Categorie archief: geometry

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 (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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Sesame street/Philip Glass: Geometry of circles

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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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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),


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.


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Vitruvian Man



The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci around the year 1487 It is accompanied by notes based on the work of Vitruvius. The drawing, which is in pen and ink on paper, depicts a nude male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called theCanon of Proportions or, less often, Proportions of Man. It is stored in the Gallerie dell’Accademia in Venice, Italy, and, like most works on paper, is displayed only occasionally.[2][3]

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Other artists had attempted to depict the concept, with less success. The drawing is traditionally named in honour of the architect.


This image exemplifies the blend of art and science during the Renaissance and provides the perfect example of Leonardo’s keen interest in proportion. In addition, this picture represents a cornerstone of Leonardo’s attempts to relate man to nature. Encyclopaedia Britannica online states, “Leonardo envisaged the great picture chart of the human body he had produced through his anatomical drawings and Vitruvian Man as a cosmografia del minor mondo (cosmography of the microcosm). He believed the workings of the human body to be an analogy for the workings of the universe.” It is also believed by some[who?] that Leonardo symbolized the material existence by the square and spiritual existence by the circle. Thus he attempted to depict the correlation between these two aspects of human existence.[4] According to Leonardo’s notes in the accompanying text, written in mirror writing, it was made as a study of the proportions of the (male) human body as described in Vitruvius, who wrote that in the human body:

  • a palm is the width of four fingers
  • a foot is the width of four palms (i.e., 12 inches)
  • a cubit is the width of six palms
  • a pace is four cubits
  • a man’s height is four cubits (and thus 24 palms)
  • the length of a man’s outspread arms is equal to his height
  • the distance from the hairline to the bottom of the chin is one-tenth of a man’s height
  • the distance from the top of the head to the bottom of the chin is one-eighth of a man’s height
  • the distance from the bottom of the neck to the hairline is one-sixth of a man’s height
  • the maximum width of the shoulders is a quarter of a man’s height
  • the distance from the middle of the chest to the top of the head is a quarter of a man’s height
  • the distance from the elbow to the tip of the hand is one-fourth of a man’s height
  • the distance from the elbow to the armpit is one-eighth of a man’s height
  • the length of the hand is one-tenth of a man’s height
  • the distance from the bottom of the chin to the nose is one-third of the length of the head
  • the distance from the hairline to the eyebrows is one-third of the length of the face
  • the length of the ear is one-third of the length of the face
  • the length of a man’s foot is one-sixth of his height

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ISO 216


ISO 216 specifies international standard (ISO) paper sizes used in most countries in the world today. It is the standard which defines the commonly available A4 paper size. The underlying principle is that when rectangles with width/length ratio 1:\sqrt{2} are cut or folded in half the rectangles thus formed retain the original width/length ratio.


The international ISO standard is based on the German DIN standard 476 (DIN 476) from 1922.

Some of the formats contained therein were independently invented in France during its revolution, but were later forgotten.



The aspect ratio used by this standard was mentioned in a letter by the German Georg Christoph Lichtenberg, written on 25 October 1786.

  • ISO 216:1975, defines two series of paper sizes: A and B
  • ISO 269:1985, defines a C series for envelopes
  • ISO 217:1995, defines two untrimmed series of raw paper sizes: RA and SRA




Paper in the A series format has a 1:\sqrt{2} \approx 0.707 aspect ratio, although this is rounded to the nearest millimetre. A0 is defined so that it has an area of 1 m², prior to the above mentioned rounding. Successive paper sizes in the series (A1, A2, A3, etc.) are defined by halving the preceding paper size, cutting parallel to its shorter side (so that the long side of A(n+1) is the same length as the short side of An, again prior to rounding).

The most frequently used of this series is the size A4 (210 × 297 mm). A4 paper is 6 mm narrower and 18 mm longer than the “Letter” paper size, 8½ × 11 inches (216 × 279 mm), commonly used in North America.

The geometric rationale behind the square root of 2 is to maintain the aspect ratio of each subsequent rectangle after cutting the sheet in half, perpendicular to the larger side. Given a rectangle with a longer side, x, and a shorter side, y, the following equation shows how the aspect ratio of a rectangle compares to that of a half rectangle: \ x/y = y/(x/2) which reduces to x/y = \sqrt{2} or an aspect ratio of 1 : \sqrt{2}

The formula that gives the larger border of the paper size An in metres and without rounding off is the geometric sequence: an = 21 / 4 − n / 2. The paper size An thus has the dimension an × an + 1.

The exact millimetre measurement of the long side of An is given by \left \lfloor 1000/(2^{(2n-1)/4})+0.2 \right \rfloor.

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