## 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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## 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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## Braille

The Braille system is a method that is widely used by blind people to read and write. Braille was devised in 1821 by Louis Braille, a Frenchman. Each Braille character or cell is made up of six dot positions, arranged in a rectangle containing two columns of three dots each. A dot may be raised at any of the six positions to form sixty-four (26) permutations, including the arrangement in which no dots are raised. For reference purposes, a particular permutation may be described by naming the positions where dots are raised, the positions being universally numbered 1 to 3, from top to bottom, on the left, and 4 to 6, from top to bottom, on the right. For example, dots 1-3-4 would describe a cell with three dots raised, at the top and bottom in the left column and on top of the right column, i.e., the letter m. The lines of horizontal Braille text are separated by a space, much like visible printed text, so that the dots of one line can be differentiated from the Braille text above and below. Punctuation is represented by its own unique set of characters. The Braille system was based on a method of communication originally developed by Charles Barbier in response to Napoleon’s demand for a code that soldiers could use to communicate silently and without light at night called night writing. Barbier’s system was too complex for soldiers to learn, and was rejected by the military. In 1821 he visited the National Institute for the Blind in Paris, France, where he met Louis Braille. Braille identified the major failing of the code, which was that the human finger could not encompass the whole symbol without moving, and so could not move rapidly from one symbol to another. His modification was to use a 6 dot cell — the Braille system — which revolutionized written communication for the blind.

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## Bora (rings)

A Bora is the name given both to an initiation ceremony of Indigenous Australians, and to the site on which the initiation is performed. At such a site, young boys are transformed into men. The initiation ceremony differs from culture to culture, but often involves circumcision and scarification, and may also involve the removal of a tooth or part of a finger. The ceremony, and the process leading up to it, involves the learning of sacred songs, stories, dances, and traditional lore. Many different clans will assemble to participate in an initiation ceremony. The word Bora was originally from South-East Australia, but is now often used throughout Australia to describe an initiation site or ceremony. It is called a Burbung in the language of the Darkinjung, to the North of Sydney. The name is said to come from that of the belt worn by initiated men. The appearance of the site varies from one culture to another, but it is often associated with stone arrangements, rock engravings, or other art works. Women are generally prohibited from entering a bora. In South East Australia, the Bora is often associated with the creator-spirit Baiame. In the Sydney region, large Earth mounds were made, shaped as long bands or simple circles. Sometimes the boys would have to pass along a path marked on the ground representing the transition from childhood to manhood, and this path might be marked by a stone arrangement or by footsteps, or mundoes, cut into the rock. In other areas of South-East Australia, a Bora site might consist of two circles of stones, and the boys would start the ceremony in the larger, public, one, and end it in the other, smaller, one, to which only initiated men are admitted. Bora rings, found in South-East Australia, are circles of foot-hardened earth surrounded by raised embankments. They were generally constructed in pairs (although some sites have three), with a bigger circle about 22 metres in diameter and a smaller one of about 14 metres. The rings are joined by a sacred walkway. Matthews (1897) gives an excellent eye-witness account of a Bora ceremony, and explains the use of the two circles.

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