry unit called the armigeri defensores seniores (“shield-bearers”) is graphically identical in all but colour to the dynamic, clockwise version of the Far Eastern tradition.[1] Another Western Romandetachment, the Pseudocomitatenses Mauri Osismiaci, featured an insignia with the same contours, but with the dot in each part kept in the same shade of colour.[1] A third infantry regiment, the Legion palatinaeThebaei, had a shield pattern comparable to the static version of the East Asian symbol: three concentric circles vertically divided into two halfs of opposite and alternating colors, so that on each side the two colors follow one another in the inverse order of the opposite half.[1] The Roman yin-yang-like symbols predate the Taoist version by several hundred years:

As for the appearance of the iconography of the “yin-yang” in the course of time, it was recorded that in China the first representations of the yin-yang, at least the ones that have reached us, go back to the eleventh century AD, even though these two principles were spoken of in the fourth or fifth century BC. With the Notitia Dignitatum we are instead in the fourth or fifth century AD, therefore from the iconographic point of view, almost seven hundred years earlier than the date of its appearance in China.

It should be mentioned that there is no academically established relationship between the ancient Roman and the later Taoist symbols.

The Taijitu or Taiji diagram is a well known symbol representing the principle of yin and yang, introduced in China by Ming period author Lai Zhide. The term taijitu (literally “diagram of the supreme ultimate”) is commonly used to mean the simple ‘divided circle’ form, but may refer to any of several schematic diagrams representing these principles, such as the one at right.

In the taijitu, the circle itself represents a whole (see wuji), while the black and white areas within it represent interacting parts or manifestations of the whole. The white area represents yang elements, and is generally depicted as rising on the left, while the dark (yin) area is shown descending on the right (though other arrangements exist, most notably the version used on the flag of South Korea). The image is designed to give the appearance of movement. Each area also contains a small circle of the opposite color at its fullest point (near the zenith and nadir of the figure) to indicate how each will transform into the other.

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## Magic circle

Magic circles were invented by the Song Dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four circles , with 9 at the center.

Yang Hui’s magic circle has the following properties

• The sum of the numbers on four diameters = 147，
• 28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147
• The sum of 8 numbers plus 9 at the center =147;
• 28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 + 9 = 147
• The sum of eight radius without 9 =magic number 69: such as 27 + 15 + 3 + 24 = 69
• The sum of all numbers on each circle (not including 9) = 2 × 69
• There exist 8 semicircles, where the sum of numbers = magic number 69; there are 16 line segments(semi circles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.

Yang Hui’s magic square was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》 (Sequel to Excerpts of Mathematical Wonders) of 1275.

Ding Yidong was a mathematician contemporary with Yang Hui, in his 6th order magic circle with 6 rings, the 5 out rings have connection with a 3rd order magic square: the unit number of the 8 numbers on any ring form a 3rd order magic square.

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## Fairy Ring

fairy ring, also known as fairy circleelf circle or pixie ring, is a naturally occurring ring or arc of mushrooms. The rings may grow over ten meters in diameter and become stable over time as the fungus grows and seeks food underground. They are found mainly in forested areas, but also appear in grasslands or rangelands. Fairy rings are not only detectable by sporocarps in rings or arcs, but also by a necrotic zone (dead grass) or a ring of dark green grass. If these manifestations are visible a fairy fungus mycelium is likely present in the ring or arc underneath.

Fairy rings also occupy a prominent place in European folklore as the location of gateways into elfin kingdoms, or places where elves gather and dance.

wikipedia

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## Sesame street/Philip Glass: Geometry of circles

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## Spirograph

Spirograph is a geometric drawing toy that produces mathematical curves of the variety technically known as hypotrochoids and epitrochoids. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced with the drawing equipment (so in this sense it may be regarded as a synonym of hypotrochoid). The name is a registered trademark of Hasbro, Inc.

The Spirograph was invented by British engineer Denys Fisher who exhibited it in 1965 at the Nuremberg International Toy Fair. It was subsequently produced by his company. Distribution rights were acquired by Kenner, Inc., which introduced it to the United States’ market in 1966.

A Spirograph consists of a set of plastic gears and other shapes such as rings, triangles, or straight bars. There are several sizes of gears and shapes, and all edges have teeth to engage any other piece. For instance, smaller gears fit inside the larger rings, but also can engage the outside of the rings in such a fashion that they rotate around the inside or along the outside edge of the rings.

To use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic pieces is pinned to the paper and cardboard. Another plastic piece is placed so that its teeth engage with those of the pinned piece. For example, a ring may be pinned to the paper and a small gear placed inside the ring – the actual number of arrangements possible by combining different gears is very large. The point of a pen is placed in one of the holes in the moving piece. As the moving part is moved the pen traces out a curve.

The pen is used both to draw and provide locomotive force; some practice is required before Spirograph can be operated without disengaging the fixed and moving pieces. More intricate and unusual-shaped patterns may be made through the use of both hands, one to draw and one to guide the pieces. It is possible to move several pieces in relation to each other (say, the triangle around the ring, with a circle “climbing” from the ring onto the triangle), but this requires concentration or even additional assistance from other artists.

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## Epicycloid

In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called an epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:

$x (\theta) = (R + r) \cos \theta - r \cos \left( \frac{R + r}{r} \theta \right)$
$y (\theta) = (R + r) \sin \theta - r \sin \left( \frac{R + r}{r} \theta \right),$

or:

$x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \,$
$y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right). \,$

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).

If k is a rational number, say k=p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R + 2r.

wikipedia

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