Categorie archief: mathematics

Magic circle

702px-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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Spirograph

spirographbox

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

 

340px-epicycloid-5-5svg

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.

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Vitruvian Man

 

441px-da_vinci_vitruve_luc_viatour

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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Tree of life

tree-of-life_flower-of-life_stage

The concept of a many-branched tree illustrating the idea that all life on earth is related has been used in science, religion, philosophy, mythology and other areas. A tree of life is variously, a) a mystical concept alluding to the interconnectedness of all life on our planet, b) a metaphor for common descent in the evolutionary sense, and c) a motif in various world theologies, mythologies and philosophies.Various trees of life are recounted in folklore, culture and fiction, often relating to immortality or fertility. They had their origin in religious symbolism.

picture: A Tree of Life, in the form of ten interconnected nodes, is an important part of the Kabbalah. As such, it resembles the ten Sephirot.

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Opgeslagen onder architecture, art, geometry, mathematics, philosophy, religion, science, spiritual, symbol

Small circle

150px-small_circlesvg

small circle of a sphere is the circle constructed by a plane crossing the sphere not in its center. Small circles always have smaller diameters than the sphere itself (compare great circle). Small circles cannot be parallel, because parallelism doesn’t exist in spherical geometry. They may look parallel but they are no more parallel than concentric circles on a plane.

The small circle does not have the smallest curvature and hence, a segment on its circumference does not represent the shortest path between two points on a spherical surface.

Except for 90 Degrees North or South and the Equator, all parallels of latitude upon the Earth are small circles (or at least close approximations, as the Earth varies from a true sphere to a relatively minor extent). An observer standing on such a circle and viewing its path toward an unobstructed horizon, would perceive it to bend away from his line of sight, an effect of the inequality between the amount of curvature to his left and right.

By contrast, all meridians of longitude are great circles.

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Opgeslagen onder geometry, mathematics, science

Great circles

600px-sphere_halve

great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere.

Great circles serve as the analogue of “straight lines” in spherical geometry. See also spherical trigonometry and geodesic.

The great circle, also known as the Riemannian circle, is the path with the smallest curvature, and hence, an arc (or an orthodrome) of a great circle is the shortest path between two points on the surface. The distance between any two points on a sphere is therefore known as the great-circle distance. The great-circle route is the shortest path between two points on a sphere; however, if one were to travel along such a route, it would be difficult to steer manually as the heading would constantly be changing (except in the case of due north, south, or along the equator). Thus, Great Circle routes are often broken into a series of shorter Rhumb lines which allow the use of constant headings between waypoints along the Great Circle.

When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would look like a straight line on the map would actually be longer. An exception is the gnomonic projection, in which all straight lines represent great circles.

On the Earth, the meridians are on great circles, and the equator is a great circle. Other lines of latitude are not great circles, because they are smaller than the equator; their centers are not at the center of the Earth — they are small circles instead. Great circles on Earth are roughly 40,000 km in length, though the Earth is not a perfect sphere; for instance, the equator is 40,075 km.

Some examples of great circles on the celestial sphere include the horizon (in the astronomical sense), the celestial equator, and the ecliptic.

Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. For aircraft traveling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic region, while easterly flights will often fly a more southerly track to take advantage of the jet stream.

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Opgeslagen onder geometry, mathematics, science