از انیمیشن کمک بگیریم

 

قضيه سوا Ceva's Theorem

Ceva's Theorem. Concurrency. Interactive proof with animation. Key concept: Menelaus Theorem. Puzzle of the Ceva's Theorem: 48 classic piece اثبات قضيه مرحله به مرحله با انيميشن.

قضيه پروانه Butterfly Theorem

Butterfly Theorem Proof with animation اثبات قضيه مرحله به مرحله با انيميشن.

قضيه مورليMorley's Theorem

Morley's Theorem. Introduction with animation. Triangle + Trisectors = Equilateral triangle. Morley's Theorem Puzzle: 22 pieces of polygons اثبات وشرح قضيه مرحله به مرحله با انيميشن.

قضيه منلائوسMenelaus' Theorem

Menelaus' Theorem. Interactive proof with animation and key concepts اثبات قضيه مرحله به مرحله با انيميشن.

اجسام افلاطونیPlatonic Solids

Platonic Solids. Interactive animation

Eyeball Theorem

Eyeball Theorem: Animated Angle to Geometry Study.

Hexagon and Lissajous Art

Hexagon and Lissajous Art

 

اثبات مرحله به مرحله

Nagel Point Theorem Proof

Butterfly Theorem

Ceva's Theorem

Clifford's Theorems

Bevan Point

بعضی از نقاط خاص درمثلث

 1. Centroid

 2.  Orthocenter

 3.  Circumcenter

 4.  Incenter

 5.  Excenter

 6. Apollonius Point

7. Nine-Point Center

 8. Gergonne Point

 9. Nagel Point.

 10. Fermat Point

 11. Napoleon Point

 12. Bevan Point V. See: Illustration with animation

 13. Vecten Point V

 14. Symmedian (Lemoine) Point 

Symmedian (Lemoine) Point

 15. Mittenpunkt or Middlespoint 

Mittenpunkt Point

 16. Steiner Point

Steiner Point

مساله3

In triangle ABC, angle B is a right angle. BD is the altitude. r1, and r2, are the inradii of the triangles ADB, and BDC respectively, the points E, F, and G are points of tangency. Prove that .

مساله2

Prove the following: if, in a triangle ABC,  mÐ BAO = mÐ OAC = 20°, mÐ ACO = 10°,and mÐ OCB = 30°, then mÐ OBC = 80°.

مساله

Prove the following: if, in a triangle ABC, median BD is such that mÐ A = mÐ DBC, and mÐ ADB = 45°, then mÐ A = 30°.

مسائل وقضایای هندسه

Table of Content

1. Triangles. Theorems and problems about triangles.

2. Circles. Theorems and problems about circles.

3. Triangle Centers

4. Archimedes' Book of Lemmas

5. Quadrilaterals

6. Butterfly Theorem. See also: Butterfly Theorem Puzzle

7. Carnot's Theorem

8. Ceva's Theorem

9. Clifford's Circles Chain Theorems

10. Equal Incircles Theorem

11. Equilic Quadrilateral.

12. Eyeball Theorem. Illustration

13. Gergonne Point Theorem

14. Heron's Formula. Key facts and a purely geometric step-by-step proof.

15. Johnson's Theorem

16. Kurschak's Tile & Theorem

17. Langley Problem: 20° Isosceles Triangle Adventitious angles.

18. Menelaus' Theorem

19. Miquel's Pentagram. Proof

20. Miquel's Pentagram. Dynamic Geometry Cabri, GSP, Cinderella, C.a.R.

21. Monge & d'Alembert Three Circles Theorem I - Dynamic Geometry Requires Java applet 1.3 or higher.

22. Monge & d'Alembert Three Circles Theorem II - Dynamic Geometry Requires Java applet 1.3 or higher.

23. Morley's Theorem

24. Napoleon's Theorem

25. Nagel Point Theorem. See also: Nagel Point Flowchart Proof, Nagel Point Puzzle.

26. Newton's Theorem: Newton's Line

27. Parallelogram with Squares theorem.

28. Pentagons and Pentagrams. Menelaus and Collinearity

29. Platonic Solids. Interactive animation.

30. Poncelet's Theorem

31. Proposed Problems about congruence of line segments, angles, and triangles. Level: High School, SAT Prep, College geometry.

32. Sangaku Problem

33. Sangaku Problem 2

34. Sangaku Problem 3

35. Semiperimeter and excircles of a triangle

36. Semiperimeter and incircle of a triangle

37. Semiperimeter, incircle and excircles of a triangle

38. Seven Circles Theorem

39. Simson Line

40. Simson Line: Angle between two Simson lines

41. Steiner's Theorem

42. 100° Isosceles Triangle Problem

43. Triangles

44. Triangle with the bisectors of the exterior angles. Collinearity

45. Triangle & squares. Fifteen theorems, visual illustrations.

46. Triangle with Squares 0 Two squares.

47. Triangle with Squares 1 Two squares.

48. Triangle with Squares 2 Two squares.

49. Triangle with Squares 3 Three squares.

50. Triangle with Squares 4 Finsler-Hadwiger Theorem.

51. Triangle with Squares 5 Two squares, median and altitude.

52. Triangle with Squares 6 Four squares.

53. Triangle Centers

54. Van Aubel's Theorem Quadrilateral with Squares

55. Varignon and Wittenbauer parallelograms

56. Adjacent Angles and Five Bisectors

Naming and Classification of Curves

Classification of Curves

How Curves are Named

Other Constructions and Definitions

Curve Family Tree

Related Web Sites

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Classification of Curves

There are many ways to classify curves. One way is to determine whether a curve is the graph of some polynomial equation p[x,y]==0. The graph of a polynomial equation are called algebraic curves. Algebraic curve are assigned an order. The order of an algebraic curve is the degree of the polynomial. For example, line (a x + b y + c == 0); circle ((x+h)^2 + (y+k)^2 -r^2 == 0), or the deltoid ((x^2+y^2)^2 - 8 a x (x^2 - 3 y^2) + 18 a^2 (x^2 + y^2) - 27 a^4 == 0), are algebraic curves. Curves may be easy to trace but are not algebraic. For example, no polynomial's graph can be any of cycloid, equiangular spiral, or quadratrix of Hippias. Algebraic curves with degree greater than 2 are called higher plane curves. Non-algebraic curves are called transcendental curves. To determine whether a curve is algebraic requires graduate level math knowledge, and is beyond this project's scope. Here we will group curves by tracing methods.

There is a special class of curves known as fractal and space-filling curves. Simply, fractal curves are curves that are not smooth. All the curves covered here are such that when you keep magnifying parts of the curve, it'll eventually looks like a line, unless you are magnifying a cusp point. Fractal curves are such that no matter how large you magnify, the curve is still corrugated. In a sense, it's all cusps! Space filling curves are a special class of fractal curves, so named because they completely fill an area. Fractal and space-filling curves are discovered in late 17th century and their existence shocked mathematicians profoundly. They are interesting as a topic but this project will not deal with fractal or space-filling curves. Space-filling curves are also known as Peano curve, after logician Giuseppe Peano. Famous fractals includes the Helge von Koch's "Koch snowflake", David Hilbert's spacing-filling curve, William Gosper's "flowsnake", Benoit Mandelbrot's Mandelbrot set, George Canter's "Canter Dust", and others. (see Related Web Sites)

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How Curves are Named

Here are some examples of how curves got named. The list is not exhaustive.

By Person's names: quadratrix of Hippias, conchoid of Nicomedes, Kappa curve, limacon of Pascal, lemniscate of Bernoulli, witch of Agnesi, folium of Descarte, trisectrix of Maclaurin, trident of Newton (aka parabola of Descartes), Bowditch curve, Tschirnhausen's cubic (aka trisectrix of Catalan, L'Hospital's cubic), Cayley's sextic, nephroid of Freeth. Euler's spiral (aka Clothoid ), Euler's curve (x^y==y^x), spiral of Cornu, Durer's conchoid, lissajous (aka Bowditch curve).

Note that in general, the name attached to a curve or a math theorem is not necessary the person who invented or worked on it.

By method of construction: caustics, derivative, integral, isoptic, orthoptic, parallel, pedal, radial, envelope, evolute, involute, cissoid, conchoid, roulette, glissette, strophoid, pursuit curve.

By property: trisectrix, quadratrix, brachistochrone (aka cycloid), tautochrone (aka cycloid), isochrone (aka semi-cubic parabola, semi-cubical parabola), anallagmatic curve.

By shape: astroid (star), deltoid (greek letter Delta) (aka tricuspid, Steiner's hypocycloid), cardioid (hear-shaped), conchoid of Nicomedes (mussel-shaped) (aka cochloid), nephroid (kidney-shaped), cycloid (circle, wheel), folia (leaf), trident of Newton (aka parabola of Descartes), serpentine (snake) (named by Newton), cissoid of Diocles (Ivy-shaped), rose (aka rhodonea), catenary (chain-shaped) (aka chainette, alysoid), tractrix (aka equitangential curve), spiral, lemniscate of Bernoulli (ribbon-shaped), lituus (crook), figure eight curve (aka lemniscate of Gerono), bullet nose, cross curve.

Historical reasons: ellipse, parabola, hyperbola, witch of Agnesi, limacon of Pascal (snail of Pascal, originated from Roberval).

By the form of the formula: semi-cubic parabola (aka isochrone, semi-cubical parabola), Parabola of Descartes (?) (aka trident of Newton).

Not exactly sure (research to do): circle, line, right strophoid (Barrow, 1670), Kampyle of Eudoxus, Kappa curve (Gutschoven's curve), Hippopede (horse fetter) (Proclus, ca.75 BC), bicorn (Sylvester, 1864), piriform (pear-shaped quartic) (De Longchamps, 1886), Clothoid, cochleoid (bernoulli, 1726), Cayley's sextic, Devil's curve (Cramer, 1750).

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Other Constructions and Definitions

Brachistochrone (from Greek, brakhus:short, chrone:time) means a curve that connects to two given points such that a particle sliding from the higher point to the lower point under ideal physical law (ideal gravitational force, no friction, no air-resistance, particle has no volume ...etc.) will descent with the fastest time, among all possible curves. The unique solution is the cycloid, not straight line.

Tautochrone (Greek, tauto:same, chrone:time) is similar to brachistochrone. It's a curve that, connects two given points such that it takes the same amount of time for a particle to slide from any point on the curve to the lower point, under ideal physical law. This may sound paradoxical. Intuitively, the longer path the particle needs to glide, the longer time it will have to take. However, consider that acceleration rate is high when the slope is sharp, therefore it is possible for a curve whose curvature balances out so that no matter where the particle starts, it always takes the same time to reach the end. The unique solution is the cycloid.

The word Isochrone (Greek, iso:equal, chrone:time) is sometimes used to mean tautochrone, but other authors use it to mean the semi-cubic parabola -- a curve which a particle will descend equal vertical distances in equal time intervals, under ideal physical law. The latter meaning will be used here.

Anallagmatic curve is any curve that has inversion that's itself. Circle and Cassinian oval are examples.

Isoptic of a given curve C and a given angle α is the locus of a point P such that P is the intersection of tangents of C that meets in angle α. If α:=π/2, then the derived curve is called an orthoptic. For example, the orthoptic of a parabola is its directrix, and the orthoptic of an ellipse, hyperbola, or deltoid is a circle.

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Curve Family Tree

The phrases in parenthesis are reminders. Not all curves in this project appears below.

• roulettes (curve rolling on curve)
  • cycloidal curves (circle rolling on circle/line)
    • epitrochoids (circle rolling outside another circle)
      • circle
      • epicycloids (tracing point on circle)
        • cardioid (1 cusp)
        • nephroid (2 cusps)
      • roses (r==Cos[p/q*θ])
        • circle
        • trifolium (3 petals)
        • quadrifolium (4 petals)
    • hypotrochoids (circle inside a circle. Either circle may be the rolling circle.)
      • hypocycloids (tracing point on circle)
        • ellipses, line segment
        • deltoid (3 cusps)
        • astroid (4 cusps)
        • cardioid (by double generation)
        • nephroid (by double generation)
    • trochoids (circle rolling on a line)
      • line (tracing point on center of circle)
      • prolate (extended) cycloids
      • curtate (contracted) cycloids
      • cycloid (tracing point on circle)
• conic sections (r==e/(1+e*Cos[θ]))
  • e>1, hyperbolas
  • e==1, parabola
  • e<1, ellipses
• Cassinian ovals (locus of points whoes product of distances to two fixed points is a constant)
  • lemniscate of Bernoulli (two loops with a double point in center)
• conchoids (shifting the curve by a constant to a point on radial line from a pole.)
  • limacon of Pascal (conchoids of a circle)
    • cardioid
    • trisectrix
  • Conchoids of Nicomedes (conchoids of a line)
• cissoids (distance of two curves by a radial line from a pole)
  • cissoid of a Line and a Circle
    • oblique cissoid
      • cissoid of Diocles
    • strophoid
  • circle
  • line
  • conchoid of Nicomedes
  • lemniscate of Bernoulli
• spirals
  • Archimedean spiral (r == θ^n)
    • (n:=1), Archimedes' spiral
    • (n:=1/2), parabolic spiral (aka Fermat's spiral)
    • (n:=-1), hyperbolic spiral (aka reciprocal spiral)
    • (n:=-1/2), lituus
    • (n:=-0), circle
  • equiangular spiral (r == E^(θ*Cot[α]))
    • (α:=π/2), circle
  • Euler's Spiral (aka Clothoid, Cornu's spiral)
• sinusoidal “spirals” (r^n==a^n*Cos[n*θ], n rational)
  • (n:=-2), rectangular hyperbola
  • (n:=-1), line
  • (n:=-1/2), parabola
  • (n:=-1/3), Tschirnhausen's cubic
  • (n:=1/3), Cayley's Sextic
  • (n:=1/2), cardioid
  • (n:=1), circle
  • (n:=2), lemniscate of Bernoulli
• lissajous
  • ellipse

Sinusoidal “spiral” is defined to be r^n==a^n*Cos[n*θ], n rational. It is so-called because it's a “spira” but the radius vector increases and decreases “sinuously”.

Strophoid

Base Curve	Pole	Fixed Point	Strophoid
line	not on line	on line	oblique strophoid
circle	center	on curve	nephroid of Freeth

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Related Web Sites

see Generic Reference Page.

Fractals

Many visual representations of iterative process are called fractals, not all of which are remotely considered curves. There are hundreds fractals resources on the web. Here are few selected ones.

William McWorter. A tutorial on L-systems -- an iterative process that can generate space-filling curves and other fractals. (7 long pages. excellent) http://spanky.triumf.ca/www/fractint/lsys/tutor.html

FractInt has a page that explains various types of fractals. http://spanky.triumf.ca/www/fractint/fractal_types.html. FractInt is an excellent (free) fractal program by coorperative efforts of programers.

Non-linear science FAQ, explains fractals, chaos and others. (excellent) http://www.faqs.org/faqs/sci/nonlinear-faq/

Robert M. Dickau has graphics on fractals and chaos. http://mathforum.org/advanced/robertd/index.html#frac Especially the page on space-filling curves. http://mathforum.org/advanced/robertd/lsys2d.html.

MacTutor short biographies on Giuseppe Peano, Helge von Koch, Benoit Mandelbrot, Gaston Maurice Julia.

Curves

MacTutor Famous Curve Index on Plane Curve terminology.

Silvio Levy/CRC Press. Geometry Formulas and Facts. http://www.geom.umn.edu/docs/reference/CRC-formulas/ In particular the page on Special Plane Curves. http://www.geom.umn.edu/docs/reference/CRC-formulas/node32.html

★ Back to Table of Contents

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Content updated: 1999.
Last modified: 2002-03-28.
© 1995-97 by Xah Lee. ([e-mail])
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html

3D-XplorMath Surface

A Photo Exhibition of Beautiful Seashells

برای رفتن به صفحه مربوط به هر تصویرروی آن کلیک کنید Click on the image to go to its page.

Beautiful Seashells Exhibition

Seashell

Seashells are a showcasing of spirals. There are great variety of spiral shapes. Suppose we start with a circle winding around a spiral.

• The circle can change size periodically, creating a corrugated shell somewhat emulate that of Paper Nautilus 04180001m-s.jpg

• If instead of a circle we have a polygon, we can simulate that of top shells 04020013m-s.jpg painted_top-s.jpg or Thather shell. thatcher-s.jpg or cones 09140046m-s.jpg DSCN0180m-s.jpg

• if the rounding shape periodically changes shape, as to become a star, then we might emulate shells that have horns such as the angaria_delphinula angaria_delphinula-s.jpg pink-mouthed murex 09130071m-s.jpg Venus's Comb 09130032m-s.jpg • the periodic change might also emulate those shell having ribs such as the wentletrap angulate_wentletrap-s.jpg the Harper shell 04170013m-s.jpg

This particular image is generated by the parametric formula:

x = 2*(1 - E^(u/(6*Pi)))*Cos[u]*Cos[v/2]^2,
y = 2*(-1 + E^(u/(6*Pi)))*Cos[v/2]^2*Sin[u],
z = 1 - E^(u/(3*Pi)) - Sin[v] + E^(u/(6*Pi))*Sin[v]}

Graphics code; seashell_wentletrap.nb; shell_para.gcf; shell_para2.gcf; shell_para3.gcf; spindle.gcf corrugated-shell.gcf seashell-tops.gcf seashell-wentletrap.gcf

I offer $5 for anyone who can come up with a parametric formula for the thatcher shell, wentletrap, or Venu's Comb including varios types of murexs that have radiating horns. (Mike Willams has sent me various formulas, see here :20050120-mike_williams.txt

This photo i took with my camera on 2002. The snail is picked up from the street in Bay Area California USA.

Tech name: angaria delphinula, of the Delphinula group. Notice the exquisite color and regular horns.	This type of shell is called wentletrap. The word is originated in German meaning spiral staircase. The characteristics are the creamy white ribs. †
This one is called rose-branched murex. It has the typical shape of murexes. ‡	Lataixis Mawae. A beauty of bizarreness. ‡
This one is caled painted top, of commonedly called “top” shells because they resembel the toy top. Top shells are characterized by a geometrical flat circular cone. †	Zebra Auger. Auger shells are long and thing. ‡
A thatcher shell. How extremely elegant. ＊	This one is called Spider Scorpian shell, of the family commonly called Spider Conch Shells. The spider family are characterized by the feet-like horny projections on their opening. One thing special about spider scorpian conch is that it has beautiful purplish openings. Here's a photo of it's spiral.
These types of shells are called Miter shells, and this is a typical shape. An auger shell are pretty much the same shape but 2 or 3 times longer. ‡	This is a Salisbury's Spindle, of the spindle shell family. One can see that spindle shells are spindle-like.
Lamp Chank ＊	Episcopal Miter ＊
Cowie shells are characterized by their shiny and beautifully patterned surface. Their spiral is not apparent if viewed from the outside. ＊	Grove snail †

The biological classification:

Kindom
 Phylum
  Class
   Order
    Family
     Genus
      Species

The shells animals are of Kindom Animalia (that is, animals), and Phylum Mollusca (they are mollusks). Mollusks have two major class: Gastropoda (gastropods) and Bivalvia (bivalves). Gastropods are those with spiral shells. Snails slugs limpets and abalone are all gastropods. Bivalves are those clam-like shells, including mussell, cockle, clam, oyster.

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See also

a Photo Exhibition of Seashells

‡ Photo from Encyclopedia of Shells, by Kenneth R. Wye, 1991. amazon.com↗.

† Photo from National Audubon Society Field Guide to North American Seashells, by Harald A. Rehder. 1981. amazon.com↗.

＊ Photo from a big art book.

• The Algorithmic Beauty of Sea Shells. by Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler. amazon.com↗

Here's a collection of Seashell icons made in 2001 by Kate England kate.html

★ Back to Table of Contents

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Content updated: 2003-05
Last modified: 2003-05
© 1995-97 by Xah Lee. ([e-mail])
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html

Famous Curves

Famous Curves Index

Click on the name of a curve below to see its history and some of its associated curves.

Astroid Bicorn Cardioid Cartesian Oval Cassinian Ovals Catenary Cayley's Sextic Circle Cissoid of Diocles Cochleoid Conchoid Conchoid of de Sluze Cycloid Devil's Curve Double Folium Dürer's Shell Curves Eight Curve Ellipse Epicycloid Epitrochoid Equiangular Spiral

Fermat's Spiral Folium Folium of Descartes Freeth's Nephroid Frequency Curve Hyperbola Hyperbolic Spiral Hypocycloid Hypotrochoid Involute of a Circle Kampyle of Eudoxus Kappa Curve Lamé Curves Lemniscate of Bernoulli Limacon of Pascal Lissajous Curves Lituus Neile's Parabola Nephroid Newton's Parabolas Parabola

Pearls of de Sluze Pear-shaped Quartic Plateau Curves Pursuit Curve Quadratrix of Hippias Rhodonea Curves Right Strophoid Serpentine Sinusoidal Spirals Spiral of Archimedes Spiric Sections Straight Line Talbot's Curve Tractrix Tricuspoid Trident of Newton Trifolium Trisectrix of Maclaurin Tschirnhaus' Cubic Watt's Curve Witch of Agnesi

Bernoulli's Illustration

  Bernoulli's Illustration

Polar Animations

how a surface area is generated by revolution

 

Replay the animation	This animation demonstrates how a surface area is generated by revolution and then how the sum of disks results in a volume. but the surface area is more complicated.	This "SA" for the surface on the left was calculated using Mathematica®.

Khayyam-Pascal's Triangle

یک تابلوی گلدوزی از مثلث خیام پاسکال که توسط "ویلیام اچ میشل۱۹۲۵  " و همسرش دوخته شده است در این تابلو هر یک ازاعداد اول به یک رنگ دوخته شده و اعداد مرکب باترکیب رنگ عاملهای اول آن عدد . به رنگ اعداد مربع کامل و توانهای اعداد اول دقت کنید!

When he retired, William H. Mitchell joined his wife in a hobby of needlepoint.  As a 1925 graduate of Rutgers University, Mitchell chose to create original designs based on a life long love of his undergraduate major, mathematics.  His son, Alexander M. "Sandy" Mitchell of Joppa, Maryland, asked the NCB to honor his father's memory by displaying one of his projects.

 

In Mitchell's design, prime numbers ( 2, 3, 5, 7, 11, 13, . . . ) are designated by a single colored square.  For example, the prime numbers 2, 3, 5, and 7 are represented respectively by the solid colors red, yellow, blue and green.
The composite number  6  is represented by its prime factor colors of red and yellow, or ( 2 x 3).  The number 4, or 2²  is represented by red with a tiny square.  For larger composite numbers, requiring several factors, Miller chose to use the colors of the prime factors in modified patterns. 

Note that 126 =  2 x 3² x 7  is composed of two parts yellow (3) and one part each of red (2) and green (7).  He finished his triangle with the 32nd row, or  2⁵, or red with a tiny blue square.  In addition, the project contains math operation symbols, Rubic's Cube, Sierpinski's Triangle and other mathematical memorabilia.

Miller's mathematical needlework required great patience spread over several years.  From a technical view point, each individual block required 120 stitches.  There are over 500 blocks of colors.

In Spring, 1984, Rutgers University honored their former math major by featuring his creation on the cover of their alumni magazine, 1766.

Pascal's version of the triangular array.

From a very early Greek manuscript found in the Vatican Library by the great classical scholar from Denmark, Johan Heiberg.

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 برای اطلاعات بیش تر به آدرس زیر مراجعه کنید:   http://curvebank.calstatela.edu/ptriangle/ptriangle.htm

ابو عبدالله محمّد بن موسی خوارزمی

 

 

ابو عبدالله محمّد بن موسی خوارزمی (قرون دوم و سوم هجری)  بزرگ‌ترین عالم عصر خود در ریاضی، جغرافی، نجوم و تاریخ بوده است. او در بیت‌الحکمه کار مي‌کرد. یکی از مهمترین پیشرفت‌ها با کارهی خوارزمی شروع شد. ین پیشرفت، شروع جبر نام دارد و حرکتی انقلابی بود در دور شدن از مفهوم یونانی ریاضی که اساساً هندسی بود.

مهمترین کتاب خوارزمی کتاب حساب ‌الجبر و المقابله است. کلمه‌یAlgebra  از نام ین کتاب گرفته شده است. البتّه فقط قسمت اوّل ین کتاب به آنچه جبر مي‌نامیم ارتباط دارد. بید بدانیم که ین کتاب به شکلی کاربردی  و بری حلّ مسائل روزمره‌ی قلمرو اسلام نوشته شده است. خوارزمی در ین کتاب ابتدا اعداد طبیعی را معرّفی مي‌کند و سپس به حلّ معادلات مي‌پردازد. او معادلات خطّی و معادلات مربّعی را بررسی مي‌کند. خوارزمی از نماد استفاده نمي‌کند و مسائل را با کلمات بیان مي‌کند. او معادلات را در شش دسته رده‌بندی مي‌کند. ین رده‌بندی با اجری جبر و مقابله انجام مي‌شود؛ جبر یعنی جابجیی جملات بری مثبت بودن همه‌ی ضریب، و مقابله یعنی حذف جملات متناظر در دوطرف تساوی. رده بندی خوارزمی به ین صورت بود:

• مربّع‌ها مساوی ریشه‌ها
• . مربّع‌ها مساوی اعداد.
• ریشه‌ها مساوی اعداد.
• جمع ریشه‌‌ها و مربّع‌ها مساوی اعداد؛ مثلاً
• جمع مربّع‌ها و اعداد مساوی ریشه‌ها؛ مثلاً
• جمع ریشه‌ها و اعداد مساوی مربّع‌ها؛ مثلاً ، و ریشه یعنی . سپس خوارزمی راه حلّ هریک از شش رده را بیان مي‌کند. او هم از روش هندسی و هم از روش جبری استفاده مي‌کند. او روش جبری خود را چنین بیان مي‌کند:

  ... مربّعی و ده ریشه برابر سي‌‌ونه واحد اند. پس مسأله در ین نوع معادله ین‌گونه است: چه مربّعی است که وقتی با ده ریشه‌اش جمع شود مجموع سي‌ونه را مي‌دهد؟ روش حلّ ین نوع معادله ین است که نصف ریشه‌هی مذکور را بگیرید، در ین مسأله پنج، که وقتی در خودش ضرب شود بیست‌وپنج مي‌شود، وقتی که وقتی با سي‌ونه جمع شود شصت‌وچهار را مي‌دهد. ریشه‌ی شصت‌وچهار را مي‌گیریم که هشت است، و نصف ریشه‌ها را از آن منها مي‌کنیم، که سه مي‌شود. پس ریشه عدد سه است و مربّع عدد ۹.

  روش هندسی در شکل زیر مشخّص است:

   

  

  خوارزمی رساله‌ای هم در زمینه‌ی شمار هندی-عربی نوشت، متن عربی گم شده‌است ولی ترجمه‌ی از ین کتاب به لاتین به نام Algoritmi de numero Indorum (به معنی الخوارزمی در باب روش حساب هندی) باعث برخاستن کلمه‌ی الگوریتم شد. البتّه ین ترجمه دقیقاً با متن کتاب خوارزمی انطباق ندارد. بسیاری از ترجمه‌هی ین کتاب با عبارت dixit Algorismi ("الخوارزمی چنین مي‌گوید") آغاز شدند، که به در قرون وسطی استفاده‌ی کلمه‌ی الگوریسم بری اشاره‌ به حساب با ارقام هندی را سبب شد. کلمه‌ی امروزی الگوریتم از ین واژه مشتق شده است.

  منابع:

• http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Khwarizmi.html

• http://www.maa.org/devlin/devlin_0708_02.html

• دائرة‌المعارف فارسی

شبکه های دنیای کوچک

http://veda.sharif.ir/articles/little-worlds/index.htm

نوار موبیوس

عجیب‌ترین نوار دنیا

 

یک تکه کاغذ بردارید، آن را نیم دور بپیچانید و دو انتهای آن را به هم بچسبانید. موجود ساده ای که ساخته اید، کلی خاصیت های عجیب و غریب دارد.

مثلا حتما می دانید که اگر سر و ته یک نوار را بدون پیچش به هم بچسبانیم، یک استوانه مانند ساخته میشود که اگر آن را از وسط ببریم، دو تکه میشود. اما اگر همین کار را روی این نوار عجیب انجام دهیم یک تکه باقی میماند و تنها طولش دو برابر می شود.

 برای اینکه با خاصیت های دیگر این موجود آشنا شوید چند تکه کاغذ و چسب نواری و قیچی بردارید و سعی کنید جواب این سوالات را پیدا کنید. به کمک جواب این سوال ها تردستی های زیادی طراحی شده است. شما هم می توانید به کمک آن ها دوستانتان را به تعجب وادارید.

فرض کنید قبل از آنکه دو سر نوار را به هم بچسبانیم، به جای یک بار، دو بار آن را بپیچانیم و بعد از وسط ببریم. چه اتفاقی خواهد افتاد؟

 

اگر نوار را سه، چهار، پانزده ....   بار بپیچانیم چه اتفاقی خواهد افتاد؟ چه فرقی بین عددهای زوج و فرد هست؟

اگر به جای یک برش از وسط نوار دو برش به فاصله یک سوم از لبه ها بزنیم چه اتفاقی خواهد افتاد؟

این موجود را Augustus Mobius ریاضیدان و منجم آلمانی در سال 1858 کشف کرد و به همین خاطر نام آن را نوار موبیوس گذاشتند.. خاصیتی که در این نوار توجه موبیوس را جلب کرد، یک طرفه و یک لبه بودن آن بود. این نوار عجیب تنها یک رو دارد، یعنی یک مورچه که در نقطه ای از یک نوار موبیوس کاغذی ایستاده می تواند بدون رد شدن از لبه کاغذ به پشت آن نقطه (در سمت دیگر کاغذ) برسد. در حقیقت این نوار اصلا پشت ندارد. این خاصیت را می توانید در نقاشی زیر ببینید. همینطور، لبه این نوار از یک تکه تشکیل شده: یک دایره که روی خودش تا شده است.

Mobius Band II  اثر   escher

 به نظر شما آیا نوار هایی که با تعداد زوجی پیچاندن ساخته می شوند هم این خاصیت ها را دارند؟

مساله هایی همراه با حل از AMC

http://www.unl.edu/amc/mathclub/04,2b-q,MWorld.html

سری فوریه

where

http://mwt.e-technik.uni-ulm.de/world/lehre/basic_mathematics/fourier/node1.php3

http://mathworld.wolfram.com/FourierSeries.html

 (the French mathematician Joseph Fourier (1768-1830

 

 

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